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changed: - \documentclass{article} \usepackage{axiom} \usepackage{makeidx} \makeindex \begin{document} \title{The Wester Problem Set} \author{Timothy Daly} \maketitle \begin{abstract} I'm starting to construct a document based on the Wester problem set (ref Wester, Michael J. (ed) "Computer Algebra Systems" Wiley 1999 ISBN 0-471-98353-5) that shows axiom in it's worst light. I've attached the proto-pamphlet. The idea is to focus on upgrading axiom to enhance the algebra. Eventually it is intended to be a "Can't Do" document of the kinds of algebra axiom still cannot do. A future effort wishlist. \end{abstract} \eject \tableofcontents \eject \section{Boolean Logic and Quantifier Elimination} \begin{verbatim} -- Simplify logical expressions => false true and false (1) false Type: Boolean Time: 0.03 (OT) + 0.02 (GC) = 0.05 sec -- => true x or (not x) Argument number 1 to "or" must be a Boolean. x : Boolean Type: Void Time: 0 sec x or (not x) x is declared as being in Boolean but has not been given a value. -- => x or y y : Boolean Type: Void Time: 0 sec x or y or (x and y) x is declared as being in Boolean but has not been given a value. -- => x xor(xor(x, y), y) x is declared as being in Boolean but has not been given a value. -- => [not (w and x)] or (y and z) w : Boolean Type: Void Time: 0 sec z : Boolean Type: Void Time: 0 sec implies(w and x, y and z) w is declared as being in Boolean but has not been given a value. -- => (x and y) or [not (x or y)] --x iff y -- => false x and 1 > 2 x is declared as being in Boolean but has not been given a value. )clear properties w x y z -- Quantifier elimination: See Richard Liska and Stanly Steinberg, ``Using -- Computer Algebra to Test Stability'', draft of September 25, 1995, and -- Hoon Hong, Richard Liska and Stanly Steinberg, ``Testing Stability by -- Quantifier Elimination'', _Journal of Symbolic Computation_, Volume 24, -- 1997, 161--187. -- => (a > 0 and b > 0 and c > 0) or (a < 0 and b < 0 and c < 0) -- [Hong, Liska and Steinberg, p. 169] --forAll y in C {implies(a*y**2 + b*y + c = 0, real(y) < 0)} -- => v > 1 [Liska and Steinberg, p. 24] --thereExists w in R suchThat _ --{v > 0 and w > 0 and -5*v**2 - 13*v + v*w - w > 0} -- => a^2 <= 1/2 [Hoon, Liska and Steinberg, p. 174] --forAll c in R _ --{implies(-1 <= c <= 1, a**2*(-c**4 - 2*c**3 + 2*c + 1) + c**2 + 2*c + 1 <= 4)} -- => v > 0 and w > |W| [Liska and Steinberg, p. 22] --forAll y in C _ --{implies(v > 0 and y**4 + 4*v*w*y**3 + 2*(2*v**2*w**2 + w**2 + W**2)*y**2 _ -- + 4*v*w*(w**2 - W**2) _ -- + (w**2 - W**2)**2 = 0, real(y) < 0)} -- This quantifier free problem was derived from the above example by QEPCAD -- => v > 0 and w > |W| [Liska and Steinberg, p. 22] v > 0 and 4*w*v > 0 and 4*w*(4*w**2*v**2 + 3*W**2 + w**2) > 0 _ and 64*w**2*v**2*(w**2 - W**2)*(w**2*v**2 + W**2) > 0 _ and 64*w**2*v**2*(w**2 - W**2)**3*(w**2*v**2 + W**2) > 0 (6) true Type: Boolean Time: 0.40 (IN) + 0.05 (EV) + 0.08 (OT) + 0.03 (GC) = 0.57 sec -- => B < 0 and a b > 0 [Liska and Steinberg, p. 49 (equation 86)] --thereExists y in C, thereExists n in C, thereExists e in R suchThat _ --{real(y) > 0 and real(n) < 0 and y + A*%i*e - B*n = 0 and a*n + b = 0} -- ---------- Quit ---------- )quit real 3.1 user 1.4 sys 0.2 \end{verbatim} \section{Set Theory} \begin{verbatim} -- ---------- Set Theory ---------- x:= set [a, b, b, c, c, c]; Type: Set OrderedVariableList [a,b,c] Time: 0.03 (IN) + 0.05 (EV) + 0.08 (OT) + 0.07 (GC) = 0.23 sec y:= set [d, c, b]; Type: Set OrderedVariableList [d,c,b] Time: 0.02 (IN) = 0.02 sec z:= set [b, e, b]; Type: Set OrderedVariableList [b,e] Time: 0.02 (EV) = 0.02 sec -- [x \/ y \/ z, x /\ y /\ z] => [{a, b, c, d, e}, {b}] [union(union(x, y), z), intersect(intersect(x, y), z)] (4) [{a,b,c,d,e},{b}] Type: List Set Symbol Time: 1.20 (IN) + 0.20 (OT) = 1.40 sec -- x \/ y \/ z - x /\ y /\ z => {a, c, d, e} difference(%.1, %.2) (5) {a,c,d,e} Type: Set Symbol Time: 0.02 (IN) + 0.02 (EV) = 0.03 sec )clear properties x y z -- Cartesian product of sets => {(a, c), (a, d), (b, c), (b, d)} Cartesian(A, B) == _ set reduce(append, @[[[A.i, B.j] for j in 1..#B] for i in 1..#A]) Type: Void Time: 0 sec Cartesian([a, b], [c, d]) Compiling function Cartesian with type (List OrderedVariableList [a, b],List OrderedVariableList [c,d]) -> Set List Symbol (7) {[a,c],[a,d],[b,c],[b,d]} Type: Set List Symbol Time: 0.12 (IN) + 0.08 (OT) = 0.20 sec -- ---------- Quit ---------- )quit real 8.4 user 2.7 sys 0.3 \end{verbatim} \section{Numbers} \begin{verbatim} -- ---------- Numbers ---------- -- Let's begin by playing with numbers: infinite precision integers -- => 30414 0932017133 7804361260 8166064768 8443776415 6896051200 0000000000 factorial(50) (1) 30414093201713378043612608166064768844377641568960512000000000000 Type: PositiveInteger Time: 0.03 (EV) + 0.02 (OT) + 0.03 (GC) = 0.08 sec -- => 2^47 3^22 5^12 7^8 11^4 13^3 17^2 19^2 23^2 29 31 37 41 43 47 factor(%) 47 22 12 8 4 3 2 2 2 (2) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 Type: Factored Integer Time: 0.03 (EV) + 0.05 (OT) = 0.08 sec -- Double factorial => 10!! = 10*8*6*4*2 = 3840, 9!! = 9*7*5*3*1 = 945 --[10!!, 9!!] -- ABC base 16 => 2748 base 10 abc (3) abc Type: Variable abc Time: 0 sec -- 123 base 10 => 234 base 7 radix(123, 7) (4) 234 Type: RadixExpansion 7 Time: 0.02 (IN) + 0.02 (EV) + 0.03 (OT) + 0.02 (GC) = 0.08 sec -- 677 base 8 => 1BF base 16 radix(447, 16) (5) 1BF Type: RadixExpansion 16 Time: 0.02 (EV) + 0.03 (OT) = 0.05 sec -- [log base 8](32768) => 5 log(32768)/log(8) log(32768) (6) ---------- log(8) Type: Expression Integer Time: 0.03 (IN) + 0.33 (EV) + 0.12 (OT) + 0.12 (GC) = 0.60 sec -- 5^(-1) mod 7 => 3; 5^(-1) mod 6 => 5 (5 :: PrimeField 7)**(-1) (7) 3 Type: PrimeField 7 Time: 0.03 (IN) + 0.03 (OT) = 0.07 sec recip(5 :: PrimeField(7)) (8) 3 Type: Union(PrimeField 7,...) Time: 0.02 (OT) = 0.02 sec invmod(5, 7) (9) 3 Type: PositiveInteger Time: 0 sec (5 :: IntegerMod 6)**(-1) Compiling function G82136 with type Integer -> Boolean There are 20 exposed and 17 unexposed library operations named ** having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op ** to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named ** with argument type(s) IntegerMod 6 Integer recip(5 :: IntegerMod 6) (10) 5 Type: Union(IntegerMod 6,...) Time: 0.02 (EV) = 0.02 sec invmod(5, 6) (11) 5 Type: PositiveInteger Time: 0 sec -- Greatest common divisor => 74 reduce(gcd, [1776, 1554, 5698]) (12) 74 Type: PositiveInteger Time: 0.03 (OT) = 0.03 sec -- Infinite precision rational numbers => 4861/2520 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 4861 (13) ---- 2520 Type: Fraction Integer Time: 0.05 (IN) = 0.05 sec -- Complete decimal expansion of a rational number => 0.142857 ... decimal(1/7) ______ (14) 0.142857 Type: DecimalExpansion Time: 0.02 (IN) + 0.02 (EV) = 0.03 sec -- Multiply two complete decimal expansions and produce an exact result => 2 decimal(7/11) * decimal(22/7) (15) 2 Type: DecimalExpansion Time: 0.02 (IN) = 0.02 sec -- This number should immediately simplify to 3^(1/3) 10/7 * (1 + 29/1000)**(1/3) 3+-+ (16) \|3 Type: AlgebraicNumber Time: 0.20 (IN) + 0.02 (EV) + 0.02 (OT) = 0.23 sec -- Simplify an expression with nested square roots => 1 + sqrt(3) sqrt(2*sqrt(3) + 4) +---------+ | +-+ (17) \|2\|3 + 4 Type: AlgebraicNumber Time: 0.03 (IN) + 0.02 (EV) + 0.05 (OT) = 0.10 sec simplify(%) +---------+ | +-+ (18) \|2\|3 + 4 Type: Expression Integer Time: 0.10 (EV) + 0.02 (OT) = 0.12 sec -- Try a more complicated example (from the Putnam exam) => 3 + sqrt(2) sqrt(14 + 3*sqrt(3 + 2*sqrt(5 - 12*sqrt(3 - 2*sqrt(2))))) +---------------------------------------+ | +------------------------------+ | | +----------------------+ | | | +-----------+ | | | | +-+ (19) \|3\|2\|- 12\|- 2\|2 + 3 + 5 + 3 + 14 Type: AlgebraicNumber Time: 0.05 (IN) + 0.03 (OT) = 0.08 sec -- See D.J. Jeffrey and A.D. Rich, ``The nesting habits of radicals'', draft of -- 1998 => sqrt(2) + sqrt(3) + sqrt(5) sqrt(10 + 2*sqrt(6) + 2*sqrt(10) + 2*sqrt(15)) +----------------------------+ | +--+ +--+ +-+ (20) \|2\|15 + 2\|10 + 2\|6 + 10 Type: AlgebraicNumber Time: 0.05 (IN) + 0.02 (EV) + 0.02 (OT) = 0.08 sec -- Rationalize the denominator => 5 + 2 sqrt(6) (sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2)) +-+ +-+ (21) 2\|2 \|3 + 5 Type: AlgebraicNumber Time: 0.03 (IN) + 0.22 (EV) + 0.02 (OT) = 0.27 sec -- A factorization of 3 in the integers extended by sqrt(-5) sqrt(-2 + sqrt(-5)) * sqrt(-2 - sqrt(-5)) +------------+ +----------+ | +---+ | +---+ (22) \|- \|- 5 - 2 \|\|- 5 - 2 Type: AlgebraicNumber Time: 0.02 (IN) + 0.02 (EV) + 0.02 (OT) = 0.05 sec -- => 3 + sqrt(7) [Jeffrey and Rich] (90 + 34*sqrt(7))^(1/3) +-----------+ 3| +-+ (23) \|34\|7 + 90 Type: AlgebraicNumber Time: 0.05 (IN) + 0.02 (OT) = 0.07 sec -- This is a nontrivial way of writing 12 ! ((135 + 78*sqrt(3))**(2/3) + 3)*sqrt(3)/(135 + 78*sqrt(3))**(1/3) +------------+2 +------------+ +-+ 3| +-+ +-+3| +-+ (24) (- 15\|3 + 26)\|78\|3 + 135 + \|3 \|78\|3 + 135 Type: AlgebraicNumber Time: 0.05 (IN) + 0.07 (EV) + 0.02 (OT) = 0.13 sec % :: Float (25) 11.9999999999 99999998 Type: Float Time: 0.17 (IN) + 0.03 (OT) = 0.20 sec -- See David Jeffrey, ``Current Problems in Computer Algebra Systems'', talk -- => 1 + sqrt(2) (49 + 21*sqrt(2))^(1/5) +-----------+ 5| +-+ (26) \|21\|2 + 49 Type: AlgebraicNumber Time: 0.05 (IN) + 0.02 (OT) = 0.07 sec -- A nasty example generated by Axiom => [log(sqrt(2) + 1) + sqrt(2)]/3 q:= ((6 - 4*sqrt(2))*log(3 - 2*sqrt(2)) + (3 - 2*sqrt(2))*log(17 - 12*sqrt(2)) _ + 32 - 24*sqrt(2)) / (48*sqrt(2) - 72) (27) +-+ +-+ +-+ +-+ +-+ (- 4\|2 + 6)log(- 2\|2 + 3) + (- 2\|2 + 3)log(- 12\|2 + 17) - 24\|2 + 32 ----------------------------------------------------------------------------- +-+ 48\|2 - 72 Type: Expression Integer Time: 0.43 (IN) + 0.05 (EV) + 0.07 (OT) = 0.55 sec q :: Float Cannot convert from type Expression Integer to Float for value +-+ +-+ +-+ +-+ +-+ (- 4\|2 + 6)log(- 2\|2 + 3) + (- 2\|2 + 3)log(- 12\|2 + 17) - 24\|2 + 32 ----------------------------------------------------------------------------- +-+ 48\|2 - 72 q :: Complex Float (28) 0.7651957164 6421269157 Type: Complex Float Time: 0.25 (IN) + 0.03 (GC) = 0.28 sec (log(sqrt(2) + 1) + sqrt(2))/3 :: Float (29) 0.7651957164 6421269135 Type: Expression Float Time: 0.35 (IN) + 0.05 (OT) = 0.40 sec )clear properties q -- Cardinal numbers => infinity 2*Aleph(0) - 3 (30) Aleph(0) Type: Union(CardinalNumber,...) Time: 0.13 (IN) = 0.13 sec -- 2^aleph_0 => aleph_1 2**Aleph(0) >> Error detected within library code: Transfinite exponentiation only implemented under GCH initial (31) -> real 12.4 user 5.8 sys 0.4 \end{verbatim} \section{Numerical Analysis} \begin{verbatim} -- ---------- Numerical Analysis ---------- -- This number should immediately simplify to 0.0 0.0/sqrt(2) (1) 0.0 Type: Expression Float Time: 0.40 (IN) + 0.20 (EV) + 0.22 (OT) + 0.18 (GC) = 1.0 sec -- This number normally produces an underflow => 3.29683e-434295 exp(-1000000.0) (2) 0.3296831478 088558579 E -434294 Type: Float Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec -- Arbitrary precision floating point numbers digits(50); Type: PositiveInteger Time: 0.02 (IN) = 0.02 sec -- This number is nearly an integer: -- 26253741 2640768743.9999999999 9925007259 7198185688 ... exp(sqrt(163.)*%pi) (4) 26253741 2640768743.9999999999 9925007259 7198185688 9 Type: Float Time: 0.20 (IN) + 0.03 (OT) + 0.03 (GC) = 0.27 sec digits(20); Type: PositiveInteger Time: 0 sec -- => [-2, -1] [floor(-5/3), ceiling(-5/3)] (6) [- 2,- 1] Type: List Integer Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec -- Generate a cubic natural spline s from x = [1, 2, 4, 5] and y = [1, 4, 2, 3] -- and then compute s(3) => 27/8 @[[1, 2, 4, 5], [1, 4, 2, 3]] (7) @[[1,2,4,5],[1,4,2,3]] Type: List List PositiveInteger Time: 0.02 (OT) = 0.02 sec -- Translation a:= operator('a); Type: BasicOperator Time: 0.05 (IN) + 0.02 (OT) = 0.07 sec p:= sum(a(i)*x**i, i = 1..n) n --+ i (9) > a(i)x --+ i= 1 Type: Expression Integer Time: 0.77 (IN) + 0.68 (EV) + 0.15 (OT) + 0.07 (GC) = 1.67 sec -- Convert into FORTRAN syntax outputAsFortran('p = p) >> Fortran translation error: No corresponding Fortran structure for: n --+ i > a(i)x --+ i= 1 Type: Void Time: 0.27 (IN) + 0.03 (EV) + 0.05 (OT) = 0.35 sec -- Convert into C syntax -- Horner's rule---this is important for numerical algorithms -- => (a[1] + (a[2] + (a[3] + (a[4] + a[5] x) x) x) x) x p:= sum(a(i)*x**i, i = 1..5) 5 4 3 2 (11) a(5)x + a(4)x + a(3)x + a(2)x + a(1)x Type: Expression Integer Time: 0.07 (IN) + 0.22 (EV) + 0.03 (OT) = 0.32 sec factor(p) 5 4 3 2 (12) a(5)x + a(4)x + a(3)x + a(2)x + a(1)x Type: Factored Expression Integer Time: 0.02 (EV) + 0.02 (OT) = 0.03 sec p:= p :: MPOLY([x], Expression Integer) 5 4 3 2 (13) a(5)x + a(4)x + a(3)x + a(2)x + a(1)x Type: MultivariatePolynomial([x],Expression Integer) Time: 0.07 (IN) + 0.03 (OT) + 0.02 (GC) = 0.12 sec p:= factor(p) WARNING (genufact): No known algorithm to factor 4 a(4) 3 a(3) 2 a(2) a(1) ? + ---- ? + ---- ? + ---- ? + ----, trying square-free. a(5) a(5) a(5) a(5) 4 a(4) 3 a(3) 2 a(2) a(1) (14) a(5)x(x + ---- x + ---- x + ---- x + ----) a(5) a(5) a(5) a(5) Type: Factored MultivariatePolynomial([x],Expression Integer) Time: 0.08 (IN) + 0.32 (EV) + 0.02 (OT) + 0.05 (GC) = 0.47 sec -- Convert the result into FORTRAN syntax -- => p = (a(1) + (a(2) + (a(3) + (a(4) + a(5)*x)*x)*x)*x)*x )set fortran ints2floats off outputAsFortran('p = p) p=a(5)*x*(x**4+(a(4)/a(5))*x**3+(a(3)/a(5))*x*x+(a(2)/a(5))*x+a(1) &/a(5)) Type: Void Time: 0.48 (IN) + 0.03 (EV) + 0.12 (OT) = 0.63 sec )clear properties a p -- Convert the result into C syntax -- => p = (a[1] + (a[2] + (a[3] + (a[4] + a[5]*x)*x)*x)*x)*x ; -- Count the number of (floating point) operations needed to compute an -- expression => {[+, n - 1], [*, (n^2 - n)/2], [f, (n^2 + n)/2]} f:= operator('f); Type: BasicOperator Time: 0.03 (IN) = 0.03 sec sum(product(f(i, k), i = 1..k), k = 1..n) n k --+ ++-++ (17) > | | f(i,i) --+ | | k= 1 i= 1 Type: Expression Integer Time: 0.13 (IN) + 0.10 (EV) + 0.05 (OT) = 0.28 sec -- Interval analysis (interval polynomial example): -- ([-4, 2] x + [1, 3])^2 => [-8, 16] x^2 + [-24, 12] x + [1, 9] -- Discretize a PDE: for example, forward differencing time (explicit Euler) -- and central differencing x on the heat equation => -- (f[i, j+1] - f[i, j])/dt = (f[i+1, j] - 2 f[i, j] + f[i-1, j])/dx^2 D(f(x, t), t) = D(f(x, t), x, 2) (18) f (x,t)= f (x,t) ,2 ,1,1 Type: Equation Expression Integer Time: 0.08 (IN) + 0.07 (EV) + 0.03 (OT) = 0.18 sec )clear properties f -- ---------- Quit ---------- )quit real 7.1 user 6.2 sys 0.4 \end{verbatim} \section{Statistics} \begin{verbatim} initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Statistics ---------- -- Compute the mean of a list of numbers => 9 mean(lst) == reduce(+, lst)/#lst Type: Void Time: 0 sec mean([3, 7, 11, 5, 19]) Compiling function mean with type List PositiveInteger -> Fraction Integer (2) 9 Type: Fraction Integer Time: 0.08 (IN) + 0.02 (OT) + 0.02 (GC) = 0.12 sec -- Compute the median of a list of numbers => 7 [3, 7, 11, 5, 19] (3) [3,7,11,5,19] Type: List PositiveInteger Time: 0.02 (OT) = 0.02 sec -- Compute the first quartile (25% quantile) of a list of numbers => 2 or 5/2 xx:= [1, 2, 3, 4, 5, 6, 7, 8]; Type: List PositiveInteger Time: 0 sec --quartiles(xx, 1) --quantile(xx, 1/4) )clear properties xx -- Compute the mode (the most frequent item) of a list of numbers => 7 [3, 7, 11, 7, 3, 5, 7] (5) [3,7,11,7,3,5,7] Type: List PositiveInteger Time: 0.02 (IN) = 0.02 sec -- Compute the unbiased sample standard deviation of a list of numbers -- => sqrt(5/2) [1, 2, 3, 4, 5] (6) [1,2,3,4,5] Type: List PositiveInteger Time: 0 sec -- Discrete distributions---what is the probability of finding exactly 12 -- switches that work from a group of 15 where the probability of a single one -- working is 75%? (Need to use the probability density function [PDF] of the -- binomial distribution.) => 0.22520 --PDF(BinomialDistribution(15, .75), 12) -- Replace `exactly' by `up through' in the above. (Need to use the cumulative -- probability density function [CDF] of the binomial distribution.) => 0.76391 --CDF(BinomialDistribution(15, .75), 12) -- Continuous distributions---if a radiation emission can be modeled by a -- normal distribution with a mean of 4.35 mrem and a standard deviation of -- 0.59 mrem, what is the probability of being exposed to anywhere from 4 to 5 -- mrem? => .5867 --CDF(Normal(4.35, 0.59), 5) - CDF(Normal(4.35, 0.59), 4) -- Hypothesis testing---how good of a guess is 5 for the mean of xx? */ xx:= [1, -2, 3, -4, 5, -6, 7, -8, 9, 10]; Type: List Integer Time: 0 sec -- Using Student's T distribution (preferred) => 0.057567 --students_t_distrib((sample_mean(xx) - 5)/(sample_standard_deviation(xx) / -- sqrt(length(xx))), length(xx) - 1) --% :: Float -- Using the normal distribution (as an alternative) => 0.040583 --standard_normal_distrib((sample_mean(xx) - 5)/(sample_standard_deviation(xx) / -- sqrt(length(xx)))) --% :: Float )clear properties xx -- Chi-square test---what is the expectation that row characteristics are -- independent of column characteristics for a two dimensional array of data? -- => 0.469859 (chi2 = 1153/252) x:= matrix(@[[41, 27, 22], [79, 53, 78]]); Type: Matrix Integer Time: 0.02 (IN) + 0.02 (EV) + 0.02 (OT) = 0.05 sec m:= nrows(x); Type: PositiveInteger Time: 0 sec n:= ncols(x); Type: PositiveInteger Time: 0 sec rowSum:= [reduce(+, row(x, i)) for i in 1..m]; Type: List Integer Time: 0.03 (IN) = 0.03 sec colSum:= [reduce(+, column(x, j)) for j in 1..n]; Type: List Integer Time: 0.02 (IN) = 0.02 sec matSum:= reduce(+, rowSum); Type: PositiveInteger Time: 0.02 (IN) = 0.02 sec e : ARRAY2 Fraction Integer := new(m, n, 0); Type: TwoDimensionalArray Fraction Integer Time: 0.02 (IN) = 0.02 sec for i in 1..m repeat for j in 1..n repeat e(i, j):= rowSum(i)*colSum(j)/matSum; Type: Void Time: 0.02 (EV) + 0.02 (OT) = 0.03 sec for i in 1..m repeat for j in 1..n repeat _ e(i, j):= (x(i, j) - e(i, j))**2/e(i, j); Type: Void Time: 0.03 (IN) + 0.02 (EV) + 0.03 (OT) + 0.02 (GC) = 0.10 sec -- chi2:= sum(sum((x(i, j) - e(i, j))**2/e(i, j), i = 1..m), j = 1..n) chi2:= reduce(+, [reduce(+, row(e, i)) for i in 1..m]) 1153 (17) ---- 252 Type: Fraction Integer Time: 0.03 (IN) = 0.03 sec --ChiSquarePValue(chi2, m*n - 1) -- or --1 - CDF(ChiSquareDistribution(m*n - 1), chi2) )clear properties chi2 colSum matSum rowSum e m n x -- Linear regression (age as a function of developmental score). See Lambert -- H. Koopmans, _Introduction to Contemporary Statistical Methods_, Second -- Edition, Duxbury Press, 1987, p. 459 => y' = 0.7365 x + 6.964 t:= @[[3.33, 3.25, 3.92, 3.50, 4.33, 4.92, 6.08, 7.42, 8.33, 8.00, 9.25, _ 10.75], _ [8.61, 9.40, 9.86, 9.91, 10.53, 10.61, 10.59, 13.28, 12.76, 13.44, 14.27, _ 14.13]]; Type: List List Float Time: 0.10 (IN) + 0.03 (OT) + 0.02 (GC) = 0.15 sec --fit(transpose(%), [1, x], x) )clear properties t -- Multiple linear regression (income as a function of age and years of -- college) => y = -16278.7 + 960.925 x1 + 2975.66 x2 @[[37, 45, 38, 42, 31], [4, 0, 5, 2, 4], [31200, 26800, 35000, 30300, 25400]]; Type: List List NonNegativeInteger Time: 0.05 (IN) + 0.02 (OT) = 0.07 sec --fit(transpose(%), [1, x1, x2], [x1, x2]) -- Multiple linear regression using the L1 or Least Absolute Deviations -- technique rather than the Least Squares technique (minimizing the sum of the -- absolute values of the residuals rather than the sum of the squares of the -- residuals). Here, the Stack-loss Data is used (percentage of ammonia lost -- times 10 from the operation of a plant over 21 days as a function of air -- flow to the plant, cooling water inlet temperature and acid concentration). -- See W. N. Venables and B. D. Ripley, _Modern Applied Statistics with -- S-plus_, Springer, 1994, p. 218. -- => y = 0.83188 x1 + 0.57391 x2 - 0.06086 x3 - 39.68984 @[[80, 80, 75, 62, 62, 62, 62, 62, 58, 58, 58, 58, 58, 58, 50, 50, 50, 50, 50, _ 56, 70], _ [27, 27, 25, 24, 22, 23, 24, 24, 23, 18, 18, 17, 18, 19, 18, 18, 19, 19, 20, _ 20, 20], _ [89, 88, 90, 87, 87, 87, 93, 93, 87, 80, 89, 88, 82, 93, 89, 86, 72, 79, 80, _ 82, 91], _ [42, 37, 37, 28, 18, 18, 19, 20, 15, 14, 14, 13, 11, 12, 8, 7, 8, 8, 9, _ 15, 15]]; Type: List List PositiveInteger Time: 0.02 (IN) = 0.02 sec --fit(transpose(%), [1, x1, x2, x3], [x1, x2, x3]) -- Nonlinear regression (Weight Loss Data from an Obese Patient consisting of -- the time in days and the weight in kilograms of a patient undergoing a -- weight rehabilitation program). Fit this using least squares to weight = -- b0 + b1 2^(- days/th), starting at (b0, b1, th) = (90, 95, 120) [Venables -- and Ripley, p. 225] => weight = 81.37375 + 102.6842 2^(- days/141.9105) @[[ 0, 4, 7, 7, 11, 18, 24, 30, 32, 43, 46, 60, 64, 70, 71, _ 71, 73, 74, 84, 88, 95, 102, 106, 109, 115, 122, 133, 137, 140, 143, _ 147, 148, 149, 150, 153, 156, 161, 164, 165, 165, 170, 176, 179, 198, 214, _ 218, 221, 225, 233, 238, 241, 246], _ [184.35, 182.51, 180.45, 179.91, 177.91, 175.81, 173.11, 170.06, 169.31, _ 165.10, 163.11, 158.30, 155.80, 154.31, 153.86, 154.20, 152.20, 152.80, _ 150.30, 147.80, 146.10, 145.60, 142.50, 142.30, 139.40, 137.90, 133.70, _ 133.70, 133.30, 131.20, 133.00, 132.20, 130.80, 131.30, 129.00, 127.90, _ 126.90, 127.70, 129.50, 128.40, 125.40, 124.90, 124.90, 118.20, 118.20, _ 115.30, 115.70, 116.00, 115.50, 112.60, 114.00, 112.60]]; Type: List List Float Time: 0.30 (IN) + 0.03 (OT) = 0.33 sec -- ---------- Quit ---------- )quit real 15.0 user 2.4 sys 0.3 \end{verbatim} \section{Combinatorial Theory} \begin{verbatim} -- ---------- Combinatorial Theory ---------- -- Pochhammer symbol (a)_n = a (a + 1) ... (a + n - 1) => a (a + 1) (a + 2) --pochhammer(a, 3) -- Binomial coefficient => n (n - 1) (n - 2)/6 binomial(n, 3) n (1) ( ) 3 Type: Expression Integer Time: 0.27 (IN) + 0.12 (EV) + 0.20 (OT) + 0.12 (GC) = 0.70 sec factorFraction(normalize(%) :: Fraction Polynomial Integer) (n - 2)(n - 1)n (2) --------------- 6 Type: Fraction Factored Polynomial Integer Time: 0.12 (IN) + 0.38 (EV) + 0.25 (OT) + 0.07 (GC) = 0.82 sec -- 2^n n! (2 n - 1)!! => (2 n)! --2**n * factorial(n) * (2*n - 1)!! -- 2^n n! product(2 k - 1, k = 1..n) => (2 n)! 2**n * factorial(n) * product(2*k - 1, k = 1..n) n n++-++ (3) n!2 | | 2k - 1 | | k= 1 Type: Expression Integer Time: 0.30 (IN) + 0.03 (EV) + 0.07 (OT) = 0.40 sec -- => (2 n)!/[2^(2 n) (n!)^2] or (2 n - 1)!!/[2^n n!] Gamma(n + 1/2)/(sqrt(%pi) * factorial(n)) _ 2n + 1 | (------) 2 (4) ---------- +---+ n!\|%pi Type: Expression Integer Time: 0.85 (IN) + 0.02 (EV) + 0.07 (OT) + 0.02 (GC) = 0.95 sec -- Partitions of an integer => {1+1+1+1, 1+1+2, 1+3, 2+2, 4} (5 in all) 4 (5) 4 Type: PositiveInteger Time: 0 sec -- Stirling numbers of the first kind: S_1(5, 2) => -50 stirling1(5, 2) (6) - 50 Type: Integer Time: 0.02 (EV) = 0.02 sec -- Euler's totient function => 576 eulerPhi(1776) (7) 576 Type: PositiveInteger Time: 0.02 (IN) = 0.02 sec -- ---------- Quit ---------- )quit real 9.9 user 3.6 sys 0.3 \end{verbatim} \section{Number Theory} \begin{verbatim} -- ---------- Number Theory ---------- -- Display the largest 6-digit prime and the smallest 7-digit prime -- => [999983, 1000003] [prevPrime(1000000), nextPrime(1000000)] (1) [999983,1000003] Type: List PositiveInteger Time: 0.02 (IN) + 0.08 (OT) + 0.02 (GC) = 0.12 sec -- Primitive root => 19 191 (2) 191 Type: PositiveInteger Time: 0.02 (OT) = 0.02 sec -- (a + b)^p mod p => a^p + b^p for p prime and a, b in Z_p [Chris Hurlburt] -- See Thomas W. Hungerford, _Algebra_, Springer-Verlag, 1974, p. 121 for a -- more general simplification: (a +- b)^(p^n) => a^(p^n) +- b^(p^n) (a + b)**p :: PrimeField(p) Cannot convert the first argument of PrimeField p to the type PositiveInteger. -- Congruence equations. See Harold M. Stark, _An Introduction to Number -- Theory_, The MIT press, 1984. -- 9 x = 15 mod 21 => x = 4 mod 7 or {4, 11, 18} mod 21 [Stark, p. 68] solve(9*x = 15 :: IntegerMod(21), x) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial IntegerMod 21 Variable x -- 7 x = 22 mod 39 => x = 5 mod 13 or 31 mod 39 [Stark, p. 69] solve(7*x = 22 :: IntegerMod(39), x) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial IntegerMod 39 Variable x -- x^2 + x + 4 = 0 mod 8 => x = {3, 4} mod 8 [Stark, p. 97] solve(x**2 + x + 4 = 0 :: IntegerMod(8), x) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial IntegerMod 8 Variable x -- x^3 + 2 x^2 + 5 x + 6 = 0 mod 11 => x = 3 mod 11 [Stark, p. 97] solve(x**3 + 2*x**2 + 5*x + 6 = 0 :: PrimeField(11), x) (3) [x= 3] Type: List Equation Fraction Polynomial PrimeField 11 Time: 0.72 (IN) + 0.18 (EV) + 0.18 (OT) = 1.08 sec -- {x = 7 mod 9, x = 13 mod 23, x = 1 mod 2} => x = 151 mod 414 [Stark, -- p. 76] chineseRemainder([7, 13, 1], [9, 23, 2]) (4) 151 Type: PositiveInteger Time: 0.02 (IN) + 0.02 (EV) + 0.03 (OT) + 0.03 (GC) = 0.10 sec -- {5 x + 4 y = 6 mod 7, 3 x - 2 y = 6 mod 7} => x = 1 mod 7, y = 2 mod 7 -- [Stark, p. 76] solve([5*x + 4*y = 6 :: PrimeField(7), 3*x - 2*y = 6 :: PrimeField(7)], [x, y]) (5) @[[x= 1,y= 2]] Type: List List Equation Fraction Polynomial PrimeField 7 Time: 1.33 (IN) + 0.08 (EV) + 0.27 (OT) + 0.02 (GC) = 1.70 sec -- 2 x + 3 y = 1 mod 5 => -- (x, y) = {(0, 2), (1, 3), (2, 4), (3, 0), (4, 1)} mod 5 solve(2*x + 3*y = 1 :: PrimeField(5), [x, y]) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial PrimeField 5 List OrderedVariableList [x,y] -- 2 x + 3 y = 1 mod 6 => [Stark, p. 76] -- (x, y) = {(2, 1), (2, 3), (2, 5), (5, 1), (5, 3), (5, 5)} mod 6 solve(2*x + 3*y = 1 :: IntegerMod(6), [x, y]) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial IntegerMod 6 List OrderedVariableList [x,y] -- Diophantine equations => x = 2, y = 5 (Wallis) [Stark, p. 147] solve(x**4 + 9 = y**2, [x, y]) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial Integer List OrderedVariableList [x,y] -- => x = 11, y = 5 (Fermat) [Stark, p. 147] solve(x**2 + 4 = y**3, [x, y]) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial Integer List OrderedVariableList [x,y] -- => (x, y, t, z, w) = (3, 4, 5, 12, 13), (7, 24, 25, 312, 313), ... -- [Stark, p. 154] system:= [x**2 + y**2 = t**2, t**2 + z**2 = w**2] 2 2 2 2 2 2 (6) [y + x = t ,z + t = w ] Type: List Equation Polynomial Integer Time: 0.02 (IN) + 0.02 (EV) + 0.03 (OT) = 0.07 sec solve(system, [x, y, t, z, w]) >> Error detected within library code: system does not have a finite number of solutions initial (7) -> real 58.4 user 14.4 sys 0.5 ------------------------------------------------------------------------------- Fri Jun 13 01:16:45 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Number Theory ---------- -- Rational approximation of sqrt(3) with an error tolerance of 1/500 => 26/15 rationalApproximation(sqrt(3.), 3) 26 (1) -- 15 Type: Fraction Integer Time: 0.02 (IN) + 0.05 (EV) + 0.07 (OT) + 0.07 (GC) = 0.20 sec -- Continued fractions => 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ... continuedFraction(3.1415926535) (2) 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 + +---+ + +----+ + +---+ + +-----+ + +---+ + +---+ + +---+ + +---+ | 7 | 15 | 1 | 292 | 1 | 1 | 6 | 2 + 1 | 1 | +----+ + +---+ + ... | 13 | 3 Type: ContinuedFraction Integer Time: 0.03 (EV) + 0.05 (OT) + 0.02 (GC) = 0.10 sec -- => 4 + 1/(1 + 1/(3 + 1/(1 + 1/(8 + 1/(1 + 1/(3 + 1/(1 + 1/(8 + ... -- [Stark, p. 340] continuedFraction(sqrt(23)) There are 2 exposed and 3 unexposed library operations named continuedFraction having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op continuedFraction to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named continuedFraction with argument type(s) AlgebraicNumber continuedFraction(sqrt(23) :: Float) (3) 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ | 1 | 3 | 1 | 8 | 1 | 3 | 1 | 8 | 1 + 1 | +---+ + ... | 3 Type: ContinuedFraction Integer Time: 0.13 (IN) + 0.03 (EV) + 0.03 (OT) + 0.03 (GC) = 0.23 sec -- => 1 + 1/(1 + 1/(1 + 1/(1 + ... See Oskar Perron, _Die Lehre von den -- Kettenbr\"uchen_, Chelsea Publishing Company, 1950, p. 52. continuedFraction((1 + sqrt(5))/2 :: Float) (4) 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 + 1 | +---+ + ... | 1 Type: ContinuedFraction Integer Time: 0.35 (IN) + 0.02 (EV) + 0.05 (OT) = 0.42 sec -- => 1/(2 x + 1/(6 x + 1/(10 x + 1/(14 x + ... [Perron, p. 353] continuedFraction((exp(1/x) - 1)/(exp(1/x) + 1)) There are 2 exposed and 3 unexposed library operations named continuedFraction having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op continuedFraction to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named continuedFraction with argument type(s) Expression Integer -- => 1/(2 x + 1/(2 x + 1/(2 x + ... (Re x > 0) From Liyang Xu, ``Method -- Derived from Continued Fraction Approximations'', draft. continuedFraction(sqrt(x**2 + 1) - x) There are 2 exposed and 3 unexposed library operations named continuedFraction having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op continuedFraction to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named continuedFraction with argument type(s) Expression Integer -- ---------- Quit ---------- )quit real 9.3 user 3.0 sys 0.4 \end{verbatim} \section{Algebra} \begin{verbatim} -- ---------- Algebra ---------- -- One would think that the simplification 2 2^n => 2^(n + 1) would happen -- automatically or at least easily ... 2*2**n n (1) 2 2 Type: Expression Integer Time: 0.18 (IN) + 0.10 (EV) + 0.17 (OT) + 0.12 (GC) = 0.57 sec -- And how about 4 2^n => 2^(n + 2)? [Richard Fateman] 4*2**n n (2) 4 2 Type: Expression Integer Time: 0.05 (IN) = 0.05 sec -- (-1)^[n (n + 1)] => 1 for integer n (-1)**(n*(n + 1)) 2 n + n (3) (- 1) Type: Expression Integer Time: 0.07 (IN) + 0.02 (EV) = 0.08 sec -- Also easy => 2 (3 x - 5) factor(6*x - 10) (4) 2(3x - 5) Type: Factored Polynomial Integer Time: 0.08 (IN) + 0.15 (EV) + 0.03 (OT) + 0.05 (GC) = 0.32 sec -- Univariate gcd: gcd(p1, p2) => 1, gcd(p1 q, p2 q) => q [Richard Liska] p1:= 64*x**34 - 21*x**47 - 126*x**8 - 46*x**5 - 16*x**60 - 81; Type: Polynomial Integer Time: 0.02 (IN) + 0.05 (OT) = 0.07 sec p2:= 72*x**60 - 25*x**25 - 19*x**23 - 22*x**39 - 83*x**52 + 54*x**10 + 81; Type: Polynomial Integer Time: 0.07 (IN) = 0.07 sec q:= 34*x**19 - 25*x**16 + 70*x**7 + 20*x**3 - 91*x - 86; Type: Polynomial Integer Time: 0.02 (IN) + 0.05 (OT) = 0.07 sec gcd(p1, p2) (8) 1 Type: Polynomial Integer Time: 0.07 (EV) = 0.07 sec gcd(expand(p1*q), expand(p2*q)) - q (9) 0 Type: Polynomial Integer Time: 31.85 (IN) + 0.08 (EV) + 0.02 (OT) + 4.53 (GC) = 36.48 sec -- resultant(p1 q, p2 q) => 0 resultant(expand(p1*q), expand(p2*q), x) (10) 0 Type: Polynomial Integer Time: 31.66 (IN) + 1.42 (EV) + 4.85 (GC) = 37.93 sec -- How about factorization? => p1 * p2 factor(expand(p1 * p2)) (11) - 120 112 107 99 94 86 85 1152x - 1328x + 1512x - 2095x - 4608x + 4850x - 400x + 83 73 72 70 68 65 60 - 304x + 1408x - 525x + 465x + 9072x + 3312x - 3330x + 59 57 52 47 44 39 34 1600x - 1468x - 6723x - 1071x - 4468x - 1782x - 5184x + 33 31 30 28 25 23 18 - 3150x - 2394x - 1150x - 874x - 2025x - 1539x + 6804x + 15 10 8 5 2484x + 4374x + 10206x + 3726x + 6561 Type: Factored Polynomial Integer Time: 182.21 (IN) + 184.93 (EV) + 0.07 (OT) + 60.45 (GC) = 427.65 sec )clear properties p1 p2 q -- Multivariate gcd: gcd(p1, p2) => 1, gcd(p1 q, p2 q) => q p1:= 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5; Type: Polynomial Integer Time: 0.08 (IN) + 0.02 (OT) = 0.10 sec p2:= 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z; Type: Polynomial Integer Time: 0.08 (IN) + 0.03 (OT) = 0.12 sec q:= 11*x**12*y**7*z**13 - 23*x**2*y**8*z**10 + 47*x**17*y**5*z**8; Type: Polynomial Integer Time: 0.05 (IN) + 0.03 (OT) = 0.08 sec gcd(p1, p2) (15) 1 Type: Polynomial Integer Time: 0.02 (EV) = 0.02 sec gcd(expand(p1*q), expand(p2*q)) - q (16) 0 Type: Polynomial Integer Time: 6.62 (IN) + 8.73 (EV) + 1.32 (GC) = 16.67 sec -- How about factorization? => p1 * p2 factor(expand(p1 * p2)) (17) 7 19 17 5 8 15 9 2 22 2y z((24x y - 47x y )z + 6x y z - 3x + 5) * 5 12 7 6 9 9 3 7 (17x y z + 10x z + 6x y z + 40y ) Type: Factored Polynomial Integer Time: 9.30 (IN) + 10.72 (EV) + 0.02 (OT) + 1.53 (GC) = 21.57 sec )clear properties p1 p2 q -- => x^n for n > 0 [Chris Hurlburt] gcd(2*x**(n + 4) - x**(n + 2), 4*x**(n + 1) + 3*x**n) (18) 1 Type: Expression Integer Time: 0.07 (IN) + 0.05 (OT) = 0.12 sec -- Resultants. If the resultant of two polynomials is zero, this implies they -- have a common factor. See Keith O. Geddes, Stephen R. Czapor and George -- Labahn, _Algorithms for Computer Algebra_, Kluwer Academic Publishers, 1992, -- p. 286 => 0 resultant(3*x**4 + 3*x**3 + x**2 - x - 2, x**3 - 3*x**2 + x + 5, x) (19) 0 Type: Polynomial Integer Time: 0.03 (IN) = 0.03 sec -- Numbers are nice, but symbols allow for variability---try some high school -- algebra: rational simplification => (x - 2)/(x + 2) (x**2 - 4)/(x**2 + 4*x + 4) x - 2 (20) ----- x + 2 Type: Fraction Polynomial Integer Time: 0.03 (IN) + 0.02 (OT) = 0.05 sec -- This example requires more sophistication => e^(x/2) - 1 [(%e**x - 1)/(%e**(x/2) + 1), (exp(x) - 1)/(exp(x/2) + 1)] x x %e - 1 %e - 1 (21) [-------,-------] x x - - 2 2 %e + 1 %e + 1 Type: List Expression Integer Time: 1.02 (IN) + 0.20 (EV) + 0.15 (OT) + 0.35 (GC) = 1.72 sec map(normalize, %) x x - - 2 2 (22) [%e - 1,%e - 1] Type: List Expression Integer Time: 0.05 (IN) + 0.45 (EV) + 0.02 (OT) = 0.52 sec -- Expand and factor polynomials (x + 1)**20 (23) 20 19 18 17 16 15 14 13 x + 20x + 190x + 1140x + 4845x + 15504x + 38760x + 77520x + 12 11 10 9 8 7 6 125970x + 167960x + 184756x + 167960x + 125970x + 77520x + 38760x + 5 4 3 2 15504x + 4845x + 1140x + 190x + 20x + 1 Type: Polynomial Integer Time: 0.02 (EV) + 0.02 (OT) = 0.03 sec D(%, x) (24) 19 18 17 16 15 14 13 20x + 380x + 3420x + 19380x + 77520x + 232560x + 542640x + 12 11 10 9 8 7 1007760x + 1511640x + 1847560x + 1847560x + 1511640x + 1007760x + 6 5 4 3 2 542640x + 232560x + 77520x + 19380x + 3420x + 380x + 20 Type: Polynomial Integer Time: 0.02 (EV) + 0.02 (OT) = 0.03 sec factor(%) 19 (25) 20(x + 1) Type: Factored Polynomial Integer Time: 0.02 (EV) = 0.02 sec -- Completely factor this polynomial, then try to multiply it back together! radicalSolve(x**3 + x**2 - 7 = 0, x) (26) +--------------+2 +--------------+ | +----+ | +----+ | |1295 | |1295 |9 |---- + 187 |9 |---- + 187 +---+ | \| 3 +---+ | \| 3 (- \|- 3 + 1) 3|-------------- + (- \|- 3 - 1) 3|-------------- - 2 \| 2 \| 2 [x= ----------------------------------------------------------------------, +--------------+ | +----+ | |1295 |9 |---- + 187 +---+ | \| 3 (3\|- 3 + 3) 3|-------------- \| 2 +--------------+2 +--------------+ | +----+ | +----+ | |1295 | |1295 |9 |---- + 187 |9 |---- + 187 +---+ | \| 3 +---+ | \| 3 (- \|- 3 - 1) 3|-------------- + (- \|- 3 + 1) 3|-------------- + 2 \| 2 \| 2 x= ----------------------------------------------------------------------, +--------------+ | +----+ | |1295 |9 |---- + 187 +---+ | \| 3 (3\|- 3 - 3) 3|-------------- \| 2 +--------------+2 +--------------+ | +----+ | +----+ | |1295 | |1295 |9 |---- + 187 |9 |---- + 187 | \| 3 | \| 3 3|-------------- - 3|-------------- + 1 \| 2 \| 2 x= ------------------------------------------] +--------------+ | +----+ | |1295 |9 |---- + 187 | \| 3 3 3|-------------- \| 2 Type: List Equation Expression Integer Time: 0.13 (IN) + 0.12 (EV) + 0.13 (OT) = 0.38 sec reduce(*, map(e +-> lhs(e) - rhs(e), %)) 3 2 (27) x + x - 7 Type: Expression Integer Time: 0.03 (IN) + 0.07 (EV) = 0.10 sec x**100 - 1 100 (28) x - 1 Type: Polynomial Integer Time: 0.02 (OT) = 0.02 sec factor(%) (29) 2 4 3 2 4 3 2 (x - 1)(x + 1)(x + 1)(x - x + x - x + 1)(x + x + x + x + 1) * 8 6 4 2 20 15 10 5 20 15 10 5 (x - x + x - x + 1)(x - x + x - x + 1)(x + x + x + x + 1) * 40 30 20 10 (x - x + x - x + 1) Type: Factored Polynomial Integer Time: 0.05 (EV) + 0.03 (OT) = 0.08 sec -- Factorization over the complex rationals -- => (2 x + 3 i) (2 x - 3 i) (x + 1 + 4 i) (x + 1 - 4 i) factor(4*x**4 + 8*x**3 + 77*x**2 + 18*x + 153, [rootOf(i**2 + 1)]) 3i 3i (30) 4(x - 4i + 1)(x - --)(x + --)(x + 4i + 1) 2 2 Type: Factored Polynomial AlgebraicNumber Time: 0.17 (IN) + 1.65 (EV) + 0.03 (OT) = 1.85 sec -- Algebraic extensions sqrt2:= rootOf(sqrt2**2 - 2); Type: AlgebraicNumber Time: 0.02 (OT) = 0.02 sec -- => sqrt2 + 1 1/(sqrt2 - 1) (32) sqrt2 + 1 Type: AlgebraicNumber Time: 0.02 (IN) + 0.03 (EV) + 0.02 (OT) = 0.07 sec -- => (x^2 - 2 x - 3)/(x - sqrt2) = (x + 1) (x - 3)/(x - sqrt2) -- [Richard Liska] (x**3 + (sqrt2 - 2)*x**2 - (2*sqrt2 + 3)*x - 3*sqrt2)/(x**2 - 2) 2 x - 2x - 3 (33) ----------- x - sqrt2 Type: Fraction Polynomial AlgebraicNumber Time: 0.40 (IN) + 0.20 (EV) + 0.12 (OT) = 0.72 sec numer(%)/ratDenom(denom(%)) 2 - x + 2x + 3 (34) ------------- sqrt2 - x Type: Expression Integer Time: 0.27 (IN) + 0.08 (OT) = 0.35 sec )clear properties sqrt2 -- Multiple algebraic extensions sqrt3:= rootOf(sqrt3**2 - 3); Type: AlgebraicNumber Time: 0.02 (EV) + 0.02 (OT) = 0.03 sec cbrt2:= rootOf(cbrt2**3 - 2); Type: AlgebraicNumber Time: 0.02 (IN) = 0.02 sec -- => 2 cbrt2 + 8 sqrt3 + 18 cbrt2^2 + 12 cbrt2 sqrt3 + 9 (cbrt2 + sqrt3)**4 2 (37) (12cbrt2 + 8)sqrt3 + 18cbrt2 + 2cbrt2 + 9 Type: AlgebraicNumber Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec )clear properties sqrt3 cbrt2 -- Factor polynomials over finite fields and field extensions p:= x**4 - 3*x**2 + 1 4 2 (38) x - 3x + 1 Type: Polynomial Integer Time: 0.02 (IN) = 0.02 sec factor(p) 2 2 (39) (x - x - 1)(x + x - 1) Type: Factored Polynomial Integer Time: 0.03 (EV) = 0.03 sec -- => (x - 2)^2 (x + 2)^2 mod 5 factor(p :: Polynomial(PrimeField(5))) 2 2 (40) (x + 2) (x + 3) Type: Factored Polynomial PrimeField 5 Time: 0.12 (IN) + 0.12 (EV) + 0.02 (OT) + 0.05 (GC) = 0.30 sec expand(%) 4 2 (41) x + 2x + 1 Type: Polynomial PrimeField 5 Time: 0.02 (IN) = 0.02 sec -- => (x^2 + x + 1) (x^9 - x^8 + x^6 - x^5 + x^3 - x^2 + 1) mod 65537 -- [Paul Zimmermann] factor(x**11 + x + 1 :: Polynomial(PrimeField(65537))) 2 9 8 6 5 3 2 (42) (x + x + 1)(x + 65536x + x + 65536x + x + 65536x + 1) Type: Factored Polynomial PrimeField 65537 Time: 0.13 (IN) + 0.13 (EV) + 0.03 (OT) = 0.30 sec -- => (x - phi) (x + phi) (x - phi + 1) (x + phi - 1) -- where phi^2 - phi - 1 = 0 or phi = (1 +- sqrt(5))/2 phi:= rootOf(phi**2 - phi - 1); Type: AlgebraicNumber Time: 0.03 (IN) = 0.03 sec factor(p, [phi]) (44) (x - phi)(x - phi + 1)(x + phi - 1)(x + phi) Type: Factored Polynomial AlgebraicNumber Time: 0.80 (EV) + 0.02 (OT) = 0.82 sec )clear properties phi p expand((x - 2*y**2 + 3*z**3)**20); Type: Polynomial Integer Time: 0.17 (IN) + 0.55 (EV) + 0.03 (OT) + 0.42 (GC) = 1.17 sec factor(%) 3 2 20 (46) (3z - 2y + x) Type: Factored Polynomial Integer Time: 0.13 (EV) = 0.13 sec expand((sin(x) - 2*cos(y)**2 + 3*tan(z)**3)**20); Type: Expression Integer Time: 0.07 (IN) + 1.30 (EV) + 0.07 (OT) = 1.43 sec factor(%) (48) 60 2 57 3486784401tan(z) + (23245229340sin(x) - 46490458680cos(y) )tan(z) + 2 2 4 (73609892910sin(x) - 294439571640cos(y) sin(x) + 294439571640cos(y) ) * 54 tan(z) + 3 2 2 147219785820sin(x) - 883318714920cos(y) sin(x) + 4 6 1766637429840cos(y) sin(x) - 1177758286560cos(y) * 51 tan(z) + 4 2 3 208561363245sin(x) - 1668490905960cos(y) sin(x) + 4 2 6 5005472717880cos(y) sin(x) - 6673963623840cos(y) sin(x) + 8 3336981811920cos(y) * 48 tan(z) + 5 2 4 222465454128sin(x) - 2224654541280cos(y) sin(x) + 4 3 6 2 8898618165120cos(y) sin(x) - 17797236330240cos(y) sin(x) + 8 10 17797236330240cos(y) sin(x) - 7118894532096cos(y) * 45 tan(z) + 6 2 5 185387878440sin(x) - 2224654541280cos(y) sin(x) + 4 4 6 3 11123272706400cos(y) sin(x) - 29662060550400cos(y) sin(x) + 8 2 10 44493090825600cos(y) sin(x) - 35594472660480cos(y) sin(x) + 12 11864824220160cos(y) * 42 tan(z) + 7 2 6 123591918960sin(x) - 1730286865440cos(y) sin(x) + 4 5 6 4 10381721192640cos(y) sin(x) - 34605737308800cos(y) sin(x) + 8 3 10 2 69211474617600cos(y) sin(x) - 83053769541120cos(y) sin(x) + 12 14 55369179694080cos(y) sin(x) - 15819765626880cos(y) * 39 tan(z) + 8 2 7 66945622770sin(x) - 1071129964320cos(y) sin(x) + 4 6 6 5 7497909750240cos(y) sin(x) - 29991639000960cos(y) sin(x) + 8 4 10 3 74979097502400cos(y) sin(x) - 119966556003840cos(y) sin(x) + 12 2 14 119966556003840cos(y) sin(x) - 68552317716480cos(y) sin(x) + 16 17138079429120cos(y) * 36 tan(z) + 9 2 8 29753610120sin(x) - 535564982160cos(y) sin(x) + 4 7 6 6 4284519857280cos(y) sin(x) - 19994426000640cos(y) sin(x) + 8 5 10 4 59983278001920cos(y) sin(x) - 119966556003840cos(y) sin(x) + 12 3 14 2 159955408005120cos(y) sin(x) - 137104635432960cos(y) sin(x) + 16 18 68552317716480cos(y) sin(x) - 15233848381440cos(y) * 33 tan(z) + 10 2 9 10909657044sin(x) - 218193140880cos(y) sin(x) + 4 8 6 7 1963738267920cos(y) sin(x) - 10473270762240cos(y) sin(x) + 8 6 10 5 36656447667840cos(y) sin(x) - 87975474402816cos(y) sin(x) + 12 4 14 3 146625790671360cos(y) sin(x) - 167572332195840cos(y) sin(x) + 16 2 18 125679249146880cos(y) sin(x) - 55857444065280cos(y) sin(x) + 20 11171488813056cos(y) * 30 tan(z) + 11 2 10 3305956680sin(x) - 72731046960cos(y) sin(x) + 4 9 6 8 727310469600cos(y) sin(x) - 4363862817600cos(y) sin(x) + 8 7 10 6 17455451270400cos(y) sin(x) - 48875263557120cos(y) sin(x) + 12 5 14 4 97750527114240cos(y) sin(x) - 139643610163200cos(y) sin(x) + 16 3 18 2 139643610163200cos(y) sin(x) - 93095740108800cos(y) sin(x) + 20 22 37238296043520cos(y) sin(x) - 6770599280640cos(y) * 27 tan(z) + 12 2 11 826489170sin(x) - 19835740080cos(y) sin(x) + 4 10 6 9 218193140880cos(y) sin(x) - 1454620939200cos(y) sin(x) + 8 8 10 7 6545794226400cos(y) sin(x) - 20946541524480cos(y) sin(x) + 12 6 14 5 48875263557120cos(y) sin(x) - 83786166097920cos(y) sin(x) + 16 4 18 3 104732707622400cos(y) sin(x) - 93095740108800cos(y) sin(x) + 20 2 22 55857444065280cos(y) sin(x) - 20311797841920cos(y) sin(x) + 24 3385299640320cos(y) * 24 tan(z) + 13 2 12 169536240sin(x) - 4407942240cos(y) sin(x) + 4 11 6 10 52895306880cos(y) sin(x) - 387898917120cos(y) sin(x) + 8 9 10 8 1939494585600cos(y) sin(x) - 6982180508160cos(y) sin(x) + 12 7 14 6 18619148021760cos(y) sin(x) - 37238296043520cos(y) sin(x) + 16 5 18 4 55857444065280cos(y) sin(x) - 62063826739200cos(y) sin(x) + 20 3 22 2 49651061391360cos(y) sin(x) - 27082397122560cos(y) sin(x) + 24 26 9027465707520cos(y) sin(x) - 1388840878080cos(y) * 21 tan(z) + 14 2 13 28256040sin(x) - 791169120cos(y) sin(x) + 4 12 6 11 10285198560cos(y) sin(x) - 82281588480cos(y) sin(x) + 8 10 10 9 452548736640cos(y) sin(x) - 1810194946560cos(y) sin(x) + 12 8 14 7 5430584839680cos(y) sin(x) - 12412765347840cos(y) sin(x) + 16 6 18 5 21722339358720cos(y) sin(x) - 28963119144960cos(y) sin(x) + 20 4 22 3 28963119144960cos(y) sin(x) - 21064086650880cos(y) sin(x) + 24 2 26 10532043325440cos(y) sin(x) - 3240628715520cos(y) sin(x) + 28 462946959360cos(y) * 18 tan(z) + 15 2 14 4 13 3767472sin(x) - 113024160cos(y) sin(x) + 1582338240cos(y) sin(x) + 6 12 8 11 - 13713598080cos(y) sin(x) + 82281588480cos(y) sin(x) + 10 10 12 9 - 362038989312cos(y) sin(x) + 1206796631040cos(y) sin(x) + 14 8 16 7 - 3103191336960cos(y) sin(x) + 6206382673920cos(y) sin(x) + 18 6 20 5 - 9654373048320cos(y) sin(x) + 11585247657984cos(y) sin(x) + 22 4 24 3 - 10532043325440cos(y) sin(x) + 7021362216960cos(y) sin(x) + 26 2 28 - 3240628715520cos(y) sin(x) + 925893918720cos(y) sin(x) + 30 - 123452522496cos(y) * 15 tan(z) + 16 2 15 4 14 392445sin(x) - 12558240cos(y) sin(x) + 188373600cos(y) sin(x) + 6 13 8 12 - 1758153600cos(y) sin(x) + 11427998400cos(y) sin(x) + 10 11 12 10 - 54854392320cos(y) sin(x) + 201132771840cos(y) sin(x) + 14 9 16 8 - 574665062400cos(y) sin(x) + 1292996390400cos(y) sin(x) + 18 7 20 6 - 2298660249600cos(y) sin(x) + 3218124349440cos(y) sin(x) + 22 5 24 4 - 3510681108480cos(y) sin(x) + 2925567590400cos(y) sin(x) + 26 3 28 2 - 1800349286400cos(y) sin(x) + 771578265600cos(y) sin(x) + 30 32 - 205754204160cos(y) sin(x) + 25719275520cos(y) * 12 tan(z) + 17 2 16 4 15 30780sin(x) - 1046520cos(y) sin(x) + 16744320cos(y) sin(x) + 6 14 8 13 - 167443200cos(y) sin(x) + 1172102400cos(y) sin(x) + 10 12 12 11 - 6094932480cos(y) sin(x) + 24379729920cos(y) sin(x) + 14 10 16 9 - 76622008320cos(y) sin(x) + 191555020800cos(y) sin(x) + 18 8 20 7 - 383110041600cos(y) sin(x) + 612976066560cos(y) sin(x) + 22 6 24 5 - 780151357440cos(y) sin(x) + 780151357440cos(y) sin(x) + 26 4 28 3 - 600116428800cos(y) sin(x) + 342923673600cos(y) sin(x) + 30 2 32 - 137169469440cos(y) sin(x) + 34292367360cos(y) sin(x) + 34 - 4034396160cos(y) * 9 tan(z) + 18 2 17 4 16 1710sin(x) - 61560cos(y) sin(x) + 1046520cos(y) sin(x) + 6 15 8 14 - 11162880cos(y) sin(x) + 83721600cos(y) sin(x) + 10 13 12 12 - 468840960cos(y) sin(x) + 2031644160cos(y) sin(x) + 14 11 16 10 - 6965637120cos(y) sin(x) + 19155502080cos(y) sin(x) + 18 9 20 8 - 42567782400cos(y) sin(x) + 76622008320cos(y) sin(x) + 22 7 24 6 - 111450193920cos(y) sin(x) + 130025226240cos(y) sin(x) + 26 5 28 4 - 120023285760cos(y) sin(x) + 85730918400cos(y) sin(x) + 30 3 32 2 - 45723156480cos(y) sin(x) + 17146183680cos(y) sin(x) + 34 36 - 4034396160cos(y) sin(x) + 448266240cos(y) * 6 tan(z) + 19 2 18 4 17 60sin(x) - 2280cos(y) sin(x) + 41040cos(y) sin(x) + 6 16 8 15 - 465120cos(y) sin(x) + 3720960cos(y) sin(x) + 10 14 12 13 - 22325760cos(y) sin(x) + 104186880cos(y) sin(x) + 14 12 16 11 - 386979840cos(y) sin(x) + 1160939520cos(y) sin(x) + 18 10 20 9 - 2837852160cos(y) sin(x) + 5675704320cos(y) sin(x) + 22 8 24 7 - 9287516160cos(y) sin(x) + 12383354880cos(y) sin(x) + 26 6 28 5 - 13335920640cos(y) sin(x) + 11430789120cos(y) sin(x) + 30 4 32 3 - 7620526080cos(y) sin(x) + 3810263040cos(y) sin(x) + 34 2 36 38 - 1344798720cos(y) sin(x) + 298844160cos(y) sin(x) - 31457280cos(y) * 3 tan(z) + 20 2 19 4 18 6 17 sin(x) - 40cos(y) sin(x) + 760cos(y) sin(x) - 9120cos(y) sin(x) + 8 16 10 15 12 14 77520cos(y) sin(x) - 496128cos(y) sin(x) + 2480640cos(y) sin(x) + 14 13 16 12 - 9922560cos(y) sin(x) + 32248320cos(y) sin(x) + 18 11 20 10 - 85995520cos(y) sin(x) + 189190144cos(y) sin(x) + 22 9 24 8 - 343982080cos(y) sin(x) + 515973120cos(y) sin(x) + 26 7 28 6 - 635043840cos(y) sin(x) + 635043840cos(y) sin(x) + 30 5 32 4 - 508035072cos(y) sin(x) + 317521920cos(y) sin(x) + 34 3 36 2 - 149422080cos(y) sin(x) + 49807360cos(y) sin(x) + 38 40 - 10485760cos(y) sin(x) + 1048576cos(y) Type: Factored Expression Integer Time: 0.02 (EV) + 0.88 (OT) = 0.90 sec -- expand[(1 - c^2)^5 (1 - s^2)^5 (c^2 + s^2)^10] => c^10 s^10 when -- c^2 + s^2 = 1 [modification of a problem due to Richard Liska] expand((1 - c**2)**5 * (1 - s**2)**5 * (c**2 + s**2)**10); Type: Polynomial Integer Time: 0.25 (IN) + 0.02 (EV) + 0.02 (OT) = 0.28 sec groebner([%, c**2 + s**2 - 1]) 2 2 20 18 16 14 12 10 (50) [s + c - 1,c - 5c + 10c - 10c + 5c - c ] Type: List Polynomial Integer Time: 0.02 (IN) + 0.17 (EV) + 0.02 (OT) = 0.20 sec map(factor, %) 2 2 5 10 5 (51) [s + c - 1,(c - 1) c (c + 1) ] Type: List Factored Polynomial Integer Time: 0.03 (IN) + 0.02 (EV) + 0.02 (OT) = 0.07 sec -- => (x + y) (x - y) mod 3 factor(4*x**2 - 21*x*y + 20*y**2 :: Polynomial(PrimeField(3))) There are 20 exposed and 17 unexposed library operations named ** having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op ** to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named ** with argument type(s) Variable y Polynomial PrimeField 3 -- => 1/4 (x + y) (2 x + y [-1 + i sqrt(3)]) (2 x + y [-1 - i sqrt(3)]) factor(x**3 + y**3, [rootOf(isqrt3**2 + 3)]) - isqrt3 - 1 isqrt3 - 1 (52) (y + ------------ x)(y + x)(y + ---------- x) 2 2 Type: Factored Polynomial AlgebraicNumber Time: 0.05 (IN) + 2.33 (EV) + 0.07 (OT) + 0.43 (GC) = 2.88 sec -- Partial fraction decomposition => 3/(x + 2) - 2/(x + 1) + 2/(x + 1)^2 (x**2 + 2*x + 3)/(x**3 + 4*x**2 + 5*x + 2) 2 x + 2x + 3 (53) ----------------- 3 2 x + 4x + 5x + 2 Type: Fraction Polynomial Integer Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec fullPartialFraction( _ % :: Fraction UnivariatePolynomial(x, Fraction Integer)) 2 2 3 (54) - ----- + -------- + ----- x + 1 2 x + 2 (x + 1) Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer)) Time: 0.13 (IN) + 0.20 (EV) + 0.05 (OT) = 0.38 sec -- Noncommutative algebra: note that (A B C)^(-1) = C^(-1) B^(-1) A^(-1) -- => A B C A C B - C^(-1) B^(-1) C B A : SquareMatrix(2, Integer); Type: Void Time: 0 sec B : SquareMatrix(2, Integer); Type: Void Time: 0 sec C : SquareMatrix(2, Integer); Type: Void Time: 0 sec (A*B*C - (A*B*C)**(-1)) * A*C*B A is declared as being in SquareMatrix(2,Integer) but has not been given a value. -- Jacobi's identity: [A, B, C] + [B, C, A] + [C, A, B] = 0 where [A, B, C] = -- [A, [B, C]] and [A, B] = A B - B A is the commutator of A and B comm2(A, B) == A * B - B * A; Type: Void Time: 0 sec comm3(A, B, C) == comm2(A, comm2(B, C)); Type: Void Time: 0 sec comm2(A, B) A is declared as being in SquareMatrix(2,Integer) but has not been given a value. comm3(A, B, C) + comm3(B, C, A) + comm3(C, A, B) A is declared as being in SquareMatrix(2,Integer) but has not been given a value. )clear properties A B C comm2 comm3 -- ---------- Quit ---------- )quit real 1461.7 user 485.5 sys 0.6 \end{verbatim} \section{Trigonometry} \begin{verbatim} -- ---------- Trigonometry ---------- -- => - [(sqrt(5) + 1) sqrt(2)]/[(sqrt(5) - 1) sqrt(sqrt(5) + 5)] -- = - sqrt[1 + 2/sqrt(5)] -- From B. F. Caviness, Robert P. Gilbert, Wolfram Koepf, Roman Shtokhamer and -- David W. Wood, _An Introduction to Applied Symbolic Computation using -- MACSYMA_, University of Delaware, draft of December 14, 1993, section 2.3.3. tan(7*%pi/10) 7%pi (1) tan(----) 10 Type: Expression Integer Time: 0.48 (IN) + 0.18 (EV) + 0.14 (OT) + 0.20 (GC) = 1.0 sec -- => - cos 3 sqrt((1 + cos(6))/2) +----------+ |cos(6) + 1 (2) |---------- \| 2 Type: Expression Integer Time: 0.18 (IN) + 0.03 (EV) = 0.22 sec simplify(normalize(%)) +-------+ | 2 (3) \|cos(3) Type: Expression Integer Time: 0.10 (IN) + 0.20 (EV) + 0.02 (OT) + 0.02 (GC) = 0.33 sec -- cos(n pi) + sin((4 n - 1)/2 pi) => (-1)^n - 1 for integer n cos(n*%pi) + sin((4*n - 1)/2 * %pi) (4n - 1)%pi (4) sin(-----------) + cos(n %pi) 2 Type: Expression Integer Time: 1.15 (IN) + 0.02 (EV) + 0.17 (OT) = 1.33 sec -- cos(cos(n pi) pi) + sin(cos(n pi) pi/2) => -1 + (-1)^n for integer n cos(cos(n*%pi)*%pi) + sin(cos(n*%pi)*%pi/2) %pi cos(n %pi) (5) sin(--------------) + cos(%pi cos(n %pi)) 2 Type: Expression Integer Time: 0.12 (IN) + 0.02 (EV) + 0.07 (OT) = 0.20 sec -- sin([n^5/5 + n^4/2 + n^3/3 - n/30] pi) => 0 for integer n -- [Paul Zimmermann] sin((n**5/5 + n**4/2 + n**3/3 - n/30) * %pi) 5 4 3 (6n + 15n + 10n - n)%pi (6) sin(--------------------------) 30 Type: Expression Integer Time: 0.37 (IN) + 0.05 (OT) = 0.42 sec -- | cos x |, | sin x | => - cos x, - sin x for - 3 pi < x < - 5/2 pi --assume(-3*%pi < x, x < -5/2*%pi) [abs(cos(x)), abs(sin(x))] (7) [abs(cos(x)),abs(sin(x))] Type: List Expression Integer Time: 0.05 (IN) + 0.03 (EV) + 0.02 (OT) = 0.10 sec --forget(-3*%pi < x, x < -5/2*%pi) -- Trigonometric manipulations---these are typically difficult for students r:= cos(3*x)/cos(x) cos(3x) (8) ------- cos(x) Type: Expression Integer Time: 0.03 (IN) + 0.02 (EV) = 0.05 sec -- => cos(x)^2 - 3 sin(x)^2 or similar real(complexNormalize(r)) 2 2 (9) - 2sin(x) + 2cos(x) - 1 Type: Expression Integer Time: 0.07 (IN) + 1.45 (EV) + 0.03 (OT) + 0.02 (GC) = 1.57 sec -- => 2 cos(2 x) - 1 real(normalize(simplify(complexNormalize(r)))) (10) 2cos(2x) - 1 Type: Expression Integer Time: 0.02 (IN) + 1.10 (EV) = 1.12 sec -- Use rewrite rules => cos(x)^2 - 3 sin(x)^2 sincosAngles:= rule ( _ cos((n | integer?(n)) * x) == _ cos((n - 1)*x) * cos(x) - sin((n - 1)*x) * sin(x); _ sin((n | integer?(n)) * x) == _ sin((n - 1)*x) * cos(x) + cos((n - 1)*x) * sin(x) ) (11) {cos(n x) == - sin(x)sin((n - 1)x) + cos(x)cos((n - 1)x), sin(n x) == cos(x)sin((n - 1)x) + cos((n - 1)x)sin(x)} Type: Ruleset(Integer,Integer,Expression Integer) Time: 0.25 (IN) + 0.15 (EV) + 0.07 (OT) + 0.02 (GC) = 0.48 sec sincosAngles r 2 2 (12) - 3sin(x) + cos(x) Type: Expression Integer Time: 0.02 (IN) + 0.20 (EV) + 0.02 (OT) + 0.02 (GC) = 0.25 sec )clear properties r -- Here is a tricky way of writing 0/0 expr:= (tan(x)**2 + 1 - sec(x)**2)/(sin(x)**2 + cos(x)**2 - 1) 2 2 tan(x) - sec(x) + 1 (13) --------------------- 2 2 sin(x) + cos(x) - 1 Type: Expression Integer Time: 0.08 (IN) + 0.02 (EV) + 0.02 (OT) = 0.12 sec -- Let's try simplifying this expression! simplify(expr) 1 (14) ------- 2 cos(x) Type: Expression Integer Time: 0.07 (EV) = 0.07 sec normalize(expr) >> Error detected within library code: catdef: division by zero initial (15) -> real 8.9 user 8.2 sys 0.4 ------------------------------------------------------------------------------- Thu Apr 17 07:23:53 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Trigonometry ---------- expr:= (tan(x)**2 + 1 - sec(x)**2)/(sin(x)**2 + cos(x)**2 - 1) 2 2 tan(x) - sec(x) + 1 (1) --------------------- 2 2 sin(x) + cos(x) - 1 Type: Expression Integer Time: 0.32 (IN) + 0.40 (EV) + 0.27 (OT) + 0.20 (GC) = 1.18 sec -- Let's try simplifying this expression! complexNormalize(expr) >> Error detected within library code: catdef: division by zero initial (2) -> real 5.5 user 2.0 sys 0.3 ------------------------------------------------------------------------------- Thu Apr 17 07:25:51 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Trigonometry ---------- expr:= (tan(x)**2 + 1 - sec(x)**2)/(sin(x)**2 + cos(x)**2 - 1) 2 2 tan(x) - sec(x) + 1 (1) --------------------- 2 2 sin(x) + cos(x) - 1 Type: Expression Integer Time: 0.30 (IN) + 0.45 (EV) + 0.23 (OT) + 0.20 (GC) = 1.18 sec -- What is its limit at zero? limit(expr, x = 0) (2) 0 Type: Union(OrderedCompletion Expression Integer,...) Time: 0.35 (IN) + 8.02 (EV) + 0.05 (OT) + 0.53 (GC) = 8.95 sec -- What is the derivative? dexpr:= D(expr, x) 3 2 2tan(x) + (- 2sec(x) + 2)tan(x) (3) --------------------------------- 2 2 sin(x) + cos(x) - 1 Type: Expression Integer Time: 0.02 (IN) + 0.13 (EV) + 0.02 (OT) = 0.17 sec simplify(dexpr) 2sin(x) (4) ------- 3 cos(x) Type: Expression Integer Time: 0.03 (IN) + 0.12 (EV) + 0.03 (OT) = 0.18 sec normalize(dexpr) >> Error detected within library code: catdef: division by zero initial (5) -> real 32.0 user 10.7 sys 0.4 ------------------------------------------------------------------------------- Thu Apr 17 07:28:07 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Trigonometry ---------- expr:= (tan(x)**2 + 1 - sec(x)**2)/(sin(x)**2 + cos(x)**2 - 1) 2 2 tan(x) - sec(x) + 1 (1) --------------------- 2 2 sin(x) + cos(x) - 1 Type: Expression Integer Time: 0.37 (IN) + 0.40 (EV) + 0.22 (OT) + 0.20 (GC) = 1.18 sec -- What is the derivative? dexpr:= D(expr, x) 3 2 2tan(x) + (- 2sec(x) + 2)tan(x) (2) --------------------------------- 2 2 sin(x) + cos(x) - 1 Type: Expression Integer Time: 0.15 (EV) + 0.02 (OT) + 0.02 (GC) = 0.18 sec complexNormalize(dexpr) >> Error detected within library code: catdef: division by zero initial (3) -> real 5.6 user 2.1 sys 0.3 \end{verbatim} \section{Special Functions} \begin{verbatim} initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Special Functions ---------- -- Bernoulli numbers: B_16 => -3617/510 [Gradshteyn and Ryzhik 9.71] bernoulli(16) 3617 (1) - ---- 510 Type: Fraction Integer Time: 0.02 (IN) + 0.07 (EV) + 0.03 (OT) + 0.03 (GC) = 0.15 sec -- d/dk E(phi, k) => [E(phi, k) - F(phi, k)]/k where F(phi, k) and E(phi, k) -- are elliptic integrals of the 1st and 2nd kind, respectively -- [Gradshteyn and Ryzhik 8.123(3)] --D(E(phi, k), k) -- Jacobian elliptic functions: d/du dn u => -k^2 sn u cn u -- [Gradshteyn and Ryzhik 8.158(3)] --D(dn(u), u) -- => -2 sqrt(pi) [Gradshteyn and Ryzhik 8.338(3)] Gamma(-1/2) (2) - 3.5449077018110313 Type: DoubleFloat Time: 0.03 (IN) + 0.03 (OT) + 0.02 (GC) = 0.08 sec % + 2*sqrt(%pi) (3) 4.4408920985006262e-16 Type: Expression DoubleFloat Time: 0.68 (IN) + 0.17 (OT) + 0.15 (GC) = 1.0 sec -- psi(1/3) => - Euler's_constant - pi/2 sqrt(1/3) - 3/2 log 3 where psi(x) -- is the psi function [= d/dx log Gamma(x)] [Gradshteyn and Ryzhik 8.366(6)] digamma(1/3) (4) - 3.1320337800208065 Type: DoubleFloat Time: 0.02 (IN) = 0.02 sec % + %pi/2*sqrt(1/3) + 3/2*log(3) (5) - 0.57721566490153275 Type: Expression DoubleFloat Time: 0.52 (IN) + 0.10 (EV) + 0.05 (OT) = 0.67 sec -- Bessel function of the first kind of order 2 => 0.04158 + 0.24740 i besselJ(2, 1 + %i) (6) 0.041579886943962127 + 0.2473976415133064%i Type: Complex DoubleFloat Time: 0.12 (IN) + 0.02 (EV) + 0.05 (OT) + 0.03 (GC) = 0.22 sec -- => 12/pi^2 [Gradshteyn and Ryzhik 8.464(6)] besselJ(-5/2, %pi/2) (7) 1.2158542037080535 Type: DoubleFloat Time: 0.07 (IN) + 0.03 (OT) = 0.10 sec % - 12/%pi**2 (8) 2.2204460492503131e-16 Type: DoubleFloat Time: 0.03 (IN) = 0.03 sec -- => sqrt(2/(pi z)) (sin z/z - cos z) [Gradshteyn and Ryzhik 8.464(3)] besselJ(3/2, z) 3 (9) besselJ(-,z) 2 Type: Expression Integer Time: 0.30 (IN) + 0.02 (EV) + 0.02 (OT) = 0.33 sec -- d/dz J_0(z) => - J_1(z) [Gradshteyn and Ryzhik 8.473(4)] D(besselJ(0, z), z) - besselJ(1,z) + besselJ(- 1,z) (10) ------------------------------- 2 Type: Expression Integer Time: 0.05 (IN) + 0.05 (EV) + 0.02 (GC) = 0.12 sec -- Associated Legendre (spherical) function of the 1st kind: P^mu_nu(0) -- => 2^mu sqrt(pi) / [Gamma([nu - mu]/2 + 1) Gamma([- nu - mu + 1]/2)] -- [Gradshteyn and Ryzhik 8.756(1)] --P(mu, nu, 0) -- P^1_3(x) => -3/2 sqrt(1 - x^2) (5 x^2 - 1) -- [Gradshteyn and Ryzhik 8.813(4)] --P(1, 3, x) -- nth Chebyshev polynomial of the 1st kind: T_n(x) => 0 -- [Gradshteyn and Ryzhik 8.941(1)] chebyshevT(1008, x) - 2*x*chebyshevT(1007, x) + chebyshevT(1006, x) (11) 0 Type: Polynomial Integer Time: 0.03 (IN) + 30.50 (EV) + 11.38 (GC) = 41.91 sec -- T_n(-1) => (-1)^n [Gradshteyn and Ryzhik 8.944(2)] chebyshevT(n, -1) There are 1 exposed and 0 unexposed library operations named chebyshevT having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op chebyshevT to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named chebyshevT with argument type(s) Variable n Integer -- => arcsin z/z [Gradshteyn and Ryzhik 9.121(26)] --hypergeometric([1/2, 1/2], [3/2], z**2) -- => sin(n z)/(n sin z cos z) [Gradshteyn and Ryzhik 9.121(17)] --hypergeometric([(n + 2)/2, -(n - 2)/2], [3/2], sin(z)**2) -- zeta'(0) => - 1/2 log(2 pi) [Gradshteyn and Ryzhik 9.542(4)] --subst(D(zeta(x), x), x = 0) -- Dirac delta distribution => 3 f(4/5) + g'(1) --f:= operator('f); --g:= operator('g); --integrate(f((x + 2)/5)*delta((x - 2)/3) - g(x)*D(delta(x - 1), x), x = 0..3) --)clear properties f g -- Define an antisymmetric function f f:= operator('f); Type: BasicOperator Time: 0.07 (IN) = 0.07 sec -- Test it out => [-f(a, b, c), 0] [f(c, b, a), f(c, b, c)] (13) [f(c,b,a),f(c,b,c)] Type: List Expression Integer Time: 0.17 (IN) + 0.02 (OT) = 0.18 sec )clear properties f -- ---------- Quit ---------- )quit real 68.3 user 34.3 sys 0.4 \end{verbatim} \section{The Complex Domain} \begin{verbatim} -- ---------- The Complex Domain ---------- -- Complex functions---separate into their real and imaginary parts. -- Here, variables default to REAL. -- [Re(x + i y), Im(x + i y)] => [Re(x) - Im(y), Im(x) + Re(y)] -- for x and y complex [real(x + %i*y), imag(x + %i*y)] (1) [x,y] Type: List Expression Integer Time: 0.55 (IN) + 0.30 (EV) + 0.22 (OT) + 0.25 (GC) = 1.32 sec x : Complex Expression Integer Type: Void Time: 0 sec y : Complex Expression Integer Type: Void Time: 0 sec [real(x + %i*y), imag(x + %i*y)] (4) [x,y] Type: List Expression Integer Time: 0.52 (IN) + 0.07 (OT) = 0.58 sec )clear properties x y -- => 1 [W. Kahan] abs(3 - sqrt(7) + %i*sqrt(6*sqrt(7) - 15)) +----------+ | +-+ +-+ (5) abs(%i\|6\|7 - 15 - \|7 + 3) Type: Expression Complex Integer Time: 0.20 (IN) + 0.10 (EV) + 0.05 (OT) = 0.35 sec complexForm(%) (6) 1 Type: Complex Expression Integer Time: 0.07 (EV) = 0.07 sec -- => 1/sqrt(a^2 + (1/a + b)^2) for real a, b abs(1/(a + %i/a + %i*b)) %i a (7) abs(---------------) 2 a b - %i a + 1 Type: Expression Complex Integer Time: 0.78 (IN) + 0.02 (EV) + 0.07 (OT) = 0.87 sec complexForm(%) +--------------------+ | 2 | a (8) |-------------------- | 2 2 4 \|a b + 2a b + a + 1 Type: Complex Expression Integer Time: 0.12 (EV) + 0.02 (GC) = 0.13 sec -- => log 5 + i arctan(4/3) complexForm(log(3 + 4*%i)) log(25) 4 (9) ------- + atan(-)%i 2 3 Type: Complex Expression Integer Time: 0.03 (IN) + 0.05 (EV) = 0.08 sec -- => [sin(x) cos(x) + i sinh(y) cosh(y)] / [cos(x)^2 + sinh(y)^2] simplify(complexForm(tan(x + %i*y))) - 2y +---+ - 4y +---+ 4cos(x)%e sin(x) - \|- 1 %e + \|- 1 (10) ------------------------------------------- 2 - 2y - 4y (4cos(x) - 2)%e + %e + 1 Type: Expression Integer Time: 0.10 (IN) + 0.85 (EV) + 0.03 (OT) + 0.03 (GC) = 1.02 sec simplify(complexNormalize(%)) +---+ +---+ 2x\|- 1 - 2y +---+ - \|- 1 %e + \|- 1 (11) -------------------------------- +---+ 2x\|- 1 - 2y %e + 1 Type: Expression Integer Time: 1.60 (EV) + 0.02 (OT) = 1.62 sec simp(e) == [simplify(e), normalize(e), complexNormalize(e)] Type: Void Time: 0 sec -- Check for branch abuse. See David R. Stoutemyer, ``Crimes and Misdemeanors -- in the Computer Algebra Trade'', _Notices of the American Mathematical -- Society_, Volume 38, Number 7, September 1991, 778--785. This first -- expression can simplify to sqrt(x y)/sqrt(x), but no further in general -- (consider what happens when x, y = -1). sqrt(x y) = sqrt(x) sqrt(y) if -- either x >= 0 or y >= 0 or both x and y lie in the right-half plane -- (Re x, Re y > 0) [considering principal values]. sqrt(x*y*abs(z)**2) / (sqrt(x)*abs(z)) +-----------+ | 2 \|x y abs(z) (13) -------------- +-+ abs(z)\|x Type: Expression Integer Time: 0.55 (IN) + 0.03 (EV) + 0.02 (OT) + 0.35 (GC) = 0.95 sec simp(%) Compiling function simp with type Expression Integer -> List Expression Integer +-----------+ +-----------+ +-----------+ | 2 | 2 | 2 \|x y abs(z) \|x y abs(z) \|x y abs(z) (14) [--------------,--------------,--------------] +-+ +-+ +-+ abs(z)\|x abs(z)\|x abs(z)\|x Type: List Expression Integer Time: 0.03 (IN) + 0.13 (EV) + 0.03 (OT) = 0.20 sec x : Complex Expression Integer Type: Void Time: 0 sec y : Complex Expression Integer Type: Void Time: 0.02 (IN) = 0.02 sec z : Complex Expression Integer Type: Void Time: 0 sec sqrt(x*y*abs(z)**2) / (sqrt(x)*abs(z)) +-----------+ | 2 \|x y abs(z) (18) -------------- +-+ abs(z)\|x Type: Complex Expression Integer Time: 0.10 (IN) + 0.05 (OT) = 0.15 sec simp(%) Compiling function simp with type Complex Expression Integer -> List Expression Complex Integer +-----------+ +-----------+ +-----------+ | 2 | 2 | 2 \|x y abs(z) \|x y abs(z) \|x y abs(z) (19) [--------------,--------------,--------------] +-+ +-+ +-+ abs(z)\|x abs(z)\|x abs(z)\|x Type: List Expression Complex Integer Time: 0.32 (IN) + 0.13 (EV) + 0.07 (OT) = 0.52 sec )clear properties x y z -- Special case: sqrt(x y |z|^2)/(sqrt(x) |z|) => sqrt(y) [PV] for y >= 0 -- sqrt(1/z) = 1/sqrt(z) except when z is real and negative, in which case -- sqrt(1/z) = - 1/sqrt(z) [considering principal values] sqrt(1/z) - 1/sqrt(z) (20) 0 Type: Expression Integer Time: 0.13 (IN) = 0.13 sec z : Complex Expression Integer Type: Void Time: 0 sec sqrt(1/z) - 1/sqrt(z) (22) 0 Type: Complex Expression Integer Time: 0.02 (IN) + 0.02 (EV) + 0.02 (OT) = 0.05 sec )clear properties z -- Special case: sqrt(1/z) - 1/sqrt(z) => 0 [PV] for z > 0 -- Special case: sqrt(1/z) + 1/sqrt(z) => 0 [PV] for z < 0 -- sqrt(e^z) = e^(z/2) if and only if Im z is contained in the interval -- ((4 n - 1) pi, (4 n + 1) pi] for n an integer: ..., (-5 pi, -3 pi], -- (-pi, pi], (3 pi, 5 pi], ...; otherwise, sqrt(e^z) = - e^(z/2) [considering -- principal values] z : Complex Expression Integer Type: Void Time: 0 sec sqrt(%e**z) - %e**(z/2) z +---+ - | z 2 (24) \|%e - %e Type: Complex Expression Integer Time: 0.15 (IN) + 0.05 (EV) + 0.02 (OT) = 0.22 sec simp(%) +------+ +------+ z | z 2 z | z 2 z +---+ - | - - | - - | z 2 | 2 2 | 2 2 (25) [\|%e - %e ,\|(%e ) - %e ,\|(%e ) - %e ] Type: List Expression Complex Integer Time: 0.22 (IN) + 0.42 (EV) + 0.03 (OT) = 0.67 sec )clear properties z -- Special case: sqrt(e^z) - e^(z/2) => 0 [PV] for z real sqrt(%e**z) - %e**(z/2) z +---+ - | z 2 (26) \|%e - %e Type: Expression Integer Time: 0.52 (IN) + 0.07 (EV) + 0.07 (OT) = 0.65 sec simp(%) +------+ +------+ z | z 2 z | z 2 z +---+ - | - - | - - | z 2 | 2 2 | 2 2 (27) [\|%e - %e ,\|(%e ) - %e ,\|(%e ) - %e ] Type: List Expression Integer Time: 0.27 (EV) + 0.02 (OT) = 0.28 sec -- The principal value of this expression is - e^(3 i) = - cos 3 - i sin 3 sqrt(%e**(6*%i)) +-----+ | 6%i (28) \|%e Type: Expression Complex Integer Time: 0.15 (IN) + 0.05 (EV) + 0.02 (OT) = 0.22 sec simplify(complexForm(%)) sin(6) sin(6) atan(------) atan(------) +---+ cos(6) cos(6) (29) \|- 1 sin(------------) + cos(------------) 2 2 Type: Expression Integer Time: 0.02 (IN) + 0.15 (EV) + 0.02 (OT) = 0.18 sec % :: Complex Float (30) 0.9899924966 0044545727 - 0.1411200080 598672221 %i Type: Complex Float Time: 0.20 (IN) = 0.20 sec -- log(e^z) = z if and only if Im z is contained in the interval (-pi, pi] -- [considering principal values] z : Complex Expression Integer Type: Void Time: 0 sec log(%e**z) (32) z Type: Complex Expression Integer Time: 0.05 (EV) = 0.05 sec )clear properties z -- Special case: log(e^z) => z [PV] for z real log(%e**z) (33) z Type: Expression Integer Time: 0.02 (IN) + 0.03 (EV) = 0.05 sec -- The principal value of this expression is (10 - 4 pi) i log(%e**(10*%i)) 10%i (34) log(%e ) Type: Expression Complex Integer Time: 0.02 (IN) + 0.03 (EV) + 0.03 (OT) = 0.08 sec simplify(complexForm(%)) +---+ sin(10) (35) \|- 1 atan(-------) cos(10) Type: Expression Integer Time: 0.02 (IN) + 0.07 (EV) + 0.02 (OT) = 0.10 sec % :: Complex Float (36) 0.5752220392 3062028461 %i Type: Complex Float Time: 0.02 (IN) = 0.02 sec -- (x y)^n = x^n y^n if either x > 0 or y > 0 or both x and y lie in the -- right-half plane (Re x, Re y > 0) or n is an integer [considering principal -- values] (x*y)**(1/n) - x**(1/n)*y**(1/n) 1 1 1 - - - n n n (37) (x y) - x y Type: Expression Integer Time: 0.35 (IN) + 0.02 (EV) + 0.05 (OT) = 0.42 sec simp(%) 1 1 1 - - - n n n (38) [(x y) - x y ,0,0] Type: List Expression Integer Time: 0.62 (EV) = 0.62 sec x : Complex Expression Integer Type: Void Time: 0 sec y : Complex Expression Integer Type: Void Time: 0 sec (x*y)**(1/n) - x**(1/n)*y**(1/n) 2 2 2 2 log(x y ) log(x ) log(y ) --------- ------- ------- 2n 2n 2n (41) %e - %e %e Type: Complex Expression Integer Time: 0.27 (IN) + 0.08 (EV) + 0.03 (OT) = 0.38 sec simp(%) 2 2 2 2 log(x y ) log(y ) + log(x ) --------- ----------------- 2n 2n (42) [%e - %e ,0,0] Type: List Expression Complex Integer Time: 0.22 (IN) + 1.18 (EV) + 0.03 (OT) + 0.35 (GC) = 1.78 sec -- Special case: (x y)^(1/n) - x^(1/n) y^(1/n) => 0 [PV] for y > 0 -- Special case: (x y)^n - x^n y^n => 0 [PV] for integer n (x*y)**n - x**n*y**n n n n (43) (x y) - x y Type: Expression Integer Time: 0.18 (IN) + 0.02 (OT) = 0.20 sec simp(%) n n n (44) [(x y) - x y ,0,0] Type: List Expression Integer Time: 0.62 (EV) = 0.62 sec )clear properties x y -- arctan(tan(z)) = z for z real if and only if z is contained in the interval -- (-pi/2, pi/2] [considering principal values] atan(tan(z)) (45) z Type: Expression Integer Time: 0.02 (IN) = 0.02 sec z : Complex Expression Integer Type: Void Time: 0 sec atan(tan(z)) (47) 2cos(z)sin(z) - atan(---------------------) 2 2 sin(z) - cos(z) - 1 + 2cos(z)sin(z) atan(---------------------------------------------------) 4 2 2 4 2 sin(z) + (2cos(z) - 1)sin(z) + cos(z) + cos(z) / 2 + 4 2 2 4 4sin(z) + 8cos(z) sin(z) + 4cos(z) - log(--------------------------------------------------------) 4 2 2 4 2 sin(z) + (2cos(z) - 2)sin(z) + cos(z) + 2cos(z) + 1 + 4 log(--------------------------------------------------------) 4 2 2 4 2 sin(z) + (2cos(z) - 2)sin(z) + cos(z) + 2cos(z) + 1 / 4 * %i Type: Complex Expression Integer Time: 0.02 (IN) + 0.17 (EV) + 0.05 (OT) = 0.23 sec simplify(%) sin(z) (48) atan(------) cos(z) Type: Expression Integer Time: 0.23 (EV) + 0.02 (OT) = 0.25 sec )clear properties z -- Special case: arctan(tan(z)) => z [PV] for -pi/2 < z < pi/2 -- The principal value of this expression is 10 - 3 pi atan(tan(10)) (49) 10 Type: Expression Integer Time: 0.02 (IN) = 0.02 sec -- The principal value of this expression is 11 - 4 pi + 30 i = -1.56637 + 30 i atan(tan(11 + 30*%i)) (50) 11 + 30%i Type: Expression Complex Integer Time: 0.03 (OT) = 0.03 sec atan(tan(11.0 + 30.0*%i)) >> Error detected within library code: log 0 generated initial (51) -> real 47.6 user 15.8 sys 0.4 ------------------------------------------------------------------------------- Wed Jan 28 05:31:43 MET 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- The Complex Domain ---------- -- This is a challenge problem proposed by W. Kahan: simplify the following -- expression for complex z. Expanding out the expression produces -- (z^2 + 1)/(2 z) +- (z + 1)*(z - 1)/(2 z) => z or 1/z in each of its branches z : Complex Expression Integer Type: Void Time: 0 sec w:= (z + 1/z)/2 2 z + 1 (2) ------ 2z Type: Complex Expression Integer Time: 0.33 (IN) + 0.08 (EV) + 0.22 (OT) + 0.15 (GC) = 0.78 sec expr:= w + sqrt(w + 1)*sqrt(w - 1) +-----------+ +-----------+ | 2 | 2 |z - 2z + 1 |z + 2z + 1 2 2z |----------- |----------- + z + 1 \| 2z \| 2z (3) --------------------------------------- 2z Type: Complex Expression Integer Time: 0.03 (IN) + 0.05 (EV) + 0.05 (OT) + 0.02 (GC) = 0.15 sec )clear properties z w expr -- ---------- Quit ---------- )quit real 3.2 user 1.4 sys 0.2 \end{verbatim} \section{Determining Zero Equivalence} \begin{verbatim} -- ---------- Determining Zero Equivalence ---------- -- The following expressions are all equal to zero sqrt(997) - (997**3)**(1/6) (1) 0 Type: AlgebraicNumber Time: 0.08 (IN) + 0.27 (EV) + 0.17 (OT) + 0.10 (GC) = 0.62 sec sqrt(999983) - (999983**3)**(1/6) (2) 0 Type: AlgebraicNumber Time: 0.03 (IN) + 0.08 (EV) + 0.02 (OT) = 0.13 sec (2**(1/3) + 4**(1/3))**3 - 6*(2**(1/3) + 4**(1/3)) - 6 3+-+3+-+2 3+-+2 3+-+ 3+-+ (3) 3\|2 \|4 + (3\|2 - 6)\|4 - 6\|2 Type: AlgebraicNumber Time: 0.07 (IN) + 0.10 (EV) + 0.05 (OT) = 0.22 sec expand(%) (4) 0 Type: AlgebraicNumber Time: 0.50 (IN) + 0.02 (OT) + 0.02 (GC) = 0.53 sec cos(x)**3 + cos(x)*sin(x)**2 - cos(x) 2 3 (5) cos(x)sin(x) + cos(x) - cos(x) Type: Expression Integer Time: 0.15 (IN) + 0.08 (EV) + 0.05 (OT) = 0.28 sec simplify(%) (6) 0 Type: Expression Integer Time: 0.02 (IN) + 0.07 (EV) + 0.03 (OT) = 0.12 sec -- See Joel Moses, ``Algebraic Simplification: A Guide for the Perplexed'', -- _Communications of the Association of Computing Machinery_, Volume 14, -- Number 8, August 1971, 527--537. This expression is zero if Re(x) is -- contained in the interval ((4 n - 1)/2 pi, (4 n + 1)/2 pi) for n an integer: -- ..., (-5/2 pi, -3/2 pi), (-pi/2, pi/2), (3/2 pi, 5/2 pi), ... expr:= log(tan(1/2*x + %pi/4)) - asinh(tan(x)) 2x + %pi (7) log(tan(--------)) - asinh(tan(x)) 4 Type: Expression Integer Time: 0.83 (IN) + 0.05 (EV) + 0.12 (OT) = 1.0 sec complexNormalize(expr) (8) - log +---+ 4 (2x + %pi)\|- 1 ---------------- 4 ((%e ) - 1) * +----------------------------------------------------+ | +---+ 4 | (2x + %pi)\|- 1 | ---------------- | 4 | 4(%e ) |- -------------------------------------------------- | +---+ 8 +---+ 4 | (2x + %pi)\|- 1 (2x + %pi)\|- 1 | ---------------- ---------------- | 4 4 \| (%e ) - 2(%e ) + 1 + +---+ 4 (2x + %pi)\|- 1 ---------------- +---+ 4 +---+ - \|- 1 (%e ) - \|- 1 / +---+ 4 (2x + %pi)\|- 1 ---------------- 4 (%e ) - 1 + +---+ 2 (2x + %pi)\|- 1 ---------------- +---+ 4 +---+ - \|- 1 (%e ) + \|- 1 log(--------------------------------------) +---+ 2 (2x + %pi)\|- 1 ---------------- 4 (%e ) + 1 Type: Expression Integer Time: 0.05 (IN) + 1.85 (EV) + 0.10 (OT) + 0.05 (GC) = 2.05 sec -- Use a roundabout method---show that expr is a constant equal to zero D(expr, x) (9) +-----------+ 2x + %pi 2 | 2 2x + %pi 2 2x + %pi (tan(--------) + 1)\|tan(x) + 1 - 2tan(--------)tan(x) - 2tan(--------) 4 4 4 --------------------------------------------------------------------------- +-----------+ 2x + %pi | 2 2tan(--------)\|tan(x) + 1 4 Type: Expression Integer Time: 0.05 (EV) + 0.02 (OT) = 0.07 sec simplify(real(complexNormalize(expand(simplify(%))))) +-----------------------+ | x 4 x 2 x 2 - |4cos(-) - 4cos(-) + 1 + 2cos(-) - 1 \| 2 2 2 (10) ------------------------------------------- x 4 x 2 4cos(-) - 4cos(-) + 1 2 2 Type: Expression Integer Time: 0.08 (IN) + 3.50 (EV) + 0.03 (OT) + 0.52 (GC) = 4.13 sec normalize(eval(expr, x = 0)) (11) 0 Type: Expression Integer Time: 0.83 (IN) + 0.02 (EV) + 0.17 (OT) = 1.02 sec )clear properties expr log((2*sqrt(r) + 1)/sqrt(4*r + 4*sqrt(r) + 1)) +-+ 2\|r + 1 (12) log(-----------------) +--------------+ | +-+ \|4\|r + 4r + 1 Type: Expression Integer Time: 0.10 (IN) + 0.02 (EV) + 0.03 (OT) = 0.15 sec simplify(%) +-+ 2\|r + 1 (13) log(-----------------) +--------------+ | +-+ \|4\|r + 4r + 1 Type: Expression Integer Time: 0.03 (EV) + 0.02 (OT) = 0.05 sec (4*r + 4*sqrt(r) + 1)**(sqrt(r)/(2*sqrt(r) + 1)) _ * (2*sqrt(r) + 1)**(1/(2*sqrt(r) + 1)) - 2*sqrt(r) - 1 +-+ 1 \|r --------- --------- +-+ +-+ +-+ 2\|r + 1 +-+ 2\|r + 1 +-+ (14) (2\|r + 1) (4\|r + 4r + 1) - 2\|r - 1 Type: Expression Integer Time: 0.22 (IN) + 0.05 (EV) + 0.02 (OT) = 0.28 sec normalize(%) (15) 0 Type: Expression Integer Time: 1.02 (EV) = 1.02 sec -- [Gradshteyn and Ryzhik 9.535(3)] --2**(1 - z)*Gamma(z)*zeta(z)*cos(z*%pi/2) - %pi^z*zeta(1 - z) -- ---------- Quit ---------- )quit real 31.5 user 12.1 sys 0.6 \end{verbatim} \section{Equations} \begin{verbatim} -- ---------- Equations ---------- -- Manipulate an equation using a natural syntax: -- (x = 2)/2 + (1 = 1) => x/2 + 1 = 2 (x = 2)/2 + (1 = 1) x + 2 (1) -----= 2 2 Type: Equation Fraction Polynomial Integer Time: 0.55 (IN) + 0.08 (EV) + 0.20 (OT) + 0.08 (GC) = 0.92 sec -- Solve various nonlinear equations---this cubic polynomial has all real roots radicalSolve(3*x**3 - 18*x**2 + 33*x - 19 = 0, x) (2) +----------+2 +----------+ | +---+ | +---+ +---+ |\|- 3 + 1 +---+ |\|- 3 + 1 (- 3\|- 3 + 3) |---------- + (6\|- 3 + 6) |---------- - 2 3| +---+ 3| +---+ \| 6\|- 3 \| 6\|- 3 [x= --------------------------------------------------------------, +----------+ | +---+ +---+ |\|- 3 + 1 (3\|- 3 + 3) |---------- 3| +---+ \| 6\|- 3 +----------+2 +----------+ | +---+ | +---+ +---+ |\|- 3 + 1 +---+ |\|- 3 + 1 (- 3\|- 3 - 3) |---------- + (6\|- 3 - 6) |---------- + 2 3| +---+ 3| +---+ \| 6\|- 3 \| 6\|- 3 x= --------------------------------------------------------------, +----------+ | +---+ +---+ |\|- 3 + 1 (3\|- 3 - 3) |---------- 3| +---+ \| 6\|- 3 +----------+2 +----------+ | +---+ | +---+ |\|- 3 + 1 |\|- 3 + 1 3 |---------- + 6 |---------- + 1 3| +---+ 3| +---+ \| 6\|- 3 \| 6\|- 3 x= ------------------------------------] +----------+ | +---+ |\|- 3 + 1 3 |---------- 3| +---+ \| 6\|- 3 Type: List Equation Expression Integer Time: 0.10 (IN) + 0.55 (EV) + 0.15 (OT) + 0.08 (GC) = 0.88 sec map(e +-> lhs(e) = simplify(complexForm(rhs(e))), %) (3) +-+ %pi %pi +-+ +-+ %pi %pi +-+ - \|3 sin(---) - cos(---) + 2\|3 \|3 sin(---) - cos(---) + 2\|3 18 18 18 18 [x= ---------------------------------, x= -------------------------------, +-+ +-+ \|3 \|3 %pi +-+ 2cos(---) + 2\|3 18 x= -----------------] +-+ \|3 Type: List Equation Expression Integer Time: 0.25 (IN) + 1.62 (EV) + 0.12 (OT) + 0.02 (GC) = 2.0 sec -- Some simple seeming problems can have messy answers: -- x = { [sqrt(5) - 1]/4 +/- 5^(1/4) sqrt(sqrt(5) + 1)/[2 sqrt(2)] i, -- - [sqrt(5) + 1]/4 +/- 5^(1/4) sqrt(sqrt(5) - 1)/[2 sqrt(2)] i} eqn:= x**4 + x**3 + x**2 + x + 1 = 0 4 3 2 (4) x + x + x + x + 1= 0 Type: Equation Polynomial Integer Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec radicalSolve(eqn, x) (5) [ x = - 2 * ROOT +-------------+2 +-------------+ | +---+ | +---+ | | 5 | | 5 |45 |- - - 25 |45 |- - - 25 | \| 3 | \| 3 (- 4 3|------------- - 10 3|------------- - 40) \| 2 \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 - 15 3|------------- \| 2 / +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 12 3|------------- \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 2 |------------------------------------------- - 1 | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 / 4 , x = 2 * ROOT +-------------+2 +-------------+ | +---+ | +---+ | | 5 | | 5 |45 |- - - 25 |45 |- - - 25 | \| 3 | \| 3 (- 4 3|------------- - 10 3|------------- - 40) \| 2 \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 - 15 3|------------- \| 2 / +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 12 3|------------- \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 2 |------------------------------------------- - 1 | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 / 4 , x = - 2 * ROOT +-------------+2 +-------------+ | +---+ | +---+ | | 5 | | 5 |45 |- - - 25 |45 |- - - 25 | \| 3 | \| 3 (- 4 3|------------- - 10 3|------------- - 40) \| 2 \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 15 3|------------- \| 2 / +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 12 3|------------- \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 - 2 |------------------------------------------- - 1 | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 / 4 , x = 2 * ROOT +-------------+2 +-------------+ | +---+ | +---+ | | 5 | | 5 |45 |- - - 25 |45 |- - - 25 | \| 3 | \| 3 (- 4 3|------------- - 10 3|------------- - 40) \| 2 \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 15 3|------------- \| 2 / +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 12 3|------------- \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 - 2 |------------------------------------------- - 1 | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 / 4 ] Type: List Equation Expression Integer Time: 0.05 (EV) + 0.52 (OT) = 0.57 sec -- Check one of the answers eval(eqn, %.1) (6) 0= 0 Type: Equation Expression Integer Time: 0.47 (IN) + 0.58 (EV) + 0.07 (OT) = 1.12 sec )clear properties eqn -- x = {2^(1/3) +- sqrt(3), +- sqrt(3) - 1/2^(2/3) +- i sqrt(3)/2^(2/3)} -- [Mohamed Omar Rayes] solve(x**6 - 9*x**4 - 4*x**3 + 27*x**2 - 36*x - 23 = 0, x) 6 4 3 2 (7) [x - 9x - 4x + 27x - 36x - 23= 0] Type: List Equation Fraction Polynomial Integer Time: 0.23 (IN) + 0.08 (EV) + 0.13 (OT) = 0.45 sec -- x = {1, e^(+- 2 pi i/7), e^(+- 4 pi i/7), e^(+- 6 pi i/7)} solve(x**7 - 1 = 0, x) 6 5 4 3 2 (8) [x= 1,x + x + x + x + x + x + 1= 0] Type: List Equation Fraction Polynomial Integer Time: 0.05 (EV) = 0.05 sec -- x = 1 +- sqrt(+-sqrt(+-4 sqrt(3) - 3) - 3)/sqrt(2) [Richard Liska] solve(x**8 - 8*x**7 + 34*x**6 - 92*x**5 + 175*x**4 - 236*x**3 + 226*x**2 _ - 140*x + 46 = 0, x) 8 7 6 5 4 3 2 (9) [x - 8x + 34x - 92x + 175x - 236x + 226x - 140x + 46= 0] Type: List Equation Fraction Polynomial Integer Time: 0.05 (IN) + 0.02 (EV) + 0.05 (OT) = 0.12 sec -- The following equations have an infinite number of solutions (let n be an -- arbitrary integer): -- x = {log(sqrt(z) - 1), log(sqrt(z) + 1) + i pi} [+ n 2 pi i, + n 2 pi i] %e**(2*x) + 2*%e**x + 1 = z 2x x (10) %e + 2%e + 1= z Type: Equation Expression Integer Time: 0.20 (IN) + 0.07 (EV) + 0.05 (OT) = 0.32 sec solve(%, x) +-+ +-+ (11) [x= log(\|z - 1),x= log(- \|z - 1)] Type: List Equation Expression Integer Time: 0.32 (IN) + 0.85 (EV) + 0.12 (OT) + 0.38 (GC) = 1.67 sec -- x = (1 +- sqrt(9 - 8 n pi i))/2. Real solutions correspond to n = 0 => -- x = {-1, 2} solve(exp(2 - x**2) = exp(-x), x) (12) [] Type: List Equation Expression Integer Time: 0.17 (EV) + 0.02 (OT) = 0.18 sec -- x = -W[n](-1) [e.g., -W[0](-1) = 0.31813 - 1.33724 i] where W[n](x) is the -- nth branch of Lambert's W function solve(exp(x) = x, x) (13) [] Type: List Equation Expression Integer Time: 0.07 (IN) + 0.07 (EV) + 0.02 (OT) = 0.15 sec -- x = {-1, 1} solve(x**x = x, x) (14) [] Type: List Equation Expression Integer Time: 0.03 (IN) + 0.08 (EV) + 0.02 (OT) = 0.13 sec -- This equation is already factored and so *should* be easy to solve: -- x = {-1, 2*{+-arcsinh(1) i + n pi}, 3*{pi/6 + n pi/3}} (x + 1) * (sin(x)**2 + 1)**2 * cos(3*x)**3 = 0 3 4 3 2 3 (15) (x + 1)cos(3x) sin(x) + (2x + 2)cos(3x) sin(x) + (x + 1)cos(3x) = 0 Type: Equation Expression Integer Time: 0.05 (IN) + 0.03 (EV) + 0.03 (OT) = 0.12 sec solve(%, x) (16) [] Type: List Equation Expression Integer Time: 2.88 (EV) + 0.02 (GC) = 2.90 sec -- x = pi/4 [+ n pi] solve(sin(x) = cos(x), x) %pi (17) [x= ---] 4 Type: List Equation Expression Integer Time: 0.02 (IN) + 0.15 (EV) = 0.17 sec solve(tan(x) = 1, x) %pi (18) [x= ---] 4 Type: List Equation Expression Integer Time: 0.03 (EV) = 0.03 sec -- x = {pi/6, 5 pi/6} [ + n 2 pi, + n 2 pi ] solve(sin(x) = 1/2, x) 1 (19) [x= asin(-)] 2 Type: List Equation Expression Integer Time: 0.08 (IN) + 0.02 (EV) + 0.02 (OT) = 0.12 sec map(e +-> lhs(e) = normalize(rhs(e)), %) %pi (20) [x= ---] 6 Type: List Equation Expression Integer Time: 0.05 (IN) + 0.02 (EV) = 0.07 sec -- x = {0, 0} [+ n pi, + n 2 pi] solve(sin(x) = tan(x), x) (21) [x= 0] Type: List Equation Expression Integer Time: 0.15 (EV) = 0.15 sec -- x = {0, 0, 0} solve(asin(x) = atan(x), x) (22) [x= 0] Type: List Equation Expression Integer Time: 0.68 (EV) + 0.02 (OT) = 0.70 sec -- x = sqrt[(sqrt(5) - 1)/2] solve(acos(x) = atan(x), x) (23) +-----------+ +-----------+ +---------+ +---------+ | +-+ | +-+ | +-+ | +-+ \|- 2\|5 - 2 \|- 2\|5 - 2 \|2\|5 - 2 \|2\|5 - 2 [x= - --------------,x= --------------,x= - ------------,x= ------------] 2 2 2 2 Type: List Equation Expression Integer Time: 0.02 (IN) + 1.08 (EV) + 0.03 (OT) + 0.38 (GC) = 1.52 sec -- x = 2 solve((x - 2)/x**(1/3) = 0, x) (24) [x= 2] Type: List Equation Expression Integer Time: 0.10 (IN) + 0.03 (EV) + 0.03 (OT) = 0.17 sec -- This equation has no solutions solve(sqrt(x**2 + 1) = x - 2, x) 3 (25) [x= -] 4 Type: List Equation Expression Integer Time: 0.05 (IN) + 0.05 (EV) = 0.10 sec -- x = 1 solve(x + sqrt(x) = 2, x) (26) [x= 4,x= 1] Type: List Equation Expression Integer Time: 0.03 (IN) + 0.03 (EV) = 0.07 sec -- x = 1/16 solve(2*sqrt(x) + 3*x**(1/4) - 2 = 0, x) +---+ +---+ 1 3\|- 7 - 31 - 3\|- 7 - 31 (27) [x= 16,x= --,x= ------------,x= --------------] 16 32 32 Type: List Equation Expression Integer Time: 0.03 (IN) + 0.18 (EV) + 0.03 (OT) = 0.25 sec -- x = {sqrt[(sqrt(5) - 1)/2], -i sqrt[(sqrt(5) + 1)/2]} solve(x = 1/sqrt(1 + x**2), x) (28) +---------+ +---------+ +-----------+ +-----------+ | +-+ | +-+ | +-+ | +-+ \|2\|5 - 2 \|2\|5 - 2 \|- 2\|5 - 2 \|- 2\|5 - 2 [x= ------------,x= - ------------,x= --------------,x= - --------------] 2 2 2 2 Type: List Equation Expression Integer Time: 0.07 (IN) + 0.17 (EV) + 0.03 (OT) = 0.27 sec -- This problem is from a computational biology talk => 1 - log_2[m (m - 1)] solve(binomial(m, 2)*2**k = 1, k) 2 log(------) 2 m - m (29) [k= -----------] log(2) Type: List Equation Expression Integer Time: 0.13 (IN) + 0.12 (EV) = 0.25 sec -- x = log(c/a) / log(b/d) for a, b, c, d != 0 and b, d != 1 [Bill Pletsch] solve(a*b**x = c*d**x, x) (30) [] Type: List Equation Expression Integer Time: 0.15 (IN) + 0.18 (EV) + 0.02 (OT) = 0.35 sec -- x = {1, e^4} solve(sqrt(log(x)) = log(sqrt(x)), x) (31) [x= 0,x= 1] Type: List Equation Expression Integer Time: 0.02 (IN) + 0.10 (EV) + 0.03 (OT) = 0.15 sec -- Recursive use of inverses, including multiple branches of rational -- fractional powers [Richard Liska] -- => x = +-(b + sin(1 + cos(1/e^2)))^(3/2) solve(log(acos(asin(x**(2/3) - b) - 1)) + 2 = 0, x) (32) +---------------------+ 1 | 1 [x= (- sin(cos(---) + 1) - b) |sin(cos(---) + 1) + b , 2 | 2 %e \| %e +---------------------+ 1 | 1 x= (sin(cos(---) + 1) + b) |sin(cos(---) + 1) + b ] 2 | 2 %e \| %e Type: List Equation Expression Integer Time: 0.05 (IN) + 0.25 (EV) + 0.03 (OT) + 0.02 (GC) = 0.35 sec -- x = {-0.784966, -0.016291, 0.802557} From Metha Kamminga-van Hulsen, -- ``Hoisting the Sails and Casting Off with Maple'', _Computer Algebra -- Nederland Nieuwsbrief_, Number 13, December 1994, ISSN 1380-1260, 27--40. eqn:= 5*x + exp((x - 5)/2) = 8*x**3 x - 5 ----- 2 3 (33) %e + 5x= 8x Type: Equation Expression Integer Time: 0.50 (IN) + 0.02 (EV) + 0.02 (OT) = 0.53 sec solve(eqn, x) (34) [] Type: List Equation Expression Integer Time: 0.08 (EV) = 0.08 sec --root_by_bisection(eqn, x, -1, -0.5) --root_by_bisection(eqn, x, -0.5, 0.5) --root_by_bisection(eqn, x, 0.5, 1) )clear properties eqn -- x = {-1, 3} solve(abs(x - 1) = 2, x) (35) [] Type: List Equation Expression Integer Time: 0.03 (IN) + 0.03 (EV) = 0.07 sec -- x = {-1, -7} solve(abs(2*x + 5) = abs(x - 2), x) (36) [] Type: List Equation Expression Integer Time: 0.02 (IN) + 0.05 (EV) + 0.02 (OT) = 0.08 sec -- x = +-3/2 solve(1 - abs(x) = max(-x - 2, x - 2), x) (37) [] Type: List Equation Expression Integer Time: 0.03 (IN) + 0.03 (EV) + 0.02 (OT) = 0.08 sec -- x = {-1, 3} solve(max(2 - x**2, x) = max(-x, x**3/9), x) (38) [x= 3,x= 0,x= - 3] Type: List Equation Fraction Polynomial Integer Time: 0.58 (IN) + 0.02 (EV) + 0.12 (OT) = 0.72 sec -- x = {+-3, -3 [1 + sqrt(3) sin t + cos t]} = {+-3, -1.554894} -- where t = (arctan[sqrt(5)/2] - pi)/3. The third answer is the root of -- x^3 + 9 x^2 - 18 = 0 in the interval (-2, -1). solve(max(2 - x**2, x) = x**3/9, x) (39) [x= 3,x= 0,x= - 3] Type: List Equation Fraction Polynomial Integer Time: 0.08 (IN) + 0.03 (OT) = 0.12 sec -- z = 2 + 3 i z : Complex Expression Integer Type: Void Time: 0 sec (1 + %i)*z + (2 - %i)*conjugate(z) = -3*%i (41) 3z= - 3%i Type: Equation Complex Expression Integer Time: 0.47 (IN) + 0.02 (EV) + 0.05 (OT) = 0.53 sec )clear properties z (1 + %i)*(x + %i*y) + (2 - %i)*conjugate(x + %i*y) = -3*%i There are 4 exposed and 1 unexposed library operations named conjugate having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op conjugate to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named conjugate with argument type(s) Polynomial Complex Integer (1 + %i)*(x + %i*y) + (2 - %i)*(x - %i*y) = -3*%i (42) (- 2 - %i)y + 3x= - 3%i Type: Equation Polynomial Complex Integer Time: 0.07 (IN) + 0.03 (OT) = 0.10 sec solve(%, [x, y]) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial Complex Integer List OrderedVariableList [x,y] -- => {f^(-1)(1), f^(-1)(-2)} assuming f is invertible f:= operator('f); Type: BasicOperator Time: 0.05 (IN) = 0.05 sec solve(f(x)**2 + f(x) - 2 = 0, x) (44) [] Type: List Equation Expression Integer Time: 0.32 (IN) + 0.02 (EV) + 0.07 (OT) = 0.40 sec )clear properties f -- Solve a 3 x 3 system of linear equations eqn1:= x + y + z = 6 (45) z + y + x= 6 Type: Equation Polynomial Integer Time: 0.08 (IN) = 0.08 sec eqn2:= 2*x + y + 2*z = 10 (46) 2z + y + 2x= 10 Type: Equation Polynomial Integer Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec eqn3:= x + 3*y + z = 10 (47) z + 3y + x= 10 Type: Equation Polynomial Integer Time: 0.02 (OT) = 0.02 sec -- Note that the solution is parametric: x = 4 - z, y = 2 solve([eqn1, eqn2, eqn3], [x, y, z]) (48) @[[x= - %DW + 4,y= 2,z= %DW]] Type: List List Equation Fraction Polynomial Integer Time: 0.68 (IN) + 0.05 (EV) + 0.08 (OT) = 0.82 sec -- A linear system arising from the computation of a truncated power series -- solution to a differential equation. There are 189 equations to be solved -- for 49 unknowns. 42 of the equations are repeats of other equations; many -- others are trivial. Solving this system directly by Gaussian elimination -- is *not* a good idea. Solving the easy equations first is probably a better -- method. The solution is actually rather simple. [Stanly Steinberg] -- => k1 = ... = k22 = k24 = k25 = k27 = ... = k30 = k32 = k33 = k35 = ... -- = k38 = k40 = k41 = k44 = ... = k49 = 0, k23 = k31 = k39, -- k34 = b/a k26, k42 = c/a k26, {k23, k26, k43} are arbitrary eqns:= [ -b*k8/a+c*k8/a = 0, -b*k11/a+c*k11/a = 0, -b*k10/a+c*k10/a+k2 = 0, _ -k3-b*k9/a+c*k9/a = 0, -b*k14/a+c*k14/a = 0, -b*k15/a+c*k15/a = 0, _ -b*k18/a+c*k18/a-k2 = 0, -b*k17/a+c*k17/a = 0, -b*k16/a+c*k16/a+k4 = 0, _ -b*k13/a+c*k13/a-b*k21/a+c*k21/a+b*k5/a-c*k5/a = 0, b*k44/a-c*k44/a = 0, _ -b*k45/a+c*k45/a = 0, -b*k20/a+c*k20/a = 0, -b*k44/a+c*k44/a = 0, _ b*k46/a-c*k46/a = 0, b**2*k47/a**2-2*b*c*k47/a**2+c**2*k47/a**2 = 0, k3 = 0, _ -k4 = 0, -b*k12/a+c*k12/a-a*k6/b+c*k6/b = 0, _ -b*k19/a+c*k19/a+a*k7/c-b*k7/c = 0, b*k45/a-c*k45/a = 0, _ -b*k46/a+c*k46/a = 0, -k48+c*k48/a+c*k48/b-c**2*k48/(a*b) = 0, _ -k49+b*k49/a+b*k49/c-b**2*k49/(a*c) = 0, a*k1/b-c*k1/b = 0, _ a*k4/b-c*k4/b = 0, a*k3/b-c*k3/b+k9 = 0, -k10+a*k2/b-c*k2/b = 0, _ a*k7/b-c*k7/b = 0, -k9 = 0, k11 = 0, b*k12/a-c*k12/a+a*k6/b-c*k6/b = 0, _ a*k15/b-c*k15/b = 0, k10+a*k18/b-c*k18/b = 0, -k11+a*k17/b-c*k17/b = 0, _ a*k16/b-c*k16/b = 0, -a*k13/b+c*k13/b+a*k21/b-c*k21/b+a*k5/b-c*k5/b = 0, _ -a*k44/b+c*k44/b = 0, a*k45/b-c*k45/b = 0, _ a*k14/c-b*k14/c+a*k20/b-c*k20/b = 0, a*k44/b-c*k44/b = 0, _ -a*k46/b+c*k46/b = 0, -k47+c*k47/a+c*k47/b-c**2*k47/(a*b) = 0, _ a*k19/b-c*k19/b = 0, -a*k45/b+c*k45/b = 0, a*k46/b-c*k46/b = 0, _ a**2*k48/b**2-2*a*c*k48/b**2+c**2*k48/b**2 = 0, _ -k49+a*k49/b+a*k49/c-a**2*k49/(b*c) = 0, k16 = 0, -k17 = 0, _ -a*k1/c+b*k1/c = 0, -k16-a*k4/c+b*k4/c = 0, -a*k3/c+b*k3/c = 0, _ k18-a*k2/c+b*k2/c = 0, b*k19/a-c*k19/a-a*k7/c+b*k7/c = 0, _ -a*k6/c+b*k6/c = 0, -a*k8/c+b*k8/c = 0, -a*k11/c+b*k11/c+k17 = 0, _ -a*k10/c+b*k10/c-k18 = 0, -a*k9/c+b*k9/c = 0, _ -a*k14/c+b*k14/c-a*k20/b+c*k20/b = 0, _ -a*k13/c+b*k13/c+a*k21/c-b*k21/c-a*k5/c+b*k5/c = 0, a*k44/c-b*k44/c = 0, _ -a*k45/c+b*k45/c = 0, -a*k44/c+b*k44/c = 0, a*k46/c-b*k46/c = 0, _ -k47+b*k47/a+b*k47/c-b**2*k47/(a*c) = 0, -a*k12/c+b*k12/c = 0, _ a*k45/c-b*k45/c = 0, -a*k46/c+b*k46/c = 0, _ -k48+a*k48/b+a*k48/c-a**2*k48/(b*c) = 0, _ a**2*k49/c**2-2*a*b*k49/c**2+b**2*k49/c**2 = 0, k8 = 0, k11 = 0, -k15 = 0, _ k10-k18 = 0, -k17 = 0, k9 = 0, -k16 = 0, -k29 = 0, k14-k32 = 0, _ -k21+k23-k31 = 0, -k24-k30 = 0, -k35 = 0, k44 = 0, -k45 = 0, k36 = 0, _ k13-k23+k39 = 0, -k20+k38 = 0, k25+k37 = 0, b*k26/a-c*k26/a-k34+k42 = 0, _ -2*k44 = 0, k45 = 0, k46 = 0, b*k47/a-c*k47/a = 0, k41 = 0, k44 = 0, _ -k46 = 0, -b*k47/a+c*k47/a = 0, k12+k24 = 0, -k19-k25 = 0, _ -a*k27/b+c*k27/b-k33 = 0, k45 = 0, -k46 = 0, -a*k48/b+c*k48/b = 0, _ a*k28/c-b*k28/c+k40 = 0, -k45 = 0, k46 = 0, a*k48/b-c*k48/b = 0, _ a*k49/c-b*k49/c = 0, -a*k49/c+b*k49/c = 0, -k1 = 0, -k4 = 0, -k3 = 0, _ k15 = 0, k18-k2 = 0, k17 = 0, k16 = 0, k22 = 0, k25-k7 = 0, _ k24+k30 = 0, k21+k23-k31 = 0, k28 = 0, -k44 = 0, k45 = 0, -k30-k6 = 0, _ k20+k32 = 0, k27+b*k33/a-c*k33/a = 0, k44 = 0, -k46 = 0, _ -b*k47/a+c*k47/a = 0, -k36 = 0, k31-k39-k5 = 0, -k32-k38 = 0, _ k19-k37 = 0, k26-a*k34/b+c*k34/b-k42 = 0, k44 = 0, -2*k45 = 0, k46 = 0, _ a*k48/b-c*k48/b = 0, a*k35/c-b*k35/c-k41 = 0, -k44 = 0, k46 = 0, _ b*k47/a-c*k47/a = 0, -a*k49/c+b*k49/c = 0, -k40 = 0, k45 = 0, -k46 = 0, _ -a*k48/b+c*k48/b = 0, a*k49/c-b*k49/c = 0, k1 = 0, k4 = 0, k3 = 0, _ -k8 = 0, -k11 = 0, -k10+k2 = 0, -k9 = 0, k37+k7 = 0, -k14-k38 = 0, _ -k22 = 0, -k25-k37 = 0, -k24+k6 = 0, -k13-k23+k39 = 0, _ -k28+b*k40/a-c*k40/a = 0, k44 = 0, -k45 = 0, -k27 = 0, -k44 = 0, _ k46 = 0, b*k47/a-c*k47/a = 0, k29 = 0, k32+k38 = 0, k31-k39+k5 = 0, _ -k12+k30 = 0, k35-a*k41/b+c*k41/b = 0, -k44 = 0, k45 = 0, _ -k26+k34+a*k42/c-b*k42/c = 0, k44 = 0, k45 = 0, -2*k46 = 0, _ -b*k47/a+c*k47/a = 0, -a*k48/b+c*k48/b = 0, a*k49/c-b*k49/c = 0, k33 = 0, _ -k45 = 0, k46 = 0, a*k48/b-c*k48/b = 0, -a*k49/c+b*k49/c = 0 _ ]; Type: List Equation Fraction Polynomial Integer Time: 6.35 (IN) + 0.72 (EV) + 1.33 (OT) + 0.87 (GC) = 9.26 sec vars:= [k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11, k12, k13, k14, k15, k16, _ k17, k18, k19, k20, k21, k22, k23, k24, k25, k26, k27, k28, k29, k30, _ k31, k32, k33, k34, k35, k36, k37, k38, k39, k40, k41, k42, k43, k44, _ k45, k46, k47, k48, k49]; Type: List OrderedVariableList [k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12,k13,k14,k15,k16,k17,k18,k19,k20,k21,k22,k23,k24,k25,k26,k27,k28,k29,k30,k31,k32,k33,k34,k35,k36,k37,k38,k39,k40,k41,k42,k43,k44,k45,k46,k47,k48,k49] Time: 0.05 (IN) = 0.05 sec solve(eqns, vars) (51) [ [k1= 0, k2= 0, k3= 0, k4= 0, k5= 0, k6= 0, k7= 0, k8= 0, k9= 0, k10= 0, k11= 0, k12= 0, k13= 0, k14= 0, k15= 0, k16= 0, k17= 0, k18= 0, k19= 0, %DY a k20= 0, k21= 0, k22= 0, k23= %DX, k24= 0, k25= 0, k26= -----, k27= 0, c %DY b k28= 0, k29= 0, k30= 0, k31= %DX, k32= 0, k33= 0, k34= -----, k35= 0, c k36= 0, k37= 0, k38= 0, k39= %DX, k40= 0, k41= 0, k42= %DY, k43= %DZ, k44= 0, k45= 0, k46= 0, k47= 0, k48= 0, k49= 0] ] Type: List List Equation Fraction Polynomial Integer Time: 0.69 (IN) + 1.42 (EV) + 0.15 (OT) = 2.25 sec )clear properties eqns vars -- Solve a 3 x 3 system of nonlinear equations eqn1:= x**2*y + 3*y*z - 4 = 0 2 (52) 3y z + x y - 4= 0 Type: Equation Polynomial Integer Time: 0.03 (OT) = 0.03 sec eqn2:= -3*x**2*z + 2*y**2 + 1 = 0 2 2 (53) - 3x z + 2y + 1= 0 Type: Equation Polynomial Integer Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec eqn3:= 2*y*z**2 - z**2 - 1 = 0 2 (54) (2y - 1)z - 1= 0 Type: Equation Polynomial Integer Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec -- Solving this by hand would be a nightmare solve([eqn1, eqn2, eqn3], [x, y, z]) (55) @[[x= 1,y= 1,z= 1], [x= - 1,y= 1,z= 1], 2 2 [- 3z + x + 2= 0,y= - 3z + 1,3z - 2z + 1= 0], 4 3 2 4 3 2 2 - 18z + 24z + 21z + 12z + 3 [12z - 12z - 30z + 7z + 3x = 0, y= ------------------------------, 2 5 4 3 2 6z - 6z - 9z - 7z - 3z - 1= 0] ] Type: List List Equation Fraction Polynomial Integer Time: 0.02 (IN) + 1.18 (EV) + 0.03 (OT) + 0.03 (GC) = 1.27 sec )clear properties eqn1 eqn2 eqn3 -- ---------- Quit ---------- )quit real 118.9 user 38.5 sys 0.9 \end{verbatim} \section{Inequalities} \begin{verbatim} -- ---------- Inequalities ---------- -- => True %e**%pi > %pi**%e (1) false Type: Boolean Time: 0.30 (IN) + 0.38 (EV) + 0.18 (OT) + 0.15 (GC) = 1.02 sec -- => [True, False] [x**4 - x + 1 > 0, x**4 - x + 1 > 1] (2) [true,true] Type: List Boolean Time: 0.05 (IN) + 0.02 (EV) + 0.02 (OT) = 0.08 sec -- => True --assume(abs(x) < 1) -1 < x and x < 1 (3) false Type: Boolean Time: 0.23 (IN) + 0.02 (OT) = 0.25 sec -- x > y > 0 and k, n > 0 => k x^n > k y^n --assume(x > y, y > 0) 2*x**2 > 2*y**2 (4) false Type: Boolean Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec --assume(k > 0) k*x**2 > k*y**2 (5) false Type: Boolean Time: 0.02 (EV) = 0.02 sec --assume(n > 0) k*x**n > k*y**n (6) false Type: Boolean Time: 0.18 (IN) + 0.03 (OT) = 0.22 sec -- x > 1 and y >= x - 1 => y > 0 --assume(x > 1, y >= x - 1) y > 0 (7) true Type: Boolean Time: 0.03 (IN) = 0.03 sec -- x >= y, y >= z, z >= x => x = y = z --assume(x >= y, y >= z, z >= x) [x = y, x = z, y = z] (8) [x= y,x= z,y= z] Type: List Equation Symbol Time: 0.13 (IN) + 0.02 (OT) = 0.15 sec -- x < -1 or x > 3 solve(abs(x - 1) > 2, x) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Boolean Variable x -- x < 1 or 2 < x < 3 or 4 < x < 5 solve((x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) < 0, x) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Boolean Variable x -- x < 3 or x >= 5 solve(6/(x - 3) <= 3, x) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Boolean Variable x -- => 0 <= x < 4 solve(sqrt(x) < 2, x) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Boolean Variable x -- => x is real solve(sin(x) < 2, x) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Boolean Variable x -- => x != pi/2 + n 2 pi solve(sin(x) < 1, x) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Boolean Variable x -- The next two examples come from Abdubrahim Muhammad Farhat, _Stability -- Analysis of Finite Difference Schemes_, Ph.D. dissertation, University of -- New Mexico, Albuquerque, New Mexico, December 1993 => 0 <= A <= 1/2 solve(abs(2*A*(cos(t) - 1) + 1) <= 1, A) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Boolean Variable A -- => 125 A^4 + 24 A^2 - 48 < 0 or |A| < 2/5 sqrt([8 sqrt(6) - 3]/5) solve(A**2*(cos(t) - 4)**2*sin(t)**2 < 9, A) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Boolean Variable A -- => |x| < y solve([x + y > 0, x - y < 0], [x, y]) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) List Boolean List OrderedVariableList [x,y] -- ---------- Quit ---------- )quit real 24.5 user 6.0 sys 0.4 \end{verbatim} \section{Vector Analysis} \begin{verbatim} -- ---------- Vector Analysis ---------- -- Vector norm => sqrt(15) Norm(v) == local V V : Matrix Complex Integer := v sqrt(map(conjugate, transpose(V)) * V) Type: Void Time: 0 sec Norm(vector([1 + %i, -2, 3*%i])) Compiling function Norm with type Vector Complex Integer -> Expression Complex Integer +--+ (2) \|15 Type: Expression Complex Integer Time: 0.97 (IN) + 0.23 (EV) + 0.33 (OT) + 0.17 (GC) = 1.70 sec )clear properties Norm Compiled code for Norm has been cleared. -- Cross product: (2, 2, -3) x (1, 3, 1) => (11, -5, 4) --cross(vector([2, 2, -3]), vector([1, 3, 1])) -- (a x b) . (c x d) => (a . c) (b . d) - (a . d) (b . c) --cross(a, b) . cross(c, d) -- => (2 y z^3 - 2 x^2 y^2 z, x y, 2 x y^2 z^2 - x z) --curl(vector([x*y*z, x**2*y**2*z**2, y**2*z**3])) -- DEL . (f x g) => g . (DEL x f) - f . (DEL x g) --div(cross(f, g)) -- Express DEL . a in spherical coordinates (r, theta, phi) for -- a = (a_r(r, theta, phi), a_theta(r, theta, phi), a_phi(r, theta, phi)). -- Here, phi is in the x-y plane and theta is the angle with the z-axis. -- => 1/r^2 d/dr[r^2 a_r] + 1/[r sin(theta)] d/dtheta[sin(theta) a_theta] -- + 1/[r sin(theta)] da_phi/dphi -- => da_r/dr + (2 a_r)/r + 1/r da_theta/dtheta + a_theta/[r tan(theta)] -- + 1/[r sin(theta)] da_phi/dphi -- See Keith R. Symon, _Mechanics_, Third Edition, Addison-Wesley Publishing -- Company, 1971, p. 103. --coordinates == spherical --div([a_r(r, theta, phi), a_theta(r, theta, phi), a_phi(r, theta, phi)]) -- Express dR/dt in spherical coordinates (r, theta, phi) where R is the -- position vector r*Rhat(theta, phi) with Rhat being the unit vector in the -- direction of R => (dr/dt, r dtheta/dt, r sin(theta) dphi/dt) -- [Symon, p. 98] r:= operator('r); Type: BasicOperator Time: 0.08 (IN) = 0.08 sec rhat:= operator('rhat); Type: BasicOperator Time: 0.03 (IN) = 0.03 sec theta:= operator('theta); Type: BasicOperator Time: 0.03 (IN) = 0.03 sec phi:= operator('phi); Type: BasicOperator Time: 0.02 (IN) + 0.03 (OT) = 0.05 sec v:= vector([r(t)*rhat(theta(t), phi(t)), 0, 0]) (7) [r(t)rhat(theta(t),phi(t)),0,0] Type: Vector Expression Integer Time: 0.58 (IN) + 0.03 (EV) + 0.14 (OT) = 0.75 sec D(v, t) There are 5 exposed and 0 unexposed library operations named D having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op D to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named D with argument type(s) Vector Expression Integer Variable t map(e +-> D(e, t), v) (8) [ , , r(t)phi (t)rhat (theta(t),phi(t)) + r(t)theta (t)rhat (theta(t),phi(t)) ,2 ,1 + , rhat(theta(t),phi(t))r (t) , 0, 0] Type: Vector Expression Integer Time: 0.02 (IN) + 0.10 (EV) + 0.03 (OT) = 0.15 sec )clear properties r rhat theta phi v -- Scalar potential => x^2 y + y + 2 z^3 --potential(vector([2*x*y, x**2 + 1, 6*z**2])) -- Vector potential => (x y z, x^2 y^2 z^2, y^2 z^3) is one possible solution. -- See Harry F. Davis and Arthur David Snider, _Introduction to Vector -- Analysis_, Third Edition, Allyn and Bacon, Inc., 1975, p. 97. --vectorpotential(vector([2*y*z**3 - 2*x**2*y**2*z, x*y, 2*x*y**2*z**2 - x*z])) --curl(%) -- Orthogonalize the following vectors (Gram-Schmidt). See Lee W. Johnson and -- R. Dean Riess, _Introduction to Linear Algebra_, Addison-Wesley Publishing -- Company, 1981, p. 104 => @[[0 1 2 1], [0 -1 1 -1], [2 1 0 -1]]^T [transpose(matrix(@[[0, 1, 2, 1]])), transpose(matrix(@[[0, 1, 3, 1]])), _ transpose(matrix(@[[1, 1, 1, 0]])), transpose(matrix(@[[1, 3, 6, 2]]))] +0+ +0+ +1+ +1+ | | | | | | | | |1| |1| |1| |3| (9) [| |,| |,| |,| |] |2| |3| |1| |6| | | | | | | | | +1+ +1+ +0+ +2+ Type: List Matrix Integer Time: 0.02 (IN) + 0.03 (OT) = 0.05 sec gramschmidt(%) + 1 + | ---- | | +-+ | | |3 | + 0 + + 0 + | |- | | | | | | \|2 | | +-+| | 1 | | | | \|3 | |----| +0+ | 1 | |- ----| | +-+| | | | ----- | | 3 | |\|6 | |0| | +-+| | | | | (10) [| |,| |3 |,| +-+ |,| 2 |] |0| | 2 |- | | \|3 | |----| | | | \|2 | | ---- | | +-+| +0+ | | | 3 | |\|6 | | 0 | | | | | | | | +-+| | 1 | | 1 | | \|3 | |----| |- -----| |- ----| | +-+| | +-| + 3 + +\|6 + | |3| | 2 |-| + \|2+ Type: List Matrix Expression Integer Time: 0.10 (IN) + 0.20 (EV) + 0.07 (OT) + 0.03 (GC) = 0.40 sec -- ---------- Quit ---------- )quit real 7.6 user 4.3 sys 0.5 \end{verbatim} \section{Matrix Theory} \begin{verbatim} -- ---------- Matrix Theory ---------- -- Extract the superdiagonal => [2, 6] matrix(@[[1, 2, 3], [4, 5, 6], [7, 8, 9]]); Type: Matrix Integer Time: 0.03 (IN) + 0.02 (EV) + 0.02 (GC) = 0.07 sec [%(j - 1, j) for j in 2..ncols(%)] (2) [2,6] Type: List Integer Time: 0.05 (OT) + 0.02 (GC) = 0.07 sec -- (2, 3)-minor => @[[1, 2], [7, 8]] --minor(matrix(@[[1, 2, 3], [4, 5, 6], [7, 8, 9]]), 2, 3) -- Create the 7 x 6 matrix B from rearrangements of the elements of the 4 x 4 -- matrix A (this is easiest to do with a MATLAB style notation): -- B = [A(1:3,2:4), A([1,2,4],[3,1,4]); A, [A(1:2,3:4); A([4,1],[3,2])]] -- => @[[12 13 14|13 11 14], -- [22 23 24|23 21 24], -- [32 33 34|43 41 44], -- [--------+--+-----] -- [11 12 13 14|13 14], -- [21 22 23 24|23 24], -- [ +-----] -- [31 32 33 34|43 42], -- [41 42 43 44|13 12]]. See Michael James Wester, _Symbolic Calculation -- and Expression Swell Analysis of Matrix Determinants and Eigenstuff_, Ph.D. -- dissertation, University of New Mexico, Albuquerque, New Mexico, December -- 1992, p. 89. A:= matrix(@[[11, 12, 13, 14], _ [21, 22, 23, 24], _ [31, 32, 33, 34], _ [41, 42, 43, 44]]); Type: Matrix Integer Time: 0.02 (IN) = 0.02 sec vertConcat(horizConcat(subMatrix(A, 1, 3, 2, 4), _ matrix(@[[A(1,3), A(1,1), A(1,4)], _ [A(2,3), A(2,1), A(2,4)], _ [A(4,3), A(4,1), A(4,4)]])), _ horizConcat(A, vertConcat(subMatrix(A, 1, 2, 3, 4), _ matrix(@[[A(4,3), A(4,2)], _ [A(1,3), A(1,2)]])))) +12 13 14 13 11 14+ | | |22 23 24 23 21 24| | | |32 33 34 43 41 44| | | (4) |11 12 13 14 13 14| | | |21 22 23 24 23 24| | | |31 32 33 34 43 42| | | +41 42 43 44 13 12+ Type: Matrix Integer Time: 0.07 (IN) + 0.03 (OT) = 0.10 sec )clear properties A -- Create a block diagonal matrix diagonalMatrix( _ @[matrix([[a, 1],[0, a]]), b, matrix([[c, 1, 0],[0, c, 1],[0, 0, c]])]) (5) diagonalMatrix +c 1 0+ +a 1+ | | | |,b,|0 c 1| +0 a+ | | +0 0 c+ Type: Symbol Time: 0.30 (IN) + 0.22 (OT) + 0.02 (GC) = 0.53 sec -- => @[[1 1], [1 0]] matrix(@[[7, 11], [3, 8]]) :: Matrix(PrimeField(2)) +1 1+ (6) | | +1 0+ Type: Matrix PrimeField 2 Time: 0.07 (IN) + 0.05 (OT) + 0.05 (GC) = 0.17 sec -- => @[[-cos t, -sin t], [sin t, -cos t]] matrix(@[[cos(t), sin(t)], [-sin(t), cos(t)]]) + cos(t) sin(t)+ (7) | | +- sin(t) cos(t)+ Type: Matrix Expression Integer Time: 0.22 (IN) + 0.23 (EV) + 0.07 (OT) + 0.10 (GC) = 0.62 sec D(%, t, 2) +- cos(t) - sin(t)+ (8) | | + sin(t) - cos(t)+ Type: SquareMatrix(2,Expression Integer) Time: 0.32 (IN) + 0.07 (EV) = 0.38 sec -- => @[[(a + 7) x + (2 a - 8) y, (3 a - 9) x + (4 a + 10) y, -- (5 a + 11) x + (6 a - 12) y]] matrix(@[[x, y]]) * (a*matrix(@[[1, 3, 5], [2, 4, 6]]) _ + matrix([[7, -9, 11], [-8, 10, -12]])) (9) @[(2a - 8)y + (a + 7)x (4a + 10)y + (3a - 9)x (6a - 12)y + (5a + 11)x] Type: Matrix Polynomial Integer Time: 1.58 (IN) + 0.35 (OT) = 1.93 sec -- Matrix norms: infinity norm => 7 norm(matrix(@[[1, -2*%i], [-3*%i, 4]]), %plusInfinity) There are 5 exposed and 6 unexposed library operations named norm having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op norm to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named norm with argument type(s) Matrix Complex Integer OrderedCompletion Integer -- Frobenius norm => (a^2 + b^2 + c^2)/(|a| |b| |c|) (a, b, c real) norm(matrix(@[[a/(b*c), 1/c, 1/b], [1/c, b/(a*c), 1/a], [1/b, 1/a, c/(a*b)]]), _ 'f) There are 5 exposed and 6 unexposed library operations named norm having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op norm to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named norm with argument type(s) Matrix Fraction Polynomial Integer Variable f -- Hermitian (complex conjugate transpose) => @[[1, f(4 + 5 i)], [2 - 3 i, 6]] -- (This assumes f is a real valued function. In general, the (1, 2) entry -- will be conjugate[f(4 - 5 i)] = conjugate(f)(4 + 5 i).) f:= operator('f); Type: BasicOperator Time: 0.07 (IN) = 0.07 sec map('conjugate, transpose(matrix(@[[1, 2 + 3*%i], [f(4 - 5*%i), 6]])) _ :: Matrix Complex Expression Integer) Cannot convert from type Matrix Expression Complex Integer to Matrix Complex Expression Integer for value + 1 f(4 - 5%i)+ | | +2 + 3%i 6 + m:= matrix(@[[a, b], [1, a*b]]) +a b + (11) | | +1 a b+ Type: Matrix Polynomial Integer Time: 0.03 (IN) + 0.02 (OT) = 0.05 sec -- Invert the matrix => 1/(a^2 - 1) @[[a, -1], [-1/b, a/b]] minv:= inverse(m) + a 1 + | ------ - ------ | | 2 2 | | a - 1 a - 1 | (12) | | | 1 a | |- --------- ---------| | 2 2 | + (a - 1)b (a - 1)b+ Type: Union(Matrix Fraction Polynomial Integer,...) Time: 0.05 (IN) + 0.03 (EV) + 0.02 (OT) = 0.10 sec m * minv +1 0+ (13) | | +0 1+ Type: Matrix Fraction Polynomial Integer Time: 0.38 (IN) + 0.07 (OT) = 0.45 sec )clear properties m minv -- Inverse of a triangular partitioned (or block) matrix -- => @[[A_11^(-1), -A_11^(-1) A_12 A_22^(-1)], [0, A_22^(-1)]]. -- See Charles G. Cullen, _Matrices and Linear Transformations_, Second -- Edition, Dover Publications Inc., 1990, p. 35. matrix(@[[A11, A12], [0, A22]])**(-1) + 1 A12 + |--- - -------| |A11 A11 A22| (14) | | | 1 | | 0 --- | + A22 + Type: Matrix Fraction Polynomial Integer Time: 0.15 (IN) + 0.02 (OT) = 0.17 sec -- LU decomposition of a symbolic matrix [David Wood] -- [ 1 0 0] [1 x-2 x-3] [ 1 x-2 x-3 ] -- [x-1 1 0] [0 4 x-5] = [x-1 x^2-3x+6 x^2-3x-2 ] -- [x-2 x-3 1] [0 0 x-7] [x-2 x^2-8 2x^2-12x+14] matrix(@[[ 1, x-2, x-3 ], _ [x-1, x**2-3*x+6, x**2-3*x-2 ], _ [x-2, x**2-8, 2*x**2-12*x+14]]); Type: Matrix Polynomial Integer Time: 0.08 (IN) = 0.08 sec -- Reduced row echelon form [Cullen, p. 43] -- => @[[1 0 -1 0 2], [0 1 2 0 -1], [0 0 0 1 3], [0 0 0 0 0]] matrix(@[[1, 2, 3, 1, 3], _ [3, 2, 1, 1, 7], _ [0, 2, 4, 1, 1], _ [1, 1, 1, 1, 4]]); Type: Matrix Integer Time: 0.03 (IN) = 0.03 sec rowEchelon(%) +1 0 - 1 0 2 + | | |0 1 2 0 - 1| (17) | | |0 0 0 1 3 | | | +0 0 0 0 0 + Type: Matrix Integer Time: 0.02 (EV) + 0.02 (OT) = 0.03 sec -- => 2. See Gerald L. Bradley, _A Primer of Linear Algebra_, Prentice-Hall, -- Inc., 1975, p. 135. rank(matrix(@[[-1, 3, 7, -5], [4, -2, 1, 3], [2, 4, 15, -7]])) (18) 2 Type: PositiveInteger Time: 0.02 (OT) = 0.02 sec -- => 1 rank(matrix(@[[2*sqrt(2), 8], [6*sqrt(6), 24*sqrt(3)]])) (19) 1 Type: PositiveInteger Time: 0.05 (IN) + 0.35 (EV) + 0.03 (OT) = 0.43 sec -- => 1 rank(matrix(@[[sin(2*t), cos(2*t)], _ [2*(1 - cos(t)**2)*cos(t), (1 - 2*sin(t)**2)*sin(t)]])) (20) 2 Type: PositiveInteger Time: 0.10 (IN) + 0.02 (EV) + 0.02 (OT) = 0.13 sec -- Null space => @[[2 4 1 0], [0 -3 0 1]]^T or variant [Bradley, p. 207] nullSpace(matrix(@[[1, 0, -2, 0], [-2, 1, 0, 3], [-1, 2, -6, 6]])) (21) @[[2,4,1,0],[0,- 3,0,1]] Type: List Vector Integer Time: 0.03 (EV) + 0.02 (OT) = 0.05 sec -- Define a Vandermonde matrix (useful for doing polynomial interpolations) matrix(@[[1, 1, 1, 1 ], _ [w, x, y, z ], _ [w**2, x**2, y**2, z**2], _ [w**3, x**3, y**3, z**3]]) +1 1 1 1 + | | |w x y z | | | (22) | 2 2 2 2| |w x y z | | | | 3 3 3 3| +w x y z + Type: Matrix Polynomial Integer Time: 0.15 (IN) + 0.05 (OT) = 0.20 sec determinant(%) (23) 2 2 2 2 2 3 ((x - w)y + (- x + w )y + w x - w x)z + 3 3 3 3 3 2 ((- x + w)y + (x - w )y - w x + w x)z + 2 2 3 3 3 2 2 3 3 2 2 2 3 ((x - w )y + (- x + w )y + w x - w x )z + (- w x + w x)y + 3 3 2 2 3 3 2 (w x - w x)y + (- w x + w x )y Type: Polynomial Integer Time: 0.07 (OT) = 0.07 sec -- The following formula implies a general result: -- => (w - x) (w - y) (w - z) (x - y) (x - z) (y - z) factor(%) (24) (x - w)(y - x)(y - w)(z - y)(z - x)(z - w) Type: Factored Polynomial Integer Time: 0.07 (IN) + 0.25 (EV) + 0.03 (OT) = 0.35 sec -- Minimum polynomial => (lambda - 1)^2 (lambda + 1) [Cullen, p. 181] matrix(@[[17, -8, -12, 14], _ [46, -22, -35, 41], _ [-2, 1, 4, -4], _ [ 4, -2, -2, 3]]); Type: Matrix Integer Time: 0 sec minimalPolynomial(% :: SquareMatrix(4, Integer)) There are 2 exposed and 1 unexposed library operations named minimalPolynomial having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op minimalPolynomial to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named minimalPolynomial with argument type(s) SquareMatrix(4,Integer) -- Compute the eigenvalues of a matrix from its characteristic polynomial -- => lambda = {1, -2, 3} m:= matrix(@[[ 5, -3, -7], _ [-2, 1, 2], _ [ 2, -3, -4]]) + 5 - 3 - 7+ | | (26) |- 2 1 2 | | | + 2 - 3 - 4+ Type: Matrix Integer Time: 0.02 (OT) = 0.02 sec characteristicPolynomial(m, lambda) 3 2 (27) - lambda + 2lambda + 5lambda - 6 Type: Polynomial Integer Time: 0.02 (IN) + 0.03 (OT) = 0.05 sec solve(% = 0, lambda) (28) [lambda= 3,lambda= 1,lambda= - 2] Type: List Equation Fraction Polynomial Integer Time: 0.47 (IN) + 0.18 (EV) + 0.07 (OT) = 0.72 sec )clear properties m -- In theory, an easy eigenvalue problem! => lambda = {2 - a} for k = 1..100 -- [Wester, p. 154] identityMatrix(n) == diagonalMatrix([1 for i in 1..n]) Type: Void Time: 0 sec eigenvalues((2 - a)*identityMatrix(100)) Compiling function identityMatrix with type PositiveInteger -> Matrix Integer (30) [- a + 2] Type: List Union(Fraction Polynomial Integer,SuchThat(Symbol,Polynomial Integer)) Time: 31.44 (IN) + 7.85 (EV) + 4.32 (OT) + 5.17 (GC) = 48.77 sec -- => lambda = {4 sin^2(pi k/[2 (n + 1)])} for k = 1..n for an n x n matrix. -- For n = 5, lambda = {2 - sqrt(3), 1, 2, 3, 2 + sqrt(3)} -- See J. H. Wilkinson, _The Algebraic Eigenvalue Problem_, Oxford University -- Press, 1965, p. 307. matrix(@[[2, 1, 0, 0, 0], _ [1, 2, 1, 0, 0], _ [0, 1, 2, 1, 0], _ [0, 0, 1, 2, 1], _ [0, 0, 0, 1, 2]]) +2 1 0 0 0+ | | |1 2 1 0 0| | | (31) |0 1 2 1 0| | | |0 0 1 2 1| | | +0 0 0 1 2+ Type: Matrix Integer Time: 0.02 (IN) = 0.02 sec radicalEigenvalues(%) +-+ +-+ (32) [- \|3 + 2,\|3 + 2,1,2,3] Type: List Expression Integer Time: 0.08 (IN) + 0.08 (EV) + 0.02 (OT) = 0.18 sec -- Eigenvalues of the Rosser matrix. This matrix is notorious for causing -- numerical eigenvalue routines to fail. [Wester, p. 146 (Cleve Moler)] -- => {-10 sqrt(10405), 0, 510 - 100 sqrt(26), 1000, 1000, -- 510 + 100 sqrt(26), 1020, 10 sqrt(10405)} = -- {-1020.049, 0, 0.098, 1000, 1000, 1019.902, 1020, 1020.049} rosser:= matrix([[ 611, 196, -192, 407, -8, -52, -49, 29], _ [ 196, 899, 113, -192, -71, -43, -8, -44], _ [-192, 113, 899, 196, 61, 49, 8, 52], _ [ 407, -192, 196, 611, 8, 44, 59, -23], _ [ -8, -71, 61, 8, 411, -599, 208, 208], _ [ -52, -43, 49, 44, -599, 411, 208, 208], _ [ -49, -8, 8, 59, 208, 208, 99, -911], _ [ 29, -44, 52, -23, 208, 208, -911, 99]]); Type: Matrix Integer Time: 0.02 (IN) = 0.02 sec radicalEigenvalues(rosser) (34) +-----+ +-----+ +--+ +--+ [10\|10405 ,- 10\|10405 ,- 100\|26 + 510,100\|26 + 510,0,1000,1000,1020] Type: List Expression Integer Time: 0.12 (IN) + 1.0 (EV) + 0.02 (OT) = 1.13 sec realEigenvalues(rosser, 1.0e-6) (35) [- 1020.0490183830 26123, 0.0, 1020.0490183830 26123, 1020.0, 1019.9019513130 187988, 1000.0, 0.0980486869 8120117187 5] Type: List Float Time: 0.07 (IN) + 0.28 (EV) + 0.05 (OT) + 0.02 (GC) = 0.42 sec eigenvalues(rosser :: Matrix(Float)) WARNING (genufact): No known algorithm to factor 125 124 123 122 ? - 86215.0 ? + 3672643562.0 ? - 10305 2524136076.0 ? + 121 120 214242065 1760625974.0 ? - 0.3519437981 9139045528 E 23 ? + 119 118 0.4757930276 0524392937 E 27 ? - 0.5443719612 8759169508 E 31 ? + 117 116 0.5379979009 8770356924 E 35 ? - 0.4664749268 2226816519 E 39 ? + 115 114 0.3592087692 2534478781 E 43 ? - 0.2480907901 4852864426 E 47 ? + 113 112 0.1549276891 3929107162 E 51 ? - 0.8807036930 5149411468 E 54 ? + 111 110 0.4583391720 1504761731 E 58 ? - 0.2194388254 7216326309 E 62 ? + 109 108 0.9705762773 604715712 E 65 ? - 0.3980292705 4424329091 E 69 ? + 107 106 0.1518264867 7783075257 E 73 ? - 0.5401752426 0724778564 E 76 ? + 105 104 0.1796955195 1515805598 E 80 ? - 0.5601295905 4406117874 E 83 ? + 103 102 0.1639113863 1096220906 E 87 ? - 0.4510439265 9444505639 E 90 ? + 101 100 0.1168814463 6800724178 E 94 ? - 0.2855827986 1382495038 E 97 ? + 99 98 0.6586280305 3086327597 E 100 ? - 0.1435005623 1770466372 E 104 ? + 97 96 0.2955815573 3367270221 E 107 ? - 0.5758845873 428492513 E 110 ? + 95 94 0.1061610141 8172155474 E 114 ? - 0.1851850768 7922934268 E 117 ? + 93 92 0.3056218491 6782132505 E 120 ? - 0.4769657381 1684487203 E 123 ? + 91 90 0.7032699330 7418158372 E 126 ? - 0.9782596656 7436389234 E 129 ? + 89 88 0.1280860874 3712484582 E 133 ? - 0.1573106414 8242673085 E 136 ? + 87 86 0.1802439290 923378293 E 139 ? - 0.1909575702 2321405039 E 142 ? + 85 84 0.1841324046 7558627001 E 145 ? - 0.1565578204 6290786175 E 148 ? + 83 82 0.1084018298 6945156255 E 151 ? - 0.4377988417 1758608725 E 153 ? + 81 80 - 0.2968496265 1894553792 E 156 ? + 0.1023002729 7224474459 E 160 ? + 79 78 - 0.1643770493 4269005245 E 163 ? + 0.2083326959 4734561364 E 166 ? + 77 76 - 0.2301999742 2965920355 E 169 ? + 0.2301309174 9423652886 E 172 ? + 75 74 - 0.2118385504 6752677393 E 175 ? + 0.1812642708 7694905254 E 178 ? + 73 72 - 0.1449610412 5042294701 E 181 ? + 0.1086839448 8924848295 E 184 ? + 71 70 - 0.7651391596 9967540208 E 186 ? + 0.5060299956 3124704326 E 189 ? + 69 68 - 0.3142367184 7713335751 E 192 ? + 0.1829574772 0415689208 E 195 ? + 67 66 - 0.9961784415 3166827826 E 197 ? + 0.5051793379 2698179236 E 200 ? + 65 64 - 0.2370885486 7640128904 E 203 ? + 0.1019224176 5460569951 E 206 ? + 63 62 - 0.3942260312 5442307776 E 208 ? + 0.1323834699 481648387 E 211 ? + 61 60 - 0.3524632481 6587139553 E 213 ? + 0.4907904491 2672068684 E 215 ? + 59 58 0.1935153311 3162677451 E 218 ? - 0.2135886927 6123692419 E 221 ? + 57 56 0.1223237295 5917448702 E 224 ? - 0.5462064456 7735058373 E 226 ? + 55 54 0.2062619424 5435174697 E 229 ? - 0.6737922494 4291403287 E 231 ? + 53 52 0.1898827082 1604329724 E 234 ? - 0.4486781196 52298524 E 236 ? + 51 50 0.8133738511 6364504076 E 238 ? - 0.7358667283 4876131493 E 240 ? + 49 48 - 0.1986000215 428423829 E 243 ? + 0.1331591187 7791407655 E 246 ? + 47 46 - 0.4492807815 4925031508 E 248 ? + 0.1106530639 7295677952 E 251 ? + 45 44 - 0.2046039509 2572215835 E 253 ? + 0.2430619124 0670333847 E 255 ? + 43 42 0.3252259818 2888244489 E 256 ? - 0.1131150344 9655271066 E 260 ? + 41 40 0.3691051035 7773753178 E 262 ? - 0.7900011265 6063011579 E 264 ? + 39 38 0.1233981952 5026790198 E 267 ? - 0.1270418736 3851064032 E 269 ? + 37 36 0.1660218706 9618209188 E 270 ? + 0.2862105879 5857092645 E 273 ? + 35 34 - 0.8110417668 3161233591 E 275 ? + 0.1484448521 8686301081 E 278 ? + 33 32 - 0.2123387008 794925024 E 280 ? + 0.2506751820 5612670323 E 282 ? + 31 30 - 0.2494329065 6728197272 E 284 ? + 0.2109806712 8889498289 E 286 ? + 29 28 - 0.1521224934 1012072561 E 288 ? + 0.9351456154 4154324412 E 289 ? + 27 26 - 0.4897636603 2042451012 E 291 ? + 0.2184259738 2222779488 E 293 ? + 25 24 - 0.8298614578 7219241205 E 294 ? + 0.2689816827 0170685903 E 296 ? + 23 22 - 0.7455272800 2234756452 E 297 ? + 0.1771957532 6653063304 E 299 ? + 21 20 - 0.3622167611 4512303391 E 300 ? + 0.6385537866 7592677628 E 301 ? + 19 18 - 0.9730487915 5887722779 E 302 ? + 0.1283818926 2924796953 E 304 ? + 17 16 - 0.1467911675 9006671118 E 305 ? + 0.1454586351 2070495541 E 306 ? + 15 14 - 0.1247986846 9975071194 E 307 ? + 0.9251453429 5406258973 E 307 ? + 13 12 - 0.5905679202 8110037735 E 308 ? + 0.3230290678 3860457938 E 309 ? + 11 10 - 0.1503623501 5158820289 E 310 ? + 0.5900666022 0150867207 E 310 ? + 9 8 - 0.1927643191 6531249396 E 311 ? + 0.5152198729 6418385783 E 311 ? + 7 6 - 0.1099680204 0615001798 E 312 ? + 0.1809295237 1502506625 E 312 ? + 5 4 - 0.2172461015 8315085132 E 312 ? + 0.1733446261 3560586487 E 312 ? + 3 2 - 0.7597264055 3053863391 E 311 ? + 0.1050398077 8828775171 E 311 ? + - 0.4443971993 2926345443 E 309 ? - 0.7155498947 0116891371 E 291 , trying square-free. (36) [ %G | 125 124 123 122 %G - 86215.0 %G + 3672643562.0 %G - 10305 2524136076.0 %G + 121 120 214242065 1760625974.0 %G - 0.3519437981 9139045528 E 23 %G + 119 118 0.4757930276 0524392937 E 27 %G - 0.5443719612 8759169508 E 31 %G + 117 116 0.5379979009 8770356924 E 35 %G - 0.4664749268 2226816519 E 39 %G + 115 114 0.3592087692 2534478781 E 43 %G - 0.2480907901 4852864426 E 47 %G + 113 112 0.1549276891 3929107162 E 51 %G - 0.8807036930 5149411468 E 54 %G + 111 110 0.4583391720 1504761731 E 58 %G - 0.2194388254 7216326309 E 62 %G + 109 108 0.9705762773 604715712 E 65 %G - 0.3980292705 4424329091 E 69 %G + 107 106 0.1518264867 7783075257 E 73 %G - 0.5401752426 0724778564 E 76 %G + 105 104 0.1796955195 1515805598 E 80 %G - 0.5601295905 4406117874 E 83 %G + 103 102 0.1639113863 1096220906 E 87 %G - 0.4510439265 9444505639 E 90 %G + 101 100 0.1168814463 6800724178 E 94 %G - 0.2855827986 1382495038 E 97 %G + 99 98 0.6586280305 3086327597 E 100 %G - 0.1435005623 1770466372 E 104 %G + 97 96 0.2955815573 3367270221 E 107 %G - 0.5758845873 428492513 E 110 %G + 95 94 0.1061610141 8172155474 E 114 %G - 0.1851850768 7922934268 E 117 %G + 93 92 0.3056218491 6782132505 E 120 %G - 0.4769657381 1684487203 E 123 %G + 91 90 0.7032699330 7418158372 E 126 %G - 0.9782596656 7436389234 E 129 %G + 89 88 0.1280860874 3712484582 E 133 %G - 0.1573106414 8242673085 E 136 %G + 87 86 0.1802439290 923378293 E 139 %G - 0.1909575702 2321405039 E 142 %G + 85 84 0.1841324046 7558627001 E 145 %G - 0.1565578204 6290786175 E 148 %G + 83 82 0.1084018298 6945156255 E 151 %G - 0.4377988417 1758608725 E 153 %G + 81 - 0.2968496265 1894553792 E 156 %G + 80 79 0.1023002729 7224474459 E 160 %G - 0.1643770493 4269005245 E 163 %G + 78 77 0.2083326959 4734561364 E 166 %G - 0.2301999742 2965920355 E 169 %G + 76 75 0.2301309174 9423652886 E 172 %G - 0.2118385504 6752677393 E 175 %G + 74 73 0.1812642708 7694905254 E 178 %G - 0.1449610412 5042294701 E 181 %G + 72 71 0.1086839448 8924848295 E 184 %G - 0.7651391596 9967540208 E 186 %G + 70 69 0.5060299956 3124704326 E 189 %G - 0.3142367184 7713335751 E 192 %G + 68 67 0.1829574772 0415689208 E 195 %G - 0.9961784415 3166827826 E 197 %G + 66 65 0.5051793379 2698179236 E 200 %G - 0.2370885486 7640128904 E 203 %G + 64 63 0.1019224176 5460569951 E 206 %G - 0.3942260312 5442307776 E 208 %G + 62 61 0.1323834699 481648387 E 211 %G - 0.3524632481 6587139553 E 213 %G + 60 59 0.4907904491 2672068684 E 215 %G + 0.1935153311 3162677451 E 218 %G + 58 - 0.2135886927 6123692419 E 221 %G + 57 56 0.1223237295 5917448702 E 224 %G - 0.5462064456 7735058373 E 226 %G + 55 54 0.2062619424 5435174697 E 229 %G - 0.6737922494 4291403287 E 231 %G + 53 52 0.1898827082 1604329724 E 234 %G - 0.4486781196 52298524 E 236 %G + 51 50 0.8133738511 6364504076 E 238 %G - 0.7358667283 4876131493 E 240 %G + 49 - 0.1986000215 428423829 E 243 %G + 48 47 0.1331591187 7791407655 E 246 %G - 0.4492807815 4925031508 E 248 %G + 46 45 0.1106530639 7295677952 E 251 %G - 0.2046039509 2572215835 E 253 %G + 44 43 0.2430619124 0670333847 E 255 %G + 0.3252259818 2888244489 E 256 %G + 42 - 0.1131150344 9655271066 E 260 %G + 41 40 0.3691051035 7773753178 E 262 %G - 0.7900011265 6063011579 E 264 %G + 39 38 0.1233981952 5026790198 E 267 %G - 0.1270418736 3851064032 E 269 %G + 37 36 0.1660218706 9618209188 E 270 %G + 0.2862105879 5857092645 E 273 %G + 35 - 0.8110417668 3161233591 E 275 %G + 34 33 0.1484448521 8686301081 E 278 %G - 0.2123387008 794925024 E 280 %G + 32 31 0.2506751820 5612670323 E 282 %G - 0.2494329065 6728197272 E 284 %G + 30 29 0.2109806712 8889498289 E 286 %G - 0.1521224934 1012072561 E 288 %G + 28 27 0.9351456154 4154324412 E 289 %G - 0.4897636603 2042451012 E 291 %G + 26 25 0.2184259738 2222779488 E 293 %G - 0.8298614578 7219241205 E 294 %G + 24 23 0.2689816827 0170685903 E 296 %G - 0.7455272800 2234756452 E 297 %G + 22 21 0.1771957532 6653063304 E 299 %G - 0.3622167611 4512303391 E 300 %G + 20 19 0.6385537866 7592677628 E 301 %G - 0.9730487915 5887722779 E 302 %G + 18 17 0.1283818926 2924796953 E 304 %G - 0.1467911675 9006671118 E 305 %G + 16 15 0.1454586351 2070495541 E 306 %G - 0.1247986846 9975071194 E 307 %G + 14 13 0.9251453429 5406258973 E 307 %G - 0.5905679202 8110037735 E 308 %G + 12 11 0.3230290678 3860457938 E 309 %G - 0.1503623501 5158820289 E 310 %G + 10 9 0.5900666022 0150867207 E 310 %G - 0.1927643191 6531249396 E 311 %G + 8 7 0.5152198729 6418385783 E 311 %G - 0.1099680204 0615001798 E 312 %G + 6 5 0.1809295237 1502506625 E 312 %G - 0.2172461015 8315085132 E 312 %G + 4 3 0.1733446261 3560586487 E 312 %G - 0.7597264055 3053863391 E 311 %G + 2 0.1050398077 8828775171 E 311 %G - 0.4443971993 2926345443 E 309 %G + - 0.7155498947 0116891371 E 291 ] Type: List Union(Fraction Polynomial Float,SuchThat(Symbol,Polynomial Float)) Time: 0.32 (IN) + 12.73 (EV) + 0.82 (OT) + 1.97 (GC) = 15.83 sec )clear properties rosser -- Eigenvalues of the generalized hypercompanion matrix of -- (x^5 + a4*x^4 + a3*x^3 + a2*x^2 + a1*x + a0)*(x^2 + x + 1)^2 -- => {[-1 +- sqrt(3) i]/2, [-1 +- sqrt(3) i]/2, -- RootsOf(x^5 + a4*x^4 + a3*x^3 + a2*x^2 + a1*x + a0)} matrix(@[[-a4, -a3, -a2, -a1, -a0, 0, 0, 0, 0], _ [ 1, 0, 0, 0, 0, 0, 0, 0, 0], _ [ 0, 1, 0, 0, 0, 0, 0, 0, 0], _ [ 0, 0, 1, 0, 0, 0, 0, 0, 0], _ [ 0, 0, 0, 1, 0, 0, 0, 0, 0], _ [ 0, 0, 0, 0, 0, -1, -1, 0, 0], _ [ 0, 0, 0, 0, 0, 1, 0, 0, 0], _ [ 0, 0, 0, 0, 0, 0, 1, -1, -1], _ [ 0, 0, 0, 0, 0, 0, 0, 1, 0]]); Type: Matrix Polynomial Integer Time: 0.15 (IN) + 0.03 (OT) = 0.18 sec radicalEigenvalues(%) +---+ +---+ +---+ +---+ - \|- 3 - 1 - \|- 3 - 1 \|- 3 - 1 \|- 3 - 1 (38) [------------,------------,----------,----------] 2 2 2 2 Type: List Expression Integer Time: 0.17 (IN) + 0.07 (EV) + 0.03 (OT) = 0.27 sec -- Eigenvalues and eigenvectors => lambda = {a, a, a, 1 - i, 1 + i}, -- eigenvectors = @[[1 0 0 0 0], [0 0 1 0 0], [0 0 0 1 0], -- [0, (1 + i)/2, 0, 0, 1], [0, (1 - i)/2, 0, 0, 1]]^T matrix(@[[a, 0, 0, 0, 0], _ [0, 0, 0, 0, 1], _ [0, 0, a, 0, 0], _ [0, 0, 0, a, 0], _ [0, -2, 0, 0, 2]]); Type: Matrix Polynomial Integer Time: 0.05 (IN) = 0.05 sec radicalEigenvectors(%) (40) + 0 + | | | +---+ | |- \|- 1 + 1| +---+ |------------| @[[radval= \|- 1 + 1,radmult= 1,radvect= [| 2 |]], | | | 0 | | | | 0 | | | + 1 + + 0 + | | | +---+ | |\|- 1 + 1| +---+ |----------| [radval= - \|- 1 + 1,radmult= 1,radvect= [| 2 |]], | | | 0 | | | | 0 | | | + 1 + +0+ +0+ +1+ | | | | | | |0| |0| |0| | | | | | | [radval= a,radmult= 3,radvect= [|0|,|1|,|0|]]] | | | | | | |1| |0| |0| | | | | | | +0+ +0+ +0+ Type: List Record(radval: Expression Integer,radmult: Integer,radvect: List Matrix Expression Integer) Time: 0.03 (IN) + 0.17 (EV) + 0.08 (OT) = 0.28 sec -- Eigenvalues and generalized eigenvectors [Johnson and Riess, p. 193] -- => lambda = {1, 1, 1}, eigenvectors = @[[4 -1 4], [1 -1 2], [3 -1 3]]^T matrix(@[[-1, -8, 1], _ [-1, -3, 2], _ [-4, -16, 7]]); Type: Matrix Integer Time: 0 sec radicalEigenvectors(%) +- 4+ | | (42) @[[radval= 1,radmult= 3,radvect= [| 1 |]]] | | + 0 + Type: List Record(radval: Expression Integer,radmult: Integer,radvect: List Matrix Expression Integer) Time: 0.02 (IN) + 0.02 (EV) + 0.02 (OT) = 0.05 sec -- Eigenvalues and generalized eigenvectors [Johnson and Riess, p. 199] -- => lambda = {1, 1, 1, 1, 2, 2}, eigenvectors = -- @[[1 -1 0 0 0 0], [-1 0 0 1 0 0], [0 0 1 -1 0 -1], -- [0 0 -1 -2 -1 3], [ 0 2 0 0 0 0], [2 0 1 1 0 0]]^T matrix(@[[1, 0, 1, 1, 0, 1], _ [1, 2, 0, 0, 0, 0], _ [0, 0, 2, 0, 1, 1], _ [0, 0, 1, 1, 0, 0], _ [0, 0, 0, 0, 1, 0], _ [0, 0, 0, 0, 1, 1]]); Type: Matrix Integer Time: 0.02 (IN) = 0.02 sec radicalEigenvectors(%) (44) +0+ | | |1| | | |0| @[[radval= 2,radmult= 2,radvect= [| |]], |0| | | |0| | | +0+ +- 1+ | | | 1 | | | | 0 | [radval= 1,radmult= 4,radvect= [| |]]] | 0 | | | | 0 | | | + 0 + Type: List Record(radval: Expression Integer,radmult: Integer,radvect: List Matrix Expression Integer) Time: 0.05 (IN) + 0.02 (EV) + 0.05 (OT) = 0.12 sec -- Jordan form => diag(@[[1 1],[0 1]], [[1 1],[0 1]], -1) [Gantmacher, p. 172] matrix(@[[1, 0, 0, 1, -1], _ [0, 1, -2, 3, -3], _ [0, 0, -1, 2, -2], _ [1, -1, 1, 0, 1], _ [1, -1, 1, -1, 2]]); Type: Matrix Integer Time: 0 sec -- Smith normal form => @[[1, 0], [0, x^4 - x^2 + 1]] [Cullen, p. 230] matrix(@[[x**2, x - 1], [x + 1, x**2]]) + 2 + | x x - 1| (46) | | | 2 | +x + 1 x + Type: Matrix Polynomial Integer Time: 0.03 (IN) = 0.03 sec -- Matrix exponential => e @[[cos 2, -sin 2], [sin 2, cos 2]] exp(matrix(@[[1, -2], [2, 1]])) There are 2 exposed and 6 unexposed library operations named exp having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op exp to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named exp with argument type(s) Matrix Integer -- Matrix exponential [Rick Niles] => -- @[[1, 4 sin(w t)/w - 3 t , 6 [w t - sin(w t)], 2/w [1 - cos(w t)] ], -- [0, 4 cos(w t) - 3 , 6 w [1 - cos(w t)], 2 sin(w t) ], -- [0, -2/w [1 - cos(w t)], 4 - 3 cos(w t) , sin(w t)/w ], -- [0, -2 sin(w t) , 3 w sin(w t) , cos(w t) ]] matrix(@[[0, 1, 0, 0 ], _ [0, 0, 0, 2*w], _ [0, 0, 0, 1 ], _ [0, -2*w, 3*w**2, 0 ]]); Type: Matrix Polynomial Integer Time: 0.05 (IN) = 0.05 sec exp(%*t) There are 2 exposed and 6 unexposed library operations named exp having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op exp to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named exp with argument type(s) Matrix Polynomial Integer -- Sine of a Jordan matrix => diag(@[[sin a, cos a],[0, sin a]], sin b, -- @[[sin c, cos c, -sin(c)/2],[0, sin c, cos c],[0, 0, sin c]]) -- See F. R. Gantmacher, _The Theory of Matrices_, Volume One, Chelsea -- Publishing Company, 1977, p. 100 to see how to do a general function. matrix(@[[a, 1, 0, 0, 0, 0], _ [0, a, 0, 0, 0, 0], _ [0, 0, b, 0, 0, 0], _ [0, 0, 0, c, 1, 0], _ [0, 0, 0, 0, c, 1], _ [0, 0, 0, 0, 0, c]]); Type: Matrix Polynomial Integer Time: 0.03 (IN) + 0.03 (OT) = 0.07 sec sin(%) There are 2 exposed and 6 unexposed library operations named sin having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op sin to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named sin with argument type(s) Matrix Polynomial Integer -- Sine of a matrix => @[[1 0 0], [0 1 0], [0 0 1]] [Cullen, p. 261] %pi/2*matrix(@[[2, 1, 1], [2, 3, 2], [1, 1, 2]]); Type: Matrix Pi Time: 0.30 (IN) + 0.05 (OT) = 0.35 sec sin(%) There are 2 exposed and 6 unexposed library operations named sin having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op sin to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named sin with argument type(s) Matrix Pi -- Matrix square root => {+-@[[3 1], [1 4]], +-1/sqrt(5) @[[-1 7], [7 6]]} matrix(@[[10, 7], [7, 17]]) +10 7 + (50) | | +7 17+ Type: Matrix Integer Time: 0 sec sqrt(%) There are 2 exposed and 0 unexposed library operations named sqrt having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op sqrt to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named sqrt with argument type(s) Matrix Integer -- Square root of a non-singular matrix [Gantmacher, p. 233] -- => @[[e, (e - n) v w + e/2, (n - e) v], [0, e, 0], [0, (e - n) w, n] -- for arbitrary v and w with arbitrary signs e and n = +-1 matrix(@[[1, 1, 0], [0, 1, 0], [0, 0, 1]]); Type: Matrix Integer Time: 0 sec sqrt(%) There are 2 exposed and 0 unexposed library operations named sqrt having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op sqrt to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named sqrt with argument type(s) Matrix Integer -- Square root of a singular matrix [Gantmacher, p. 239] -- => @[[0 a b], [0 0 0], [0 1/b 0]] for arbitrary a and b matrix(@[[0, 1, 0], [0, 0, 0], [0, 0, 0]]); Type: Matrix Integer Time: 0.02 (IN) = 0.02 sec sqrt(%) There are 2 exposed and 0 unexposed library operations named sqrt having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op sqrt to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named sqrt with argument type(s) Matrix Integer -- Singular value decomposition -- => [1/sqrt(14) 3/sqrt(10) 1/sqrt(35) ] [2 sqrt(7) 0] [1/sqrt(2) 1/sqrt(2)] -- [2/sqrt(14) 0 -sqrt(5/7)] [0 0] [1/sqrt(2) -1/sqrt(2)] -- [3/sqrt(14) -1/sqrt(10) 3/sqrt(35) ] [0 0] -- = U Sigma V^T --- singular values are [2 sqrt(7), 0] matrix(@[[1, 1], [2, 2], [3, 3]]) +1 1+ | | (53) |2 2| | | +3 3+ Type: Matrix Integer Time: 0.02 (OT) = 0.02 sec -- Jacobian of (r cos t, r sin t) => @[[cos t, -r sin t], [sin t, r cos t]] [r*cos(t), r*sin(t)] (54) [r cos(t),r sin(t)] Type: List Expression Integer Time: 0.07 (IN) + 0.02 (OT) = 0.08 sec -- Hessian of r^2 sin t => @[[2 sin t, 2 r cos t], [2 r cos t, -r^2 sin t]] r**2*sin(t) 2 (55) r sin(t) Type: Expression Integer Time: 0.08 (IN) + 0.02 (OT) = 0.10 sec -- Wronskian of (cos t, sin t) => @[[cos t, sin t], [-sin t, cos t]] [cos(t), sin(t)] (56) [cos(t),sin(t)] Type: List Expression Integer Time: 0.02 (IN) = 0.02 sec -- How easy is it to define functions to do the last three operations? -- Jacobian of (r cos t, r sin t) => @[[cos t, -r sin t], [sin t, r cos t]] MYjacobian(e, v) == matrix(@[[D(f, x) for x in v] for f in e]) Type: Void Time: 0 sec MYjacobian([r*cos(t), r*sin(t)], [r, t]) Compiling function MYjacobian with type (List Expression Integer, List OrderedVariableList [r,t]) -> Matrix Expression Integer +cos(t) - r sin(t)+ (58) | | +sin(t) r cos(t) + Type: Matrix Expression Integer Time: 0.08 (IN) + 0.02 (EV) + 0.05 (OT) = 0.15 sec -- Hessian of r^2 sin t => @[[2 sin t, 2 r cos t], [2 r cos t, -r^2 sin t]] MYhessian(f, x) == local n n:= #x matrix(@[[D(f, [x . i, x . j]) for j in 1..n] for i in 1..n]) Type: Void Time: 0 sec MYhessian(r**2*sin(t), [r, t]) Compiling function MYhessian with type (Expression Integer,List OrderedVariableList [r,t]) -> Matrix Expression Integer + 2sin(t) 2r cos(t) + (60) | | | 2 | +2r cos(t) - r sin(t)+ Type: Matrix Expression Integer Time: 0.13 (IN) + 0.03 (EV) + 0.03 (OT) = 0.20 sec -- Wronskian of (cos t, sin t) => @[[cos t, sin t], [-sin t, cos t]] MYwronskian(f, x) == local n n:= #f matrix(@[[D(f . j, x, i-1) for j in 1..n] for i in 1..n]) Type: Void Time: 0 sec MYwronskian([cos(t), sin(t)], t) Your expression cannot be fully compiled because it contains an integer expression (for - ) whose sign cannot be determined (in general) and so must be specified by you. Perhaps you can try substituting something like (- :: PI) or (- :: NNI) into your expression for - . AXIOM will attempt to step through and interpret the code. + cos(t) sin(t)+ (62) | | +- sin(t) cos(t)+ Type: Matrix Expression Integer Time: 0.15 (IN) + 0.05 (EV) = 0.20 sec -- ---------- Quit ---------- )quit real 160.6 user 78.2 sys 2.1 \end{verbatim} \section{Tensor Analysis} \begin{verbatim} -- ---------- Tensor Analysis ---------- -- Generalized Kronecker delta: delta([j, h], [i, k]) = -- delta(j, i) delta(h, k) - delta(j, k) delta(h, i). See David Lovelock and -- Hanno Rund, _Tensors, Differential Forms, & Variational Principles_, John -- Wiley & Sons, Inc., 1975, p. 109. delta : CartesianTensor(1, 4, Integer) := kroneckerDelta() +1 0 0 0+ | | |0 1 0 0| (1) | | |0 0 1 0| | | +0 0 0 1+ Type: CartesianTensor(1,4,Integer) Time: 0.02 (EV) + 0.03 (OT) + 0.02 (GC) = 0.07 sec delta([i, k], [j, h]) There are no exposed library operations named delta but there is one unexposed operation with that name. Use HyperDoc Browse or issue )display op delta to learn more about the available operation. Cannot find a definition or applicable library operation named delta with argument type(s) List OrderedVariableList [i,k] List OrderedVariableList [j,h] )clear properties delta -- Levi-Civita symbol: [epsilon(2,1,3), epsilon(1,3,1)] => [-1, 0] epsilon : CartesianTensor(1, 3, Integer) := leviCivitaSymbol() +0 0 0+ +0 0 - 1+ + 0 1 0+ | | | | | | (2) [|0 0 1|,|0 0 0 |,|- 1 0 0|] | | | | | | +0 - 1 0+ +1 0 0 + + 0 0 0+ Type: CartesianTensor(1,3,Integer) Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec [epsilon(2, 1, 3), epsilon(1, 3, 1)] (3) [- 1,0] Type: List Integer Time: 0.02 (IN) + 0.02 (EV) + 0.02 (GC) = 0.05 sec )clear properties epsilon -- Tensor outer product: @[[ 5 6] [-10 -12]] -- [1 -2] [ 5 6] @[[ -7 8] [ 14 -16]] -- ij ij [3 4] X [-7 8] = [ ] -- c = a b @[[ 15 18] [ 20 24]] -- kl kl @[[-21 24] [-28 32]] a:= matrix(@[[1, -2], [3, 4]]); Type: Matrix Integer Time: 0.02 (IN) + 0.02 (EV) + 0.02 (OT) + 0.03 (GC) = 0.09 sec b:= matrix(@[[5, 6], [-7, 8]]); Type: Matrix Integer Time: 0.02 (IN) = 0.02 sec product(a :: CartesianTensor(1, 2, Integer), _ b :: CartesianTensor(1, 2, Integer)) + + 5 6+ +- 10 - 12++ | | | | || | +- 7 8+ + 14 - 16+| (6) | | |+ 15 18+ + 20 24+ | || | | | | ++- 21 24+ +- 28 32+ + Type: CartesianTensor(1,2,Integer) Time: 0.15 (IN) + 0.08 (OT) = 0.23 sec )clear properties a b -- Definition of the Christoffel symbol of the first kind (a is the metric -- tensor) [Lovelock and Rund, p. 81] -- d a d a d a -- 1 kh hl lk -- Chr1 = - (----- + ----- - -----) -- lhk 2 l k h -- d x d x d x -- Partial covariant derivative of a type (1, 1) tensor field (Chr2 is the -- Christoffel symbol of the second kind) [Lovelock and Rund, p. 77] -- i d i i m m i -- T = ---- T + Chr2 T - Chr2 T -- j|k k j m k j j k m -- d x T:= operator('T); Type: BasicOperator Time: 0.08 (IN) + 0.02 (EV) + 0.02 (GC) = 0.12 sec T([i], [j]) (8) T([i],[j]) Type: Expression List Symbol Time: 0.65 (IN) + 0.02 (EV) + 0.12 (OT) + 0.05 (GC) = 0.83 sec -- Verify the Bianchi identity for a symmetric connection (K is the Riemann -- curvature tensor) [Lovelock and Rund, p. 94] -- h h h -- K + K + K = 0 -- i jk|l i kl|j i lj|k -- ---------- Quit ---------- )quit real 4.6 user 2.7 sys 0.2 \end{verbatim} \section{Sums} \begin{verbatim} -- ---------- Sums ---------- -- Simplify the sum below to sum(x[i]^2, i = 1..n) - sum(x[i], i = 1..n)^2/n x:= operator('x); Type: BasicOperator Time: 0.07 (IN) + 0.10 (OT) = 0.17 sec xbar:= sum(x(j), j = 1..n) / n n --+ > x(j) --+ j= 1 (2) ---------- n Type: Expression Integer Time: 0.98 (IN) + 0.41 (EV) + 0.32 (OT) + 0.15 (GC) = 1.87 sec sum((x(i) - xbar)**2, i = 1..n) n 2 n --+ --+ 2 2 > x(j) - 2n x(i)> x(j) + n x(i) n --+ --+ --+ j= 1 j= 1 (3) > ----------------------------------------- --+ 2 i= 1 n Type: Expression Integer Time: 0.18 (IN) + 0.22 (EV) + 0.07 (OT) + 0.02 (GC) = 0.48 sec )clear properties x xbar -- Derivation of the least squares fitting of data points (x[i], y[i]) to a -- line y = m x + b. See G. Keady, ``Using Maple's linalg package with Zill -- and Cullen _Advanced Engineering Mathematics_, Part II: Vectors, Matrices -- and Vector Calculus'', University of Western Australia, -- ftp://maths.uwa.edu.au/pub/keady/ x:= operator('x); Type: BasicOperator Time: 0.02 (EV) = 0.02 sec y:= operator('y); Type: BasicOperator Time: 0.05 (IN) = 0.05 sec f:= sum((y(i) - m*x(i) - b)**2, i = 1..n); Type: Expression Integer Time: 0.17 (IN) + 0.15 (EV) + 0.02 (OT) = 0.33 sec solve([D(f, m) = 0, D(f, b) = 0], [m, b]) (7) @[[]] Type: List List Equation Expression Integer Time: 1.45 (IN) + 0.30 (EV) + 0.20 (OT) + 0.08 (GC) = 2.03 sec )clear properties x y f -- Indefinite sum => (-1)^n binomial(2 n, n). See Herbert S, Wilf, -- ``IDENTITIES and their computer proofs'', University of Pennsylvania. sum((-1)**k * binomial(2*n, k)**2, k) k 2 --+ %A 2n (8) > (- 1) ( ) --+ %A %A Type: Expression Integer Time: 0.17 (IN) + 0.90 (EV) + 0.03 (OT) + 0.03 (GC) = 1.13 sec -- Check whether the full Gosper algorithm is implemented -- => 1/2^(n + 1) binomial(n, k - 1) sum(binomial(n, k)/2**n - binomial(n + 1, k)/2**(n + 1), k) n n + 1 n + 1 n (- n + k - 2)2 ( ) + (n - k + 2)2 ( ) k - 1 k - 1 (9) ------------------------------------------------- n n + 1 (n - 2k + 3)2 2 Type: Expression Integer Time: 0.12 (IN) + 0.92 (EV) + 0.02 (OT) = 1.05 sec normalize(%) n! (10) ----------------------------------- (n + 1)log(2) %e (k - 1)!(n - k + 1)! Type: Expression Integer Time: 0.03 (IN) + 0.40 (EV) + 0.03 (OT) = 0.47 sec -- Dixon's identity (check whether Zeilberger's algorithm is implemented). -- Note that the indefinite sum is equivalent to the definite -- sum(..., k = -min(a, b, c)..min(a, b, c)) => (a + b + c)!/(a! b! c!) -- [Wilf] sum((-1)**k * binomial(a+b, a+k) * binomial(b+c, b+k) _ * binomial(c+a, c+k), k) k --+ %A b + a c + a c + b (11) > (- 1) ( )( )( ) --+ a + %A c + %A b + %A %A Type: Expression Integer Time: 0.27 (IN) + 6.12 (EV) + 0.05 (OT) + 1.02 (GC) = 7.45 sec -- Telescoping sum => g(n + 1) - g(0) g:= operator('g); Type: BasicOperator Time: 0.05 (IN) = 0.05 sec sum(g(k + 1) - g(k), k = 0..n) n --+ (13) > g(k + 1) - g(k) --+ k= 0 Type: Expression Integer Time: 0.25 (IN) + 0.05 (EV) + 0.03 (OT) = 0.33 sec )clear properties g -- => n^2 (n + 1)^2 / 4 sum(k**3, k = 1..n) 4 3 2 n + 2n + n (14) ------------- 4 Type: Fraction Polynomial Integer Time: 0.03 (IN) + 0.12 (EV) + 0.02 (OT) = 0.17 sec factorFraction(%) 2 2 n (n + 1) (15) ---------- 4 Type: Fraction Factored Polynomial Integer Time: 0.02 (IN) + 0.08 (EV) + 0.02 (OT) = 0.12 sec -- See Daniel I. A. Cohen, _Basic Techniques of Combinatorial Theory_, John -- Wiley and Sons, 1978, p. 60. The following two sums can be derived directly -- from the binomial theorem: -- sum(k^2 * binomial(n, k) * x^k, k = 1..n) = n x (1 + n x) (1 + x)^(n - 2) -- => n (n + 1) 2^(n - 2) [Cohen, p. 60] sum(k**2 * binomial(n, k), k = 1..n) n --+ 2 n (16) > k ( ) --+ k k= 1 Type: Expression Integer Time: 0.10 (IN) + 0.20 (EV) + 0.02 (OT) = 0.32 sec -- => [2^(n + 1) - 1]/(n + 1) [Cohen, p. 83] sum(binomial(n, k)/(k + 1), k = 0..n) n ( ) n k --+ (17) > ----- --+ k + 1 k= 0 Type: Expression Integer Time: 0.03 (IN) + 0.20 (EV) + 0.02 (OT) = 0.25 sec -- Vandermonde's identity => binomial(n + m, r) [Cohen, p. 31] sum(binomial(n, k) * binomial(m, r - k), k = 0..r) r --+ m n (18) > ( )( ) --+ r - k k k= 0 Type: Expression Integer Time: 0.08 (IN) + 0.55 (EV) + 0.03 (OT) = 0.67 sec -- => Fibonacci[2 n] [Cohen, p. 88] sum(binomial(n, k) * fibonacci(k), k = 0..n) There are 1 exposed and 0 unexposed library operations named fibonacci having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op fibonacci to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named fibonacci with argument type(s) Variable k -- => Fibonacci[n] Fibonacci[n + 1] [Cohen, p. 65] sum(fibonacci(k)**2, k = 1..n) There are 1 exposed and 0 unexposed library operations named fibonacci having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op fibonacci to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named fibonacci with argument type(s) Variable k -- => 1/2 cot(x/2) - cos([2 n + 1] x/2)/[2 sin(x/2)] -- See Konrad Knopp, _Theory and Application of Infinite Series_, Dover -- Publications, Inc., 1990, p. 480. sum(sin(k*x), k = 1..n) n --+ (19) > sin(k x) --+ k= 1 Type: Expression Integer Time: 0.07 (IN) + 0.27 (EV) + 0.03 (OT) = 0.37 sec -- => sin(n x)^2/sin x [Gradshteyn and Ryzhik 1.342(3)] sum(sin((2*k - 1)*x), k = 1..n) n --+ (20) > sin((2k - 1)x) --+ k= 1 Type: Expression Integer Time: 0.05 (IN) + 0.20 (EV) + 0.03 (OT) = 0.28 sec -- => Fibonacci[n + 1] [Cohen, p. 87] sum(binomial(n - k, k), k = 0..floor(n/2)) There are 2 exposed and 0 unexposed library operations named floor having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op floor to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named floor with argument type(s) Polynomial Fraction Integer -- => pi^2 / 6 + zeta(3) =~ 2.84699 limit(sum(1/k**2 + 1/k**3, k = 1..n), n = %plusInfinity) (21) "failed" Type: Union("failed",...) Time: 0.45 (IN) + 1.0 (EV) + 0.05 (OT) + 0.55 (GC) = 2.05 sec -- => pi^2/12 - 1/2 (log 2)^2 [Gradshteyn and Ryzhik 0.241(2)] limit(sum(1/(2**k*k**2), k = 1..n), n = %plusInfinity) (22) "failed" Type: Union("failed",...) Time: 0.10 (IN) + 0.32 (EV) + 0.07 (OT) = 0.48 sec -- => pi/12 sqrt(3) - 1/4 log 3 [Knopp, p. 268] limit(sum(1/((3*k + 1)*(3*k + 2)*(3*k + 3)), k = 0..n), n = %plusInfinity) (23) "failed" Type: Union("failed",...) Time: 0.10 (IN) + 0.30 (EV) + 0.02 (OT) = 0.42 sec -- => 1/2 (2^(n - 1) + 2^(n/2) cos(n pi/4)) [Gradshteyn and Ryzhik 0.153(1)] limit(sum(binomial(n, 4*k), k = 0..nn), nn = %plusInfinity) (24) "failed" Type: Union("failed",...) Time: 0.13 (IN) + 4.83 (EV) + 0.02 (OT) + 0.53 (GC) = 5.52 sec -- => 1 [Knopp, p. 233] limit(sum(1/(sqrt(k*(k + 1)) * (sqrt(k) + sqrt(k + 1))), k = 1..n), _ n = %plusInfinity) (25) "failed" Type: Union("failed",...) Time: 0.10 (IN) + 0.58 (EV) + 0.03 (OT) = 0.72 sec -- => 1/sqrt([1 - x y]^2 - 4 x^2) (| x y | < 1 and -1 <= x < 1). -- From Evangelos A. Coutsias, Michael J. Wester and Alan S. Perelson, ``A -- Nucleation Theory of Cell Surface Capping'', draft. limit(sum(sum(binomial(n, k)*binomial(n - k, n - 2*k)*x**n*y**(n - 2*k), _ k = 0..floor(n/2)), _ n = 0..nn), nn = %plusInfinity) There are 2 exposed and 0 unexposed library operations named floor having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op floor to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named floor with argument type(s) Polynomial Fraction Integer -- An equivalent summation to the above is: limit(limit(sum(sum(factorial(n)/(factorial(k)**2*factorial(n - 2*k))* _ (x/y)**k*(x*y)**(n - k), _ n = 2*k..nn), _ k = 0..mm), _ mm = %plusInfinity), _ nn = %plusInfinity) There are 3 exposed and 0 unexposed library operations named limit having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op limit to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named limit with argument type(s) failed Equation OrderedCompletion Polynomial Integer -- => pi/2 [Knopp, p. 269] limit(sum(product(k/(2*k - 1), k = 1..m), m = 2..n), n = %plusInfinity) (26) "failed" Type: Union("failed",...) Time: 0.37 (IN) + 0.33 (EV) + 0.05 (OT) = 0.75 sec -- ---------- Quit ---------- )quit real 210.8 user 33.1 sys 1.6 \end{verbatim} \section{Products} \begin{verbatim} -- ---------- Products ---------- -- => [640 pi^3]/[2187 sqrt(3)] [Gradshteyn and Ryzhik 8.338(5)] product(Gamma(k/3), k = 1..8) _ 1 _ 2 _ 4 _ 5 _ 7 _ 8 (1) | (-)| (-)| (-)| (-)| (-)| (-) 3 3 3 3 3 3 Type: Expression Integer Time: 1.18 (IN) + 0.25 (EV) + 0.42 (OT) + 0.18 (GC) = 2.03 sec -- => n! = gamma(n + 1) product(k, k = 1..n) n ++-++ (2) | | k | | k= 1 Type: Expression Integer Time: 0.45 (IN) + 0.07 (OT) = 0.52 sec -- => x^[n (n + 1)/2] product(x**k, k = 1..n) n ++-++ k (3) | | x | | k= 1 Type: Expression Integer Time: 0.07 (IN) + 0.02 (EV) + 0.03 (OT) = 0.12 sec -- => n product((1 + 1/k), k = 1..n - 1) n - 1 ++-++ k + 1 (4) | | ----- | | k k= 1 Type: Expression Integer Time: 0.27 (IN) + 0.03 (EV) + 0.03 (OT) = 0.33 sec -- => 1/2^(2 n) binomial(2 n, n) [Knopp, p. 385] product((2*k - 1)/(2*k), k = 1..n) n ++-++ 2k - 1 (5) | | ------ | | 2k k= 1 Type: Expression Integer Time: 0.07 (IN) + 0.02 (EV) + 0.02 (OT) = 0.10 sec -- => [x^(2 n) - 1]/(x^2 - 1) [Gradshteyn and Ryzhik 1.396(1)] product(x**2 - 2*x*cos(k*%pi/n) + 1, k = 1..n - 1) n - 1 ++-++ k %pi 2 (6) | | - 2x cos(-----) + x + 1 | | n k= 1 Type: Expression Integer Time: 0.56 (IN) + 0.07 (EV) + 0.18 (OT) + 0.03 (GC) = 0.85 sec -- => 2/3 [Knopp, p. 228] limit(product((k**3 - 1)/(k**3 + 1), k = 2..n), n = %plusInfinity) (7) "failed" Type: Union("failed",...) Time: 0.45 (IN) + 0.27 (EV) + 0.09 (OT) + 0.03 (GC) = 0.83 sec -- => 2/pi [Gradshteyn and Ryzhik 0.262(2)] limit(product(1 - 1/(2*k)**2, k = 1..n), n = %plusInfinity) (8) "failed" Type: Union("failed",...) Time: 0.10 (IN) + 0.08 (EV) + 0.03 (OT) = 0.22 sec -- => sqrt(2) [Gradshteyn and Ryzhik 0.261] limit(product(1 + (-1)**(k + 1)/(2*k - 1), k = 1..n), n = %plusInfinity) (9) "failed" Type: Union("failed",...) Time: 0.08 (IN) + 0.17 (EV) = 0.25 sec -- => -1 [Knopp, p. 436] limit(product((k*(k + 1) + 1 + %i)/(k*(k + 1) + 1 - %i), k = 0..n), _ n = %plusInfinity) (10) "failed" Type: Union("failed",...) Time: 0.98 (IN) + 0.50 (EV) + 0.20 (OT) + 0.03 (GC) = 1.72 sec -- ---------- Quit ---------- )quit real 23.4 user 7.8 sys 0.5 \end{verbatim} \section{Limits} \begin{verbatim} -- ---------- Limits ---------- -- Start with a famous example => e limit((1 + 1/n)**n, n = %plusInfinity) (1) %e Type: Union(OrderedCompletion Expression Integer,...) Time: 1.07 (IN) + 0.58 (EV) + 0.28 (OT) + 0.25 (GC) = 2.18 sec -- => 1/2 limit((1 - cos(x))/x**2, x = 0) 1 (2) - 2 Type: Union(OrderedCompletion Expression Integer,...) Time: 0.38 (IN) + 0.12 (EV) + 0.03 (OT) = 0.53 sec -- See Dominik Gruntz, _On Computing Limits in a Symbolic Manipulation System_, -- Ph.D. dissertation, Swiss Federal Institute of Technology, Zurich, -- Switzerland, 1996. => 5 limit((3**x + 5**x)**(1/x), x = %plusInfinity) (3) "failed" Type: Union("failed",...) Time: 0.21 (IN) + 0.45 (EV) + 0.02 (OT) + 0.02 (GC) = 0.70 sec -- => 1 limit(log(x)/(log(x) + sin(x)), x = %plusInfinity) (4) "failed" Type: Union("failed",...) Time: 0.05 (IN) + 1.62 (EV) + 0.03 (OT) + 0.02 (GC) = 1.72 sec -- => - e^2 [Gruntz] limit((exp(x*exp(-x)/(exp(-x) + exp(-2*x**2/(x + 1)))) - exp(x))/x, _ x = %plusInfinity) (5) "failed" Type: Union("failed",...) Time: 0.17 (IN) + 4.52 (EV) + 0.03 (OT) + 0.52 (GC) = 5.23 sec -- => 1/3 [Gruntz] limit(x*log(x)*log(x*exp(x) - x**2)**2/log(log(x**2 + _ 2*exp(exp(3*x**3*log(x))))), _ x = %plusInfinity) (6) "failed" Type: Union("failed",...) Time: 0.17 (IN) + 2.21 (EV) + 0.03 (OT) = 2.42 sec -- => 1/e [Knopp, p. 73] limit(1/n * factorial(n)**(1/n), n = %plusInfinity) (7) "failed" Type: Union("failed",...) Time: 0.08 (IN) + 1.45 (EV) + 0.07 (OT) = 1.60 sec -- Rewrite the above problem slightly => 1/e limit(1/n * Gamma(n + 1)**(1/n), n = %plusInfinity) (8) "failed" Type: Union("failed",...) Time: 0.12 (IN) + 1.50 (EV) + 0.05 (OT) + 0.02 (GC) = 1.68 sec -- => 1 [Gradshteyn and Ryzhik 8.328(2)] limit(Gamma(z + a)/Gamma(z)*exp(-a*log(z)), z = %plusInfinity) (9) "failed" Type: Union("failed",...) Time: 0.32 (IN) + 2.22 (EV) + 0.08 (OT) = 2.62 sec -- => e^z [Gradshteyn and Ryzhik 9.121(8)] --limit(hypergeometric([1, k], [1], z/k), k = %plusInfinity) -- => Euler's_constant [Gradshteyn and Ryzhik 9.536] --limit(zeta(x) - 1/(x - 1), x = 1) -- => gamma(x) [Knopp, p. 385] limit(n**x/(x * product((1 + x/k), k = 1..n)), n = %plusInfinity) (10) "failed" Type: Union("failed",...) Time: 0.40 (IN) + 3.75 (EV) + 0.13 (OT) + 0.47 (GC) = 4.75 sec -- See Angus E. Taylor and W. Robert Mann, _Advanced Calculus_, Second Edition, -- Xerox College Publishing, 1972, p. 125 => 1 limit(x * integrate(exp(-t**2), t = 0..x)/(1 - exp(-x**2)), x = 0) (11) "failed" Type: Union("failed",...) Time: 0.65 (IN) + 2.63 (EV) + 0.08 (OT) + 0.05 (GC) = 3.42 sec -- => [-1, 1] [limit(x/abs(x), x = 0, "left"), limit(x/abs(x), x = 0, "right")] (12) [- 1,1] Type: List Union(OrderedCompletion Expression Integer,"failed") Time: 0.23 (IN) + 0.30 (EV) + 0.05 (OT) = 0.58 sec -- => pi/2 [Richard Q. Chen] limit(atan(-log(x)), x = 0, "right") %pi (13) --- 2 Type: Union(OrderedCompletion Expression Integer,...) Time: 0.03 (IN) + 0.72 (EV) = 0.75 sec -- ---------- Quit ---------- )quit real 85.9 user 28.1 sys 0.7 \end{verbatim} \section{Calculus} \begin{verbatim} -- ---------- Calculus ---------- -- Calculus on a non-smooth (but well defined) function => x/|x| or sign(x) D(abs(x), x) abs(x) (1) ------ x Type: Expression Integer Time: 0.20 (IN) + 0.30 (EV) + 0.22 (OT) + 0.20 (GC) = 0.92 sec -- Calculus on a piecewise defined function a(x) == if x < 0 then -x else x Type: Void Time: 0 sec -- => if x < 0 then -1 else 1 D(a(x), x) Compiling function a with type Variable x -> Polynomial Integer (3) 1 Type: Polynomial Integer Time: 0.28 (IN) + 0.03 (OT) = 0.32 sec )clear properties a Compiled code for a has been cleared. -- Derivative of a piecewise defined function at a point [Herbert Fischer]. -- f(x) = x^2 - 1 for x = 1 otherwise x^3. f(1) = 0 and f'(1) = 3 f(x) == if x = 1 then x**2 - 1 else x**3 Type: Void Time: 0.02 (IN) = 0.02 sec f(1) Compiling function f with type PositiveInteger -> Integer (5) 0 Type: NonNegativeInteger Time: 0.10 (IN) + 0.02 (EV) = 0.12 sec D(f(x), x) Compiling function f with type Variable x -> Polynomial Integer 2 (6) 3x Type: Polynomial Integer Time: 0.18 (IN) + 0.08 (OT) = 0.27 sec subst(%, x = 1) (7) 3 Type: Expression Integer Time: 0.25 (IN) + 0.03 (EV) + 0.05 (OT) = 0.33 sec )clear properties f Compiled code for f has been cleared. -- d^n/dx^n(x^n) => n! D(x**n, x, n) There are 3 exposed and 0 unexposed library operations named D having 3 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op D to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named D with argument type(s) Expression Integer Variable x Variable n -- Apply the chain rule---this is important for PDEs and many other -- applications => y_xx (x_t)^2 + y_x x_tt x:= operator('x); Type: BasicOperator Time: 0.05 (IN) + 0.02 (OT) = 0.07 sec y:= operator('y); Type: BasicOperator Time: 0.05 (IN) = 0.05 sec D(y(x(t)), t, 2) , 2 ,, , ,, (10) x (t) y (x(t)) + y (x(t))x (t) Type: Expression Integer Time: 0.42 (IN) + 0.05 (EV) + 0.12 (OT) = 0.58 sec )clear properties x y -- => f(h(x)) dh/dx - f(g(x)) dg/dx f:= operator('f); Type: BasicOperator Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec g:= operator('g); Type: BasicOperator Time: 0.05 (IN) = 0.05 sec h:= operator('h); Type: BasicOperator Time: 0.03 (IN) + 0.02 (OT) = 0.05 sec 'integrate(f(y), y = g(x)..h(x)) (14) integrate(f(y),y= g(x),h(x) ) Type: OutputForm Time: 0 sec D(%, x) There are 5 exposed and 0 unexposed library operations named D having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op D to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named D with argument type(s) OutputForm Variable x )clear properties f g h -- Exact differential => d(V(P, T)) => dV/dP DP + dV/dT DT V:= operator('V); Type: BasicOperator Time: 0.03 (IN) + 0.02 (OT) = 0.05 sec D(V(P, T)) There are 2 exposed and 0 unexposed library operations named D having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op D to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named D with argument type(s) Expression Integer )clear properties V -- Implicit differentiation => dy/dx = [1 - y sin(x y)] / [1 + x sin(x y)] y = cos(x*y) + x (16) y= cos(x y) + x Type: Equation Expression Integer Time: 0.22 (IN) + 0.12 (EV) + 0.05 (OT) + 0.02 (GC) = 0.40 sec -- => 2 (x + y) g'(x^2 + y^2) f:= operator('f); Type: BasicOperator Time: 0.02 (OT) = 0.02 sec g:= operator('g); Type: BasicOperator Time: 0.02 (OT) = 0.02 sec D(f(x, y), x) + D(f(x, y), y) (19) f (x,y) + f (x,y) ,2 ,1 Type: Expression Integer Time: 0.05 (IN) + 0.03 (EV) + 0.02 (OT) = 0.10 sec subst(%, f(x, y) = g(x**2 + y**2)) (20) f (x,y) + f (x,y) ,2 ,1 Type: Expression Integer Time: 0.10 (IN) + 0.02 (OT) = 0.12 sec )clear properties f g -- Residue => - 9/4 --residue((z**3 + 5)/((z**4 - 1)*(z + 1)), z, -1) -- Differential forms DeRham:= DERHAM(Integer, [x, y, z]) (21) DeRhamComplex(Integer,[x,y,z]) Type: Domain Time: 0.02 (IN) = 0.02 sec [dx, dy, dz]:= [generator(i)$DeRham for i in 1..3] (22) [dx,dy,dz] Type: List DeRhamComplex(Integer,[x,y,z]) Time: 0.02 (EV) + 0.02 (OT) = 0.03 sec -- (2 dx + dz) /\ (3 dx + dy + dz) /\ (dx + dy + 4 dz) => 8 dx /\ dy /\ dz (2*dx + dz) * (3*dx + dy + dz) * (dx + dy + 4*dz) (23) 8dx dy dz Type: DeRhamComplex(Integer,[x,y,z]) Time: 0.03 (IN) + 0.02 (EV) = 0.05 sec -- d(3 x^5 dy /\ dz + 5 x y^2 dz /\ dx + 8 z dx /\ dy) -- => (15 x^4 + 10 x y + 8) dx /\ dy /\ dz totalDifferential( _ 3*x**5 * dy * dz + 5*x*y**2 * dz * dx + 8*z * dx * dy :: DeRham) Internal Error The function * with signature hashcode is missing from domain Polynomial(DeRhamComplex (Integer) (x y z)) initial (24) -> real 12.1 user 5.6 sys 0.4 ------------------------------------------------------------------------------- Sat Jul 11 23:35:58 MET DST 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Calculus ---------- -- => 1 - 3/8 2^(1/3) = 0.5275296 --minimize(x**4 - x + 1) -- => [0, 1] --[minimize(1/(x**2 + y**2 + 1)), maximize(1/(x**2 + y**2 + 1))] -- Minimize on [-1, 1] x [-1, 1]: -- => min(a - b - c + d, a - b + c - d, a + b - c - d, a + b + c + d) --minimize(a + b*x + c*y + d*x*y, [x = -1..1, y = -1..1]) -- => [-1, 1] --[minimize(x**2*y**3, [x = -1..1, y = -1..1]), _ -- maximize(x**2*y**3, [x = -1..1, y = -1..1])] -- Linear programming: minimize the objective function z subject to the -- variables xi being non-negative along with an additional set of constraints. -- See William R. Smythe, Jr. and Lynwood A. Johnson, _Introduction to Linear -- Programming, with Applications_, Prentice Hall, Inc., 1966, p. 117: -- minimize z = 4 x1 - x2 + 2 x3 - 2 x4 => {x1, x2, x3, x4} = {2, 0, 2, 4} -- with zmin = 4 --simplex(-(4*x1 - x2 + 2*x3 - 2*x4), [2*x1 + x2 + x3 + x4 <= 10, _ -- x1 - 2*x2 - x3 + x4 >= 4, x1 + x2 + 3*x3 - x4 >= 4]) -- ---------- Quit ---------- )quit real 0.9 user 0.7 sys 0.1 \end{verbatim} \section{Indefinite Integrals} \begin{verbatim} -- ---------- Indefinite Integrals ---------- -- This integral only makes sense for x real => x |x|/2 integrate(abs(x), x) x ++ (1) | abs(%I)d%I ++ Type: Union(Expression Integer,...) Time: 0.33 (IN) + 0.62 (EV) + 0.13 (OT) + 0.22 (GC) = 1.30 sec -- Calculus on a piecewise defined function a(x) == if x < 0 then -x else x Type: Void Time: 0 sec -- => if x < 0 then -x^2/2 else x^2/2 integrate(a(x), x) Compiling function a with type Variable x -> Polynomial Integer 1 2 (3) - x 2 Type: Polynomial Fraction Integer Time: 0.30 (IN) + 0.02 (EV) + 0.02 (OT) = 0.33 sec )clear properties a Compiled code for a has been cleared. -- This would be very difficult to do by hand -- => 2^(1/3)/6 [1/2 log([x + 2^(1/3)]^2/[x^2 - 2^(1/3) x + 2^(2/3)]) -- + sqrt(3) arctan({[sqrt(3) x]/[2^(4/3) - x] or -- [2 x - 2^(1/3)]/[2^(1/3) sqrt(3)]}) -- [Gradshteyn and Ryzhik 2.126(1)] 1/(x**3 + 2) 1 (4) ------ 3 x + 2 Type: Fraction Polynomial Integer Time: 0.08 (IN) + 0.03 (OT) = 0.12 sec integrate(%, x) (5) +-+ 2 3+-+2 3+-+ +-+ 3+-+ - \|3 log(x \|4 - 2x\|4 + 4) + 2\|3 log(x\|4 + 2) + +-+3+-+ +-+ x\|3 \|4 - \|3 6atan(----------------) 3 / +-+3+-+ 6\|3 \|4 Type: Union(Expression Integer,...) Time: 0.10 (IN) + 0.75 (EV) + 0.07 (OT) + 0.07 (GC) = 0.98 sec D(%, x) 1 (6) ------ 3 x + 2 Type: Expression Integer Time: 0.12 (EV) = 0.12 sec -- This integral is easy if one realizes that 4^x = (2^x)^2 -- => arcsinh(2^x)/log(2) [Robert Israel in sci.math.symbolic] integrate(2**x/sqrt(1 + 4**x), x) >> Error detected within library code: integrate: implementation incomplete (constant residues) initial (7) -> real 15.2 user 3.8 sys 0.3 ------------------------------------------------------------------------------- Mon Feb 9 04:37:53 MET 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Indefinite Integrals ---------- -- => (-9 x^2 + 16 x - 41/5)/(2 x - 1)^(5/2) -- [Gradshteyn and Ryzhik 2.244(8)] integrate((3*x - 5)**2/(2*x - 1)**(7/2), x) 2 - 45x + 80x - 41 (1) ------------------------- 2 +------+ (20x - 20x + 5)\|2x - 1 Type: Union(Expression Integer,...) Time: 0.49 (IN) + 0.67 (EV) + 0.20 (OT) + 0.20 (GC) = 1.55 sec -- => 1/[2 m sqrt(10)] log([-5 + e^(m x) sqrt(10)]/[-5 - e^(m x) sqrt(10)]) -- [Gradshteyn and Ryzhik 2.314] integrate(1/(2*exp(m*x) - 5*exp(-m*x)), x) +--+ m x 2 m x +--+ 2\|10 (%e ) - 20%e + 5\|10 log(---------------------------------) m x 2 2(%e ) - 5 (2) -------------------------------------- +--+ 2m\|10 Type: Union(Expression Integer,...) Time: 0.15 (IN) + 0.75 (EV) + 0.08 (OT) + 0.03 (GC) = 1.02 sec -- => -3/2 x + 1/4 sinh(2 x) + tanh x [Gradshteyn and Ryzhik 2.423(24)] integrate(sinh(x)**4/cosh(x)**2, x) 3 2 sinh(x) + (3cosh(x) + 9)sinh(x) + (- 12x - 8)cosh(x) (3) ------------------------------------------------------ 8cosh(x) Type: Union(Expression Integer,...) Time: 0.13 (IN) + 0.17 (EV) + 0.02 (OT) = 0.32 sec simplify(%) 2 (cosh(x) + 2)sinh(x) + (- 3x - 2)cosh(x) (4) ----------------------------------------- 2cosh(x) Type: Expression Integer Time: 0.03 (IN) + 0.05 (EV) + 0.02 (OT) = 0.10 sec -- This example involves several symbolic parameters -- => 1/sqrt(b^2 - a^2) log([sqrt(b^2 - a^2) tan(x/2) + a + b]/ -- [sqrt(b^2 - a^2) tan(x/2) - a - b]) (a^2 < b^2) -- [Gradshteyn and Ryzhik 2.553(3)] integrate(1/(a + b*cos(x)), x) (5) +-------+ | 2 2 2 2 (- a cos(x) - b)\|b - a + (- b + a )sin(x) log(----------------------------------------------) b cos(x) + a [---------------------------------------------------, +-------+ | 2 2 \|b - a +---------+ | 2 2 sin(x)\|- b + a 2atan(---------------------) (b + a)cos(x) + b + a ----------------------------] +---------+ | 2 2 \|- b + a Type: Union(List Expression Integer,...) Time: 0.12 (IN) + 0.67 (EV) + 0.05 (OT) = 0.83 sec map(simplify, map(f +-> D(f, x), %)) 1 1 (6) [------------,------------] b cos(x) + a b cos(x) + a Type: List Expression Integer Time: 0.32 (EV) + 0.02 (OT) = 0.33 sec -- The integral of 1/(a + 3 cos x + 4 sin x) can have 4 different forms -- depending on the value of a ! [Gradshteyn and Ryzhik 2.558(4)] -- => (a = 3) 1/4 log[3 + 4 tan(x/2)] integrate(1/(3 + 3*cos(x) + 4*sin(x)), x) 4sin(x) + 3cos(x) + 3 log(---------------------) cos(x) + 1 (7) -------------------------- 4 Type: Union(Expression Integer,...) Time: 0.05 (IN) + 0.20 (EV) + 0.02 (OT) = 0.27 sec -- => (a = 4) 1/3 log([tan(x/2) + 1]/[tan(x/2) + 7]) integrate(1/(4 + 3*cos(x) + 4*sin(x)), x) sin(x) + 7cos(x) + 7 sin(x) + cos(x) + 1 - log(--------------------) + log(-------------------) cos(x) + 1 cos(x) + 1 (8) ------------------------------------------------------ 3 Type: Union(Expression Integer,...) Time: 0.27 (EV) + 0.02 (OT) = 0.28 sec -- => (a = 5) -1/[2 + tan(x/2)] integrate(1/(5 + 3*cos(x) + 4*sin(x)), x) - cos(x) - 1 (9) -------------------- sin(x) + 2cos(x) + 2 Type: Union(Expression Integer,...) Time: 0.03 (IN) + 0.08 (EV) + 0.02 (OT) = 0.13 sec -- => (a = 6) 2/sqrt(11) arctan([3 tan(x/2) + 4]/sqrt(11)) integrate(1/(6 + 3*cos(x) + 4*sin(x)), x) +--+ +--+ +--+ 3\|11 sin(x) + 4\|11 cos(x) + 4\|11 2atan(------------------------------------) 11cos(x) + 11 (10) ------------------------------------------- +--+ \|11 Type: Union(Expression Integer,...) Time: 0.22 (EV) + 0.05 (OT) = 0.27 sec -- => x log|x^2 - a^2| - 2 x + a log|(x + a)/(x - a)| -- [Gradshteyn and Ryzhik 2.736(1)] integrate(log(abs(x**2 - a**2)), x) 2 2 (11) x log(abs(x - a )) + a log(x + a) - a log(x - a) - 2x Type: Union(Expression Integer,...) Time: 0.05 (IN) + 0.20 (EV) + 0.02 (OT) = 0.27 sec -- => (a x)/2 + (pi x^2)/4 - 1/2 (x^2 + a^2) arctan(x/a) -- [Gradshteyn and Ryzhik 2.822(4)] or -- (a x)/2 + 1/2 (x^2 + a^2) arccot(x/a) [Gradshteyn and Ryzhik 2.853(2)] integrate(x*acot(x/a), x) 2 2 2a x (x + a )atan(-------) + 2a x 2 2 x - a (12) ----------------------------- 4 Type: Union(Expression Integer,...) Time: 0.07 (IN) + 0.80 (EV) + 0.02 (OT) + 0.03 (GC) = 0.92 sec -- => [sin(5 x) Ci(2 x)]/5 - [Si(7 x) + Si(3 x)]/10 -- [Gradshteyn and Ryzhik 5.31(1)] integrate(cos(5*x)*real(Ei(%i*2*x)), x) >> Error detected within library code: ker2trigs: cannot convert kernel to gaussian function initial (13) -> real 30.1 user 7.6 sys 0.3 ------------------------------------------------------------------------------- Mon Feb 9 04:40:40 MET 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Indefinite Integrals ---------- -- => 1/2 [f(x) - g(x)]/[f(x) + g(x)] [Gradshteyn and Ryzhik 2.02(25)] f:= operator('f); Type: BasicOperator Time: 0.03 (IN) + 0.08 (OT) + 0.03 (GC) = 0.15 sec g:= operator('g); Type: BasicOperator Time: 0.02 (OT) = 0.02 sec integrate((D(f(x), x)*g(x) - f(x)*D(g(x), x))/(f(x)**2 - g(x)**2), x) , , x f(%I)g (%I) - g(%I)f (%I) ++ (3) | ------------------------- d%I ++ 2 2 g(%I) - f(%I) Type: Union(Expression Integer,...) Time: 0.72 (IN) + 0.82 (EV) + 0.20 (OT) + 0.07 (GC) = 1.80 sec )clear properties f g -- ---------- Quit ---------- )quit real 9.1 user 2.5 sys 0.2 \end{verbatim} \section{Definite Integrals} \begin{verbatim} -- ---------- Definite Integrals ---------- -- The following two functions have a pole at a. The first integral has a -- principal value of zero; the second is divergent integrate(1/(x - a), x = (a - 1)..(a + 1)) >> Error detected within library code: integrate: pole in path of integration initial (1) -> real 5.3 user 2.5 sys 0.3 ------------------------------------------------------------------------------- Fri Jun 20 00:05:11 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Definite Integrals ---------- integrate(1/(x - a)**2, x = (a - 1)..(a + 1)) (1) potentialPole Type: Union(pole: potentialPole,...) Time: 0.63 (IN) + 0.73 (EV) + 0.17 (OT) + 0.22 (GC) = 1.75 sec -- Different branches of the square root need to be chosen in the intervals -- [0, 1] and [1, 2]. The correct results are 4/3, [4 - sqrt(8)]/3, -- [8 - sqrt(8)]/3, respectively integrate(sqrt(x + 1/x - 2), x = 0..1) (2) potentialPole Type: Union(pole: potentialPole,...) Time: 0.13 (IN) + 0.13 (EV) + 0.02 (OT) + 0.02 (GC) = 0.30 sec integrate(sqrt(x + 1/x - 2), x = 0..1, "noPole") 4 (3) - - 3 Type: Union(f1: OrderedCompletion Expression Integer,...) Time: 0.25 (IN) + 0.83 (EV) + 0.02 (OT) + 0.08 (GC) = 1.19 sec integrate(sqrt(x + 1/x - 2), x = 1..2) (4) potentialPole Type: Union(pole: potentialPole,...) Time: 0.12 (IN) + 0.02 (EV) + 0.02 (OT) = 0.15 sec integrate(sqrt(x + 1/x - 2), x = 1..2, "noPole") +-+ - 2\|2 + 4 (5) ----------- 3 Type: Union(f1: OrderedCompletion Expression Integer,...) Time: 0.07 (IN) + 0.20 (EV) + 0.03 (OT) = 0.30 sec integrate(sqrt(x + 1/x - 2), x = 0..2) (6) potentialPole Type: Union(pole: potentialPole,...) Time: 0.05 (IN) + 0.02 (OT) = 0.07 sec integrate(sqrt(x + 1/x - 2), x = 0..2, "noPole") +-+ 2\|2 (7) - ----- 3 Type: Union(f1: OrderedCompletion Expression Integer,...) Time: 0.05 (IN) + 0.18 (EV) + 0.02 (OT) = 0.25 sec -- => sqrt(2) [a modification of a problem due to W. Kahan] integrate(sqrt(2 - 2*cos(2*x))/2, x = -3*%pi/4..-%pi/4) (8) potentialPole Type: Union(pole: potentialPole,...) Time: 0.50 (IN) + 0.07 (OT) = 0.57 sec integrate(sqrt(2 - 2*cos(2*x))/2, x = -3*%pi/4..-%pi/4, "noPole") (9) 0 Type: Union(f1: OrderedCompletion Expression Integer,...) Time: 0.07 (IN) + 2.90 (EV) + 0.02 (OT) + 0.10 (GC) = 3.08 sec -- Contour integrals => pi/a e^(-a) for a > 0. See Norman Levinson and -- Raymond M. Redheffer, _Complex Variables_, Holden-Day, Inc., 1970, p. 198. integrate(cos(x)/(x**2 + a**2), x = %minusInfinity..%plusInfinity) (10) potentialPole Type: Union(pole: potentialPole,...) Time: 0.23 (IN) + 0.03 (EV) + 0.10 (OT) = 0.37 sec integrate(cos(x)/(x**2 + a**2), x = %minusInfinity..%plusInfinity, "noPole") (11) "failed" Type: Union(fail: failed,...) Time: 0.07 (IN) + 1.07 (EV) + 0.60 (GC) = 1.73 sec -- Integrand with a branch point => pi/sin(pi a) for 0 < a < 1 -- [Levinson and Redheffer, p. 212] integrate(t**(a - 1)/(1 + t), t = 0..%plusInfinity) (12) potentialPole Type: Union(pole: potentialPole,...) Time: 0.08 (IN) + 0.25 (EV) + 0.02 (GC) = 0.35 sec integrate(t**(a - 1)/(1 + t), t = 0..%plusInfinity, "noPole") (13) "failed" Type: Union(fail: failed,...) Time: 0.07 (IN) + 0.62 (EV) + 0.03 (GC) = 0.72 sec -- Integrand with a residue at infinity => -2 pi [sin(pi/5) + sin(2 pi/5)] -- (principal value) [Levinson and Redheffer, p. 234] integrate(5*x**3/(1 + x + x**2 + x**3 + x**4), _ x = %minusInfinity..%plusInfinity) (14) "failed" Type: Union(fail: failed,...) Time: 0.08 (IN) + 46.28 (EV) + 0.05 (OT) + 3.97 (GC) = 50.38 sec -- integrate(1/[1 + x + x^2 + ... + x^(2 n)], x = -infinity..infinity) -- = 2 pi/(2 n + 1) [1 + cos(pi/[2 n + 1])] csc(2 pi/[2 n + 1]) -- [Levinson and Redheffer, p. 255] => 2 pi/5 [1 + cos(pi/5)] csc(2 pi/5) integrate(1/(1 + x + x**2 + x**4), x = %minusInfinity..%plusInfinity) /usr/local/bin/axiomb: 21068 Terminated ------------------------------------------------------------------------------- Mon Jun 16 06:14:26 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Definite Integrals ---------- -- Integrand with a residue at infinity and a branch cut => pi [sqrt(2) - 1] -- [Levinson and Redheffer, p. 234] integrate(sqrt(1 - x**2)/(1 + x**2), x = -1..1) (1) potentialPole Type: Union(pole: potentialPole,...) Time: 0.55 (IN) + 0.37 (EV) + 0.20 (OT) + 0.18 (GC) = 1.30 sec integrate(sqrt(1 - x**2)/(1 + x**2), x = -1..1, "noPole") +-+ 2 +-+ 1 (2) 2\|2 atan(----) + 2\|2 atan(----) - %pi +-+ +-+ \|2 \|2 Type: Union(f1: OrderedCompletion Expression Integer,...) Time: 0.20 (IN) + 2.62 (EV) + 0.05 (OT) + 0.18 (GC) = 3.05 sec -- This is a common integral in many physics calculations -- => q/p sqrt(pi/p) e^(q^2/p) (Re p > 0) [Gradshteyn and Ryzhik 3.462(6)] integrate(x*exp(-p*x**2 + 2*q*x), x = %minusInfinity..%plusInfinity) (3) "failed" Type: Union(fail: failed,...) Time: 0.32 (IN) + 0.41 (EV) + 0.05 (OT) + 0.03 (GC) = 0.82 sec -- => 2 Euler's_constant [Gradshteyn and Ryzhik 8.367(5-6)] integrate(1/log(t) + 1/(1 - t) - log(log(1/t)), t = 0..1) (4) potentialPole Type: Union(pole: potentialPole,...) Time: 0.37 (IN) + 0.02 (EV) + 0.07 (OT) = 0.45 sec integrate(1/log(t) + 1/(1 - t) - log(log(1/t)), t = 0..1, "noPole") (5) "failed" Type: Union(fail: failed,...) Time: 0.12 (IN) + 0.75 (EV) + 0.07 (GC) = 0.93 sec -- This integral comes from atomic collision theory => 0 [John Prentice] integrate(sin(t)/t*exp(2*%i*t), t = %minusInfinity..%plusInfinity) >> Error detected within library code: integrate: pole in path of integration initial (6) -> real 18.4 user 8.2 sys 0.6 ------------------------------------------------------------------------------- Mon Jun 16 06:19:54 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Definite Integrals ---------- -- => 1/12 [Gradshteyn and Ryzhik 6.443(3)] integrate(log(Gamma(x))*cos(6*%pi*x), x = 0..1) (1) potentialPole Type: Union(pole: potentialPole,...) Time: 0.67 (IN) + 0.48 (EV) + 0.32 (OT) + 0.20 (GC) = 1.67 sec integrate(log(Gamma(x))*cos(6*%pi*x), x = 0..1, "noPole") (2) "failed" Type: Union(fail: failed,...) Time: 0.22 (IN) + 1.67 (EV) + 0.02 (OT) + 0.12 (GC) = 2.02 sec -- => 36/35 [Gradshteyn and Ryzhik 7.222(2)] integrate((1 + x)**3*legendreP(1, x)*legendreP(2, x), x = -1..1) 36 (3) -- 35 Type: Union(f1: OrderedCompletion Expression Integer,...) Time: 0.92 (IN) + 0.40 (EV) + 0.12 (OT) + 0.03 (GC) = 1.47 sec -- => 1/sqrt(a^2 + b^2) (a > 0 and b real) -- [Gradshteyn and Ryzhik 6.611(1)] integrate(exp(-a*x)*besselJ(0, b*x), x = 0..%plusInfinity) (4) potentialPole Type: Union(pole: potentialPole,...) Time: 0.37 (IN) + 0.23 (EV) + 0.07 (OT) = 0.67 sec integrate(exp(-a*x)*besselJ(0, b*x), x = 0..%plusInfinity, "noPole") (5) "failed" Type: Union(fail: failed,...) Time: 0.05 (IN) + 0.37 (EV) + 0.03 (OT) = 0.45 sec -- Integrand contains a special function => 4/(3 pi) [Tom Hagstrom] integrate((besselJ(1, x)/x)**2, x = 0..%plusInfinity) (6) potentialPole Type: Union(pole: potentialPole,...) Time: 0.12 (IN) + 0.22 (EV) + 0.05 (OT) + 0.02 (GC) = 0.40 sec integrate((besselJ(1, x)/x)**2, x = 0..%plusInfinity, "noPole") (7) "failed" Type: Union(fail: failed,...) Time: 0.03 (IN) + 0.25 (EV) + 0.02 (OT) = 0.30 sec -- => (cos 7 - 1)/7 [Gradshteyn and Ryzhik 6.782(3)] integrate(real(Ei(%i*x))*besselJ(0, 2*sqrt(7*x)), x = 0..%plusInfinity) >> Error detected within library code: ker2trigs: cannot convert kernel to gaussian function initial (8) -> real 16.2 user 8.1 sys 0.5 ------------------------------------------------------------------------------- Mon Jun 16 06:38:04 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Definite Integrals ---------- -- This integral comes from doing a two loop Feynman diagram for a QCD problem -- => - [17/3 + pi^2]/36 + log 2/9 [35/3 - pi^2/2 - 4 log 2 + log(2)^2] -- + zeta(3)/4 = 0.210883... [Rolf Mertig] --integrate(x**2*polylog(3, 1/(x + 1)), x = 0..1) --integrate(x**2*polylog(3, 1/(x + 1)), x = 0..1) :: Complex Float -- - (17/3 + %pi**2)/36 + log(2)/9*(35/3 - %pi**2/2 - 4*log(2) + log(2)**2) -- + zeta(3)/4 :: Complex Float -- Integrate a piecewise defined step function s(t) multiplied by cos t, where -- s(t) = 0 (t < 1); 1 (1 <= t <= 2); 0 (t > 2) -- => 0 (u < 1); sin u - sin 1 (1 <= u <= 2); sin 2 - sin 1 (u > 2) s(t) == if 1 <= t and t <= 2 then 1 else 0; Type: Void Time: 0 sec integrate(s(t)*cos(t), t = 0..u) Compiling function s with type Variable t -> NonNegativeInteger (2) 0 Type: Union(f1: OrderedCompletion Expression Integer,...) Time: 0.88 (IN) + 0.58 (EV) + 0.37 (OT) + 0.33 (GC) = 2.17 sec )clear properties s Compiled code for s has been cleared. -- Integrating first with respect to y and then x is much easier than -- integrating first with respect to x and then y -- => (|b| - |a|) pi [W. Kahan] integrate(integrate(x/(x**2 + y**2), y = %minusInfinity..%plusInfinity), _ x = a..b) There are 11 exposed and 7 unexposed library operations named integrate having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op integrate to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named integrate with argument type(s) Union(f1: OrderedCompletion Expression Integer,f2: List OrderedCompletion Expression Integer,fail: failed,pole: potentialPole) SegmentBinding Symbol integrate(integrate(x/(x**2 + y**2), y = %minusInfinity..%plusInfinity, _ "noPole"), x = a..b, "noPole") (3) "failed" Type: Union(fail: failed,...) Time: 0.72 (IN) + 1.10 (EV) + 0.12 (OT) + 0.12 (GC) = 2.05 sec integrate(integrate(x/(x**2 + y**2), x = a..b), _ y = %minusInfinity..%plusInfinity) There are 11 exposed and 7 unexposed library operations named integrate having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op integrate to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named integrate with argument type(s) Union(f1: OrderedCompletion Expression Integer,f2: List OrderedCompletion Expression Integer,fail: failed,pole: potentialPole) SegmentBinding OrderedCompletion Integer integrate(integrate(x/(x**2 + y**2), x = a..b, "noPole"), _ y = %minusInfinity..%plusInfinity, "noPole") 2 1 2 1 (4) - a %pi abs(-) + b %pi abs(-) a b Type: Union(f1: OrderedCompletion Expression Integer,...) Time: 0.10 (IN) + 2.18 (EV) + 0.05 (OT) + 0.05 (GC) = 2.38 sec -- => [log(sqrt(2) + 1) + sqrt(2)]/3 [Caviness et all, section 2.10.1] integrate(integrate(sqrt(x**2 + y**2), x = 0..1), y = 0..1) There are 11 exposed and 7 unexposed library operations named integrate having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op integrate to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named integrate with argument type(s) Union(f1: OrderedCompletion Expression Integer,f2: List OrderedCompletion Expression Integer,fail: failed,pole: potentialPole) SegmentBinding NonNegativeInteger integrate(integrate(sqrt(x**2 + y**2), x = 0..1, "noPole"), y = 0..1, "noPole") (5) +-+ +-+ +-+ +-+ +-+ (- 4\|2 + 6)log(- 2\|2 + 3) + (- 2\|2 + 3)log(- 12\|2 + 17) - 24\|2 + 32 ----------------------------------------------------------------------------- +-+ 48\|2 - 72 Type: Union(f1: OrderedCompletion Expression Integer,...) Time: 0.10 (IN) + 6.17 (EV) + 0.03 (OT) + 0.03 (GC) = 6.33 sec -- => (pi a)/2 [Gradshteyn and Ryzhik 4.621(1)] integrate(integrate(sin(a)*sin(y)/sqrt(1 - sin(a)**2*sin(x)**2*sin(y)**2), _ x = 0..%pi/2), _ y = 0..%pi/2) There are 11 exposed and 7 unexposed library operations named integrate having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op integrate to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named integrate with argument type(s) Union(f1: OrderedCompletion Expression Integer,f2: List OrderedCompletion Expression Integer,fail: failed,pole: potentialPole) SegmentBinding Pi integrate(integrate(sin(a)*sin(y)/sqrt(1 - sin(a)**2*sin(x)**2*sin(y)**2), _ x = 0..%pi/2, "noPole"), _ y = 0..%pi/2, "noPole") There are 4 exposed and 1 unexposed library operations named integrate having 3 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op integrate to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named integrate with argument type(s) Union(f1: OrderedCompletion Expression Integer,f2: List OrderedCompletion Expression Integer,fail: failed,pole: potentialPole) SegmentBinding Pi String -- => 46/15 [Paul Zimmermann] integrate(integrate(abs(y - x**2), y = 0..2), x = -1..1) There are 11 exposed and 7 unexposed library operations named integrate having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op integrate to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named integrate with argument type(s) Union(f1: OrderedCompletion Expression Integer,f2: List OrderedCompletion Expression Integer,fail: failed,pole: potentialPole) SegmentBinding Integer integrate(integrate(abs(y - x**2), y = 0..2, "noPole"), x = -1..1, "noPole") There are 4 exposed and 1 unexposed library operations named integrate having 3 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op integrate to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named integrate with argument type(s) Union(f1: OrderedCompletion Expression Integer,f2: List OrderedCompletion Expression Integer,fail: failed,pole: potentialPole) SegmentBinding Integer String -- Multiple integrals: volume of a tetrahedron => a b c / 6 integrate(integrate(integrate(1, z = 0..c*(1 - x/a - y/b)), _ y = 0..b*(1 - x/a)), _ x = 0..a) a b c (6) ----- 6 Type: Union(f1: OrderedCompletion Expression Integer,...) Time: 1.73 (IN) + 0.55 (EV) + 0.40 (OT) + 0.02 (GC) = 2.70 sec -- ---------- Quit ---------- )quit real 47.7 user 23.7 sys 0.7 \end{verbatim} \section{Series} \begin{verbatim} -- ---------- Series ---------- -- Taylor series---this first example comes from special relativity -- => 1 + 1/2 (v/c)^2 + 3/8 (v/c)^4 + 5/16 (v/c)^6 + O((v/c)^8) 1/sqrt(1 - (v/c)**2) 1 (1) ------------ +---------+ | 2 2 |- v + c |--------- | 2 \| c Type: Expression Integer Time: 0.43 (IN) + 0.18 (EV) + 0.22 (OT) + 0.22 (GC) = 1.05 sec series(%, v = 0) 1 2 3 4 5 6 8 (2) 1 + --- v + --- v + ---- v + O(v ) 2 4 6 2c 8c 16c Type: UnivariatePuiseuxSeries(Expression Integer,v,0) Time: 0.38 (IN) + 0.27 (EV) + 0.12 (OT) + 0.05 (GC) = 0.82 sec 1/%**2 1 2 8 (3) 1 - -- v + O(v ) 2 c Type: UnivariatePuiseuxSeries(Expression Integer,v,0) Time: 0.27 (IN) + 0.10 (OT) = 0.37 sec -- Note: sin(x) = x - x^3/6 + x^5/120 - x^7/5040 + O(x^9) -- cos(x) = 1 - x^2/2 + x^4/24 - x^6/720 + O(x^8) -- tan(x) = x + x^3/3 + 2/15 x^5 + 17/315 x^7 + O(x^9) tsin:= series(sin(x), x = 0) 1 3 1 5 1 7 9 (4) x - - x + --- x - ---- x + O(x ) 6 120 5040 Type: UnivariatePuiseuxSeries(Expression Integer,x,0) Time: 0.12 (IN) + 0.18 (EV) + 0.05 (OT) + 0.03 (GC) = 0.38 sec tcos:= series(cos(x), x = 0) 1 2 1 4 1 6 8 (5) 1 - - x + -- x - --- x + O(x ) 2 24 720 Type: UnivariatePuiseuxSeries(Expression Integer,x,0) Time: 0.03 (IN) + 0.02 (OT) = 0.05 sec -- Note that additional terms will be computed as needed tsin/tcos 1 3 2 5 17 7 9 (6) x + - x + -- x + --- x + O(x ) 3 15 315 Type: UnivariatePuiseuxSeries(Expression Integer,x,0) Time: 0.08 (IN) + 0.03 (OT) = 0.12 sec series(tan(x), x = 0) 1 3 2 5 17 7 9 (7) x + - x + -- x + --- x + O(x ) 3 15 315 Type: UnivariatePuiseuxSeries(Expression Integer,x,0) Time: 0.02 (IN) + 0.02 (EV) + 0.03 (OT) = 0.07 sec )clear properties tsin tcos -- => -x^2/6 - x^4/180 - x^6/2835 - O(x^8) series(log(sin(x)/x), x = 0) 1 2 1 4 1 6 1 8 10 (8) - - x - --- x - ---- x - ----- x + O(x ) 6 180 2835 37800 Type: UnivariatePuiseuxSeries(Expression Integer,x,0) Time: 0.02 (IN) + 0.05 (EV) + 0.05 (OT) = 0.12 sec series(sin(x)/x, x = 0, 7) 1 2 1 4 1 6 8 (9) 1 - - x + --- x - ---- x + O(x ) 6 120 5040 Type: UnivariatePuiseuxSeries(Expression Integer,x,0) Time: 0.05 (IN) + 0.03 (EV) + 0.02 (OT) = 0.10 sec log(%) 1 2 1 4 1 6 1 8 10 (10) - - x - --- x - ---- x - ----- x + O(x ) 6 180 2835 37800 Type: UnivariatePuiseuxSeries(Expression Integer,x,0) Time: 0.02 (EV) + 0.02 (OT) = 0.03 sec -- => [a f'(a d) + g(b d) + integrate(h(c y), y = 0..d)] -- + [a^2 f''(a d) + b g'(b d) + h(c d)] (x - d) f:= operator('f); Type: BasicOperator Time: 0.08 (IN) = 0.08 sec g:= operator('g); Type: BasicOperator Time: 0.03 (IN) = 0.03 sec h:= operator('h); Type: BasicOperator Time: 0.05 (IN) = 0.05 sec D(f(a*x), x) + g(b*x) + 'integrate(h(c*y), y = 0..x) , (14) af (a x) + g(b x) + integrate(h(c y),y= 0,x ) Type: OutputForm Time: 0.57 (IN) + 0.05 (EV) + 0.20 (OT) = 0.82 sec series(%, x = d, 1) There are 3 exposed and 0 unexposed library operations named series having 3 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op series to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named series with argument type(s) OutputForm Equation Symbol PositiveInteger )clear properties f g h -- Taylor series of nonscalar objects (noncommutative multiplication) -- => (B A - A B) t^2/2 + O(t^3) [Stanly Steinberg] A : SquareMatrix(2, Integer); Type: Void Time: 0.02 (IN) = 0.02 sec B : SquareMatrix(2, Integer); Type: Void Time: 0 sec %e**((A + B)*t) - %e**(A*t) * %e**(B*t) A is declared as being in SquareMatrix(2,Integer) but has not been given a value. )clear properties A B series(%e**((A + B)*t) - %e**(A*t) * %e**(B*t), t = 0, 4) 15 (17) O(t ) Type: UnivariatePuiseuxSeries(Expression Integer,t,0) Time: 0.20 (IN) + 0.23 (EV) + 0.28 (OT) = 0.72 sec -- Laurent series: -- => sum( Bernoulli[k]/k! x^(k - 2), k = 1..infinity ) -- = 1/x^2 - 1/(2 x) + 1/12 - x^2/720 + x^4/30240 + O(x^6) -- [Levinson and Redheffer, p. 173] series(1/(x*(exp(x) - 1)), x = 0) - 2 1 - 1 1 1 2 1 4 6 (18) x - - x + -- - --- x + ----- x + O(x ) 2 12 720 30240 Type: UnivariatePuiseuxSeries(Expression Integer,x,0) Time: 0.07 (IN) + 0.05 (EV) + 0.05 (OT) = 0.17 sec -- Puiseux series (terms with fractional degree): -- => 1/sqrt(x - 3/2 pi) + (x - 3/2 pi)^(3/2) / 12 + O([x - 3/2 pi]^(7/2)) series(sqrt(sec(x)), x = 3/2*%pi, 2) 1 3 7 - - - - 3%pi 2 1 3%pi 2 3%pi 2 (19) (x - ----) + -- (x - ----) + O((x - ----) ) 2 12 2 2 Type: UnivariatePuiseuxSeries(Expression Integer,x,(3*pi)/2) Time: 0.20 (IN) + 0.25 (EV) + 0.11 (OT) + 0.02 (GC) = 0.58 sec -- Generalized Taylor series => sum( [x log x]^k/k!, k = 0..infinity ) series(x**x, x = 0) (20) 2 3 4 5 log(x) 2 log(x) 3 log(x) 4 log(x) 5 1 + log(x)x + ------- x + ------- x + ------- x + ------- x 2 6 24 120 + 6 7 log(x) 6 log(x) 7 8 ------- x + ------- x + O(x ) 720 5040 Type: GeneralUnivariatePowerSeries(Expression Integer,x,0) Time: 0.03 (IN) + 0.07 (EV) + 0.07 (OT) + 0.02 (GC) = 0.18 sec -- Compare the generalized Taylor series of two different formulations of a -- function => log(z) + log(cosh(w)) + tanh(w) z + O(z^2) )set streams calculate 1 s1:= series(log(sinh(z)) + log(cosh(z + w)), z = 0) w 2 w 2 (%e ) + 1 (%e ) - 1 2 (21) log(----------) + log(z) + ---------- z + O(z ) w w 2 2%e (%e ) + 1 Type: GeneralUnivariatePowerSeries(Expression Integer,z,0) Time: 0.08 (IN) + 0.53 (EV) + 0.07 (OT) = 0.68 sec s2:= series(log(sinh(z) * cosh(z + w)), z = 0) w 2 w 2 (%e ) + 1 (%e ) - 1 2 (22) log(----------) + log(z) + ---------- z + O(z ) w w 2 2%e (%e ) + 1 Type: GeneralUnivariatePowerSeries(Expression Integer,z,0) Time: 0.53 (EV) + 0.07 (OT) = 0.60 sec s1 - s2 3 (23) O(z ) Type: GeneralUnivariatePowerSeries(Expression Integer,z,0) Time: 0.02 (IN) + 0.07 (OT) = 0.08 sec )clear properties s1 s2 )set streams calculate 7 -- Look at the generalized Taylor series around x = 1 -- => (x - 1)^a/e^b [1 - (a + 2 b) (x - 1) / 2 + O((x - 1)^2)] log(x)**a*exp(-b*x) - b x a (24) %e log(x) Type: Expression Integer Time: 0.07 (IN) + 0.02 (EV) + 0.02 (OT) = 0.10 sec series(%, x = 1, 1) >> Error detected within library code: No series expansion initial (25) -> real 29.4 user 9.7 sys 0.5 ------------------------------------------------------------------------------- Tue Aug 19 07:01:32 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected )set streams calculate 7 -- ---------- Series ---------- -- Asymptotic expansions => sqrt(2) x + O(1/x) series(sqrt(2*x**2 + 1), x = %plusInfinity, 0) There are 3 exposed and 0 unexposed library operations named series having 3 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op series to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named series with argument type(s) Expression Integer Equation OrderedCompletion Polynomial Integer NonNegativeInteger -- Wallis' product => 1/sqrt(pi n) + ... [Knopp, p. 385] series(1/2**(2*n) * binomial(2*n, n), n = %plusInfinity, 0) There are 3 exposed and 0 unexposed library operations named series having 3 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op series to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named series with argument type(s) Expression Integer Equation OrderedCompletion Polynomial Integer NonNegativeInteger -- => 0!/x - 1!/x^2 + 2!/x^3 - 3!/x^4 + O(1/x^5) [Knopp, p. 544] exp(x) * 'integrate(exp(-t)/t, t = x..%plusInfinity) x exp(- t) (1) %e integrate(--------,t= x,%plusInfinity ) t Type: OutputForm Time: 0.27 (IN) + 0.12 (EV) + 0.03 (OT) + 0.02 (GC) = 0.43 sec series(%, x = %plusInfinity, 5) There are 3 exposed and 0 unexposed library operations named series having 3 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op series to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named series with argument type(s) OutputForm Equation OrderedCompletion Polynomial Integer PositiveInteger -- Multivariate Taylor series expansion => 1 - (x^2 + 2 x y + y^2)/2 + O(x^4) )set streams calculate 2 series(cos(x + y), y = 0) cos(x) 2 3 (2) cos(x) - sin(x)y - ------ y + O(y ) 2 Type: UnivariatePuiseuxSeries(Expression Integer,y,0) Time: 0.32 (IN) + 0.22 (EV) + 0.10 (OT) + 0.10 (GC) = 0.73 sec series(%, x = 0) There are 4 exposed and 0 unexposed library operations named series having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op series to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named series with argument type(s) Any Equation Polynomial Integer )set streams calculate 7 -- Power series (compute the general formula) log(sin(x)/x) sin(x) (3) log(------) x Type: Expression Integer Time: 0.03 (IN) + 0.02 (OT) = 0.05 sec exp(-x)*sin(x) - x (4) %e sin(x) Type: Expression Integer Time: 0.03 (IN) + 0.02 (EV) = 0.05 sec series(%, x = 0) 2 1 3 1 5 1 6 1 7 9 (5) x - x + - x - -- x + -- x - --- x + O(x ) 3 30 90 630 Type: UnivariatePuiseuxSeries(Expression Integer,x,0) Time: 0.03 (IN) + 0.08 (EV) + 0.05 (OT) = 0.17 sec -- Derive an explicit Taylor series solution of y as a function of x from the -- following implicit relation: -- y = x - 1 + (x - 1)^2/2 + 2/3 (x - 1)^3 + (x - 1)^4 + 17/10 (x - 1)^5 + ... y:= operator('y); Type: BasicOperator Time: 0.08 (IN) = 0.08 sec x = sin(y(x)) + cos(y(x)) (7) x= sin(y(x)) + cos(y(x)) Type: Equation Expression Integer Time: 0.47 (IN) + 0.02 (EV) + 0.14 (OT) = 0.62 sec seriesSolve(%, y, x = 1, 0) >> Error detected within library code: Improper initial value initial (8) -> real 27.3 user 9.0 sys 0.6 ------------------------------------------------------------------------------- Tue Aug 19 07:21:02 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected )set streams calculate 7 -- ---------- Series ---------- -- Pade (rational function) approximation => (2 - x)/(2 + x) pade(1, 1, series(exp(-x), x = 0)) Compiling function G82130 with type Integer -> Boolean - x + 2 (1) ------- x + 2 Type: Union(Fraction UnivariatePolynomial(x,Expression Integer),...) Time: 0.69 (IN) + 0.67 (EV) + 0.28 (OT) + 0.37 (GC) = 2.0 sec -- Fourier series of f(x) of period 2 p over the interval [-p, p] -- => - (2 p / pi) sum( (-1)^n sin(n pi x / p) / n, n = 1..infinity ) x (2) x Type: Variable x Time: 0 sec -- => p / 2 -- - (2 p / pi^2) sum( [1 - (-1)^n] cos(n pi x / p) / n^2, n = 1..infinity ) abs(x) (3) abs(x) Type: Expression Integer Time: 0.12 (IN) + 0.03 (EV) + 0.02 (OT) = 0.17 sec -- ---------- Quit ---------- )quit real 7.4 user 2.6 sys 0.3 \end{verbatim} \section{Transforms} \begin{verbatim} -- ---------- Transforms ---------- -- Laplace and inverse Laplace transforms -- => s/[s^2 + (w - 1)^2] (Re s > |Im(w - 1)|) -- [Gradshteyn and Ryzhik 17.13(33)] laplace(cos((w - 1)*t), t, s) s (1) ---------------- 2 2 w - 2w + s + 1 Type: Expression Integer Time: 0.37 (IN) + 1.22 (EV) + 0.15 (OT) + 0.18 (GC) = 1.92 sec inverseLaplace(%, s, t) +-----------+ | 2 (2) cos(t\|w - 2w + 1 ) Type: Union(Expression Integer,...) Time: 0.02 (IN) + 0.08 (EV) + 0.02 (OT) = 0.12 sec -- => w/(s^2 - 4 w^2) (Re s > |Re w|) [Gradshteyn and Ryzhik 17.13(84)] laplace(sinh(w*t)*cosh(w*t), t, s) w (3) - -------- 2 2 4w - s Type: Expression Integer Time: 0.22 (IN) + 0.07 (EV) + 0.03 (OT) = 0.32 sec -- e^(-6 sqrt(s))/s (Re s > 0) [Gradshteyn and Ryzhik 17.13(102)] laplace(erf(3/sqrt(t)), t, s) >> Error detected within library code: Sorry - cannot handle that integrand yet initial (4) -> real 10.3 user 3.5 sys 0.3 ------------------------------------------------------------------------------- Fri Jun 13 03:43:05 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Transforms ---------- -- Solve y'' + y = 4 [H(t - 1) - H(t - 2)], y(0) = 1, y'(0) = 0 where H is the -- Heaviside (unit step) function (the RHS describes a pulse of magnitude 4 and -- duration 1). See David A. Sanchez, Richard C. Allen, Jr. and Walter T. -- Kyner, _Differential Equations: An Introduction_, Addison-Wesley Publishing -- Company, 1983, p. 211. First, take the Laplace transform of the ODE -- => s^2 Y(s) - s + Y(s) = 4/s [e^(-s) - e^(-2 s)] -- where Y(s) is the Laplace transform of y(t) y:= operator('y); Type: BasicOperator Time: 0.08 (IN) + 0.10 (OT) + 0.02 (GC) = 0.20 sec heaviside:= operator('heaviside); Type: BasicOperator Time: 0.05 (IN) = 0.05 sec map(e +-> laplace(e, t, s), _ D(y(t), t, 2) + y(t) = 4*(heaviside(t - 1) - heaviside(t - 2))) (3) 2 , (s + 1)laplace(y(t),t,s) - y (0) - y(0)s = 4laplace(heaviside(t - 1),t,s) - 4laplace(heaviside(t - 2),t,s) Type: Equation Expression Integer Time: 1.0 (IN) + 1.17 (EV) + 0.31 (OT) + 0.27 (GC) = 2.75 sec -- Now, solve for Y(s) and then take the inverse Laplace transform -- => Y(s) = s/(s^2 + 1) + 4 [1/s - s/(s^2 + 1)] [e^(-s) - e^(-2 s)] -- => y(t) = cos t + 4 {[1 - cos(t - 1)] H(t - 1) - [1 - cos(t - 2)] H(t - 2)} -- What is the Laplace transform of an infinite square wave? -- => 1/s + 2 sum( (-1)^n e^(- s n a)/s, n = 1..infinity ) -- [Sanchez, Allen and Kyner, p. 213] laplace(1 + 2*limit(sum((-1)**n*heaviside(t - n*a), n = 1..N), _ N = %plusInfinity), t, s) There are 31 exposed and 17 unexposed library operations named * having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op * to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named * with argument type(s) PositiveInteger failed laplace(1 + 2*'limit(sum((-1)**n*heaviside(t - n*a), n = 1..N), _ N = %plusInfinity), t, s) There are 1 exposed and 0 unexposed library operations named laplace having 3 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op laplace to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named laplace with argument type(s) OutputForm Variable t Variable s -- Fourier transforms => sqrt(2 pi) delta(z) [Gradshteyn and Ryzhik 17.23(1)] FourierTransform(f, x, z) == _ [integrate(1/sqrt(2*%pi) * f * %e**(%i*z*x), _ x = %minusInfinity..%plusInfinity), _ integrate(1/sqrt(2*%pi) * f * %e**(%i*z*x), _ x = %minusInfinity..%plusInfinity, "noPole")]; Type: Void Time: 0 sec FourierTransform(1, x, z) Compiling function FourierTransform with type (PositiveInteger, Variable x,Variable z) -> List Union(f1: OrderedCompletion Expression Complex Integer,f2: List OrderedCompletion Expression Complex Integer,fail: failed,pole: potentialPole) (5) ["failed","failed"] Type: List Union(f1: OrderedCompletion Expression Complex Integer,f2: List OrderedCompletion Expression Complex Integer,fail: failed,pole: potentialPole) Time: 2.50 (IN) + 2.65 (EV) + 0.38 (OT) + 0.67 (GC) = 6.20 sec -- => e^(-z^2/36) / [3 sqrt(2)] [Gradshteyn and Ryzhik 17.23(13)] FourierTransform(exp(-9*x**2), x, z) Compiling function FourierTransform with type (Expression Integer, Variable x,Variable z) -> List Union(f1: OrderedCompletion Expression Complex Integer,f2: List OrderedCompletion Expression Complex Integer,fail: failed,pole: potentialPole) (6) ["failed","failed"] Type: List Union(f1: OrderedCompletion Expression Complex Integer,f2: List OrderedCompletion Expression Complex Integer,fail: failed,pole: potentialPole) Time: 0.37 (IN) + 1.42 (EV) + 0.18 (OT) + 0.02 (GC) = 1.99 sec -- => sqrt(2 / pi) (9 - z^2)/(9 + z^2)^2 [Gradshteyn and Ryzhik 17.23(11)] FourierTransform(abs(x)*exp(-3*abs(x)), x, z) >> Error detected within library code: Function not supported by Risch d.e. initial (7) -> real 44.4 user 14.3 sys 0.7 ------------------------------------------------------------------------------- Fri Jun 13 04:04:37 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Transforms ---------- -- Mellin transforms -- => pi cot(pi s) (0 < Re s < 1) [Gradshteyn and Ryzhik 17.43(5)] MellinTransform(f, x, s) == _ [integrate(f * x**(s - 1), x = 0..%plusInfinity), _ integrate(f * x**(s - 1), x = 0..%plusInfinity, "noPole")]; Type: Void Time: 0.02 (IN) = 0.02 sec MellinTransform(1/(1 - x), x, s) Compiling function MellinTransform with type (Fraction Polynomial Integer,Variable x,Variable s) -> List Union(f1: OrderedCompletion Expression Integer,f2: List OrderedCompletion Expression Integer,fail: failed,pole: potentialPole) (2) [potentialPole,"failed"] Type: List Union(f1: OrderedCompletion Expression Integer,f2: List OrderedCompletion Expression Integer,fail: failed,pole: potentialPole) Time: 1.10 (IN) + 1.95 (EV) + 0.38 (OT) + 0.37 (GC) = 3.80 sec -- => 2^(s - 4) gamma(s/2)/gamma(4 - s/2) (0 < Re s < 1) -- [Gradshteyn and Ryzhik 17.43(16)] MellinTransform(besselJ(3, x)/x**3, x, s) Compiling function MellinTransform with type (Expression Integer, Variable x,Variable s) -> List Union(f1: OrderedCompletion Expression Integer,f2: List OrderedCompletion Expression Integer, fail: failed,pole: potentialPole) >> Error detected within library code: Function not supported by Risch d.e. initial (3) -> real 14.4 user 4.9 sys 0.4 ------------------------------------------------------------------------------- Fri Jun 13 04:06:47 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Transforms ---------- -- Z transforms. See _CRC Standard Mathematical Tables_, Twenty-first Edition, -- The Chemical Rubber Company, 1973, p. 518. -- Z[H(t - m T)] => z/[z^m (z - 1)] (H is the Heaviside (unit step) function) --heaviside(t - 3) --heaviside(t - m) -- ---------- Quit ---------- )quit real 1.1 user 0.7 sys 0.1 \end{verbatim} \section{Ordinary Difference and Differential Equations} \begin{verbatim} -- ---------- Ordinary Difference and Differential Equations ---------- -- Second order linear recurrence equation: r(n) = (n - 1)^2 + m n r:= operator('r); Type: BasicOperator Time: 0.07 (IN) + 0.05 (OT) + 0.02 (GC) = 0.13 sec [r(n + 2) - 2 * r(n + 1) + r(n) = 2, r(0) = 1, r(1) = m] (2) [r(n + 2) - 2r(n + 1) + r(n)= 2,r(0)= 1,r(1)= m] Type: List Equation Expression Integer Time: 1.17 (IN) + 0.05 (EV) + 0.24 (OT) + 0.07 (GC) = 1.52 sec -- => r(n) = 3^n - 2^n [Cohen, p. 67] [r(n) = 5*r(n - 1) - 6*r(n - 2), r(0) = 0, r(1) = 1] (3) [r(n)= 5r(n - 1) - 6r(n - 2),r(0)= 0,r(1)= 1] Type: List Equation Expression Integer Time: 0.10 (IN) + 0.05 (OT) = 0.15 sec -- => r(n) = Fibonacci[n + 1] [Cohen, p. 83] [r(n) = r(n - 1) + r(n - 2), r(1) = 1, r(2) = 2] (4) [r(n)= r(n - 1) + r(n - 2),r(1)= 1,r(2)= 2] Type: List Equation Expression Integer Time: 0.05 (IN) + 0.02 (OT) = 0.07 sec -- => [c^(n+1) [c^(n+1) - 2 c - 2] + (n+1) c^2 + 2 c - n] / [(c-1)^3 (c+1)] -- [Joan Z. Yu and Robert Israel in sci.math.symbolic] [r(n) = (1 + c - c**(n-1) - c**(n+1))/(1 - c**n)*r(n - 1) _ - c*(1 - c**(n-2))/(1 - c**(n-1))*r(n - 2) + 1, _ r(1) = 1, r(2) = (2 + 2*c + c**2)/(1 + c)] (5) [ r(n) = n - 1 n + 1 (r(n - 1)c - r(n - 1))c + n - 1 n - 2 n n - 1 2 (c - c r(n - 2)c + c r(n - 2) - 1)c + r(n - 1)(c ) + n - 1 n - 2 ((- c - 2)r(n - 1) - 1)c + c r(n - 2)c + (c + 1)r(n - 1) + - c r(n - 2) + 1 / n - 1 n n - 1 (c - 1)c - c + 1 , 2 c + 2c + 2 r(1)= 1, r(2)= -----------] c + 1 Type: List Equation Expression Integer Time: 0.48 (IN) + 0.13 (EV) + 0.13 (OT) + 0.03 (GC) = 0.78 sec )clear properties r -- Second order ODE with initial conditions---solve first using Laplace -- transforms: f(t) = sin(2 t)/8 - t cos(2 t)/4 f:= operator('f); Type: BasicOperator Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec ode:= D(f(t), t, 2) + 4*f(t) = sin(2*t) ,, (7) f (t) + 4f(t)= sin(2t) Type: Equation Expression Integer Time: 0.17 (IN) + 0.15 (EV) + 0.02 (OT) + 0.02 (GC) = 0.35 sec map(e +-> laplace(e, t, s), %) 2 , 2 (8) (s + 4)laplace(f(t),t,s) - f (0) - f(0)s= ------ 2 s + 4 Type: Equation Expression Integer Time: 0.92 (EV) + 0.02 (OT) + 0.07 (GC) = 1.0 sec subst(subst(%, f(0) = 0), subst(D(f(x), x), x = 0) = 0) 2 2 (9) (s + 4)laplace(f(t),t,s)= ------ 2 s + 4 Type: Equation Expression Integer Time: 0.47 (IN) + 0.02 (EV) + 0.07 (OT) = 0.55 sec map(e +-> e/(s**2 + 4), %) 2 (10) laplace(f(t),t,s)= ------------- 4 2 s + 8s + 16 Type: Equation Expression Integer Time: 0.08 (IN) + 0.02 (OT) = 0.10 sec map(e +-> inverseLaplace(e, s, t), %) sin(2t) - 2t cos(2t) (11) "failed"= -------------------- 8 Type: Equation Union(Expression Integer,"failed") Time: 0.02 (IN) + 0.10 (EV) + 0.03 (OT) = 0.15 sec -- Now, solve the ODE directly solve(ode, f, t = 0, [0, 0]) sin(2t) - 2t cos(2t) (12) -------------------- 8 Type: Union(Expression Integer,...) Time: 0.30 (IN) + 0.87 (EV) + 0.07 (OT) = 1.23 sec )clear properties f ode -- Separable equation => y(x)^2 = 2 log(x + 1) + (4 x + 3)/(x + 1)^2 + 2 A y:= operator('y); Type: BasicOperator Time: 0.03 (IN) = 0.03 sec D(y(x), x) = x**2/(y(x)*(1 + x)**3) 2 , x (14) y (x)= ----------------------- 3 2 (x + 3x + 3x + 1)y(x) Type: Equation Expression Integer Time: 0.08 (IN) + 0.02 (EV) + 0.07 (OT) = 0.17 sec solve(%, y, x) 2 2 2 (- 2x - 4x - 2)log(x + 1) + (x + 2x + 1)y(x) - 4x - 3 (15) -------------------------------------------------------- 2 2x + 4x + 2 Type: Union(Expression Integer,...) Time: 0.02 (IN) + 0.82 (EV) + 0.02 (OT) + 0.42 (GC) = 1.27 sec -- Homogeneous equation. See Emilio O. Roxin, _Ordinary Differential -- Equations_, Wadsworth Publishing Company, 1972, p. 11 -- => y(x)^2 = 2 x^2 log|A x| D(y(x), x) = y(x)/x + x/y(x) 2 2 , y(x) + x (16) y (x)= ---------- x y(x) Type: Equation Expression Integer Time: 0.17 (IN) + 0.02 (EV) + 0.02 (OT) = 0.20 sec solve(%, y, x) 2 2 - 2x log(x) + y(x) (17) ------------------- 2 2x Type: Union(Expression Integer,...) Time: 0.15 (EV) = 0.15 sec -- First order linear ODE: y(x) = [A - cos(x)]/x^3 y:= operator('y); Type: BasicOperator Time: 0 sec x**2 * D(y(x), x) + 3*x*y(x) = sin(x)/x 2 , sin(x) (19) x y (x) + 3x y(x)= ------ x Type: Equation Expression Integer Time: 0.08 (IN) = 0.08 sec solve(%, y, x) cos(x) 1 (20) [particular= - ------,basis= [--]] 3 3 x x Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) Time: 0.33 (EV) + 0.02 (OT) + 0.02 (GC) = 0.37 sec -- Exact equation => x + x^2 sin y(x) + y(x) = A [Roxin, p. 15] D(y(x), x) = -(1 + 2*x*sin(y(x)))/(1 + x**2*cos(y(x))) , - 2x sin(y(x)) - 1 (21) y (x)= ------------------ 2 x cos(y(x)) + 1 Type: Equation Expression Integer Time: 0.05 (IN) + 0.05 (OT) = 0.10 sec solve(%, y, x) 2 (22) x sin(y(x)) + y(x) + x Type: Union(Expression Integer,...) Time: 0.08 (EV) + 0.02 (OT) = 0.10 sec -- Nonlinear ODE => y(x)^3/6 + A y(x) = x + B ode:= D(y(x), x, 2) + y(x)*D(y(x), x)**3 = 0 ,, , 3 (23) y (x) + y(x)y (x) = 0 Type: Equation Expression Integer Time: 0.05 (IN) + 0.03 (EV) = 0.08 sec solve(%, y, x) >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation initial (24) -> real 20.4 user 9.1 sys 0.6 ------------------------------------------------------------------------------- Tue Mar 24 19:56:43 MET 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Ordinary Difference and Differential Equations ---------- y:= operator('y); Type: BasicOperator Time: 0.05 (IN) + 0.08 (OT) + 0.02 (GC) = 0.15 sec -- Nonlinear ODE => y(x)^3/6 + A y(x) = x + B ode:= D(y(x), x, 2) + y(x)*D(y(x), x)**3 = 0 ,, , 3 (2) y (x) + y(x)y (x) = 0 Type: Equation Expression Integer Time: 0.60 (IN) + 0.22 (EV) + 0.18 (OT) + 0.08 (GC) = 1.08 sec -- => y(x) = [3 x + sqrt(1 + 9 x^2)]^(1/3) - 1/[3 x + sqrt(1 + 9 x^2)]^(1/3) -- [Pos96] solve(ode, y, x = 0, [0, 2]) >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation initial (3) -> real 5.1 user 2.3 sys 0.2 ------------------------------------------------------------------------------- Tue Mar 24 19:56:49 MET 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Ordinary Difference and Differential Equations ---------- y:= operator('y); Type: BasicOperator Time: 0.05 (IN) + 0.08 (OT) + 0.02 (GC) = 0.15 sec -- A simple parametric ODE: y(x, a) = A e^(a x) D(y(x, a), x) = a*y(x, a) (2) y (x,a)= a y(x,a) ,1 Type: Equation Expression Integer Time: 0.47 (IN) + 0.15 (EV) + 0.18 (OT) + 0.07 (GC) = 0.86 sec solve(%, y, x); >> Error detected within library code: parseODE: equation has order 0 initial (3) -> real 2.2 user 1.6 sys 0.2 ------------------------------------------------------------------------------- Tue Mar 24 20:03:01 MET 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Ordinary Difference and Differential Equations ---------- y:= operator('y); Type: BasicOperator Time: 0.05 (IN) + 0.08 (OT) = 0.13 sec -- ODE with boundary conditions. This problem has nontrivial solutions -- y(x) = A sin([pi/2 + n pi] x) for n an arbitrary integer solve(D(y(x), x, 2) + k**2*y(x) = 0, y, x = 0, [0]) (2) 0 Type: Union(Expression Integer,...) Time: 1.15 (IN) + 0.97 (EV) + 0.17 (OT) + 0.17 (GC) = 2.45 sec --bc(%, x = 0, y = 0, x = 1, D(y(x), x) = 0) -- => y(x) = Z_v[sqrt(x)] where Z_v is an arbitrary Bessel function of order v -- [Gradshteyn and Ryzhik 8.491(9)] D(y(x), x, 2) + 1/x*D(y(x), x) + 1/(4*x)*(1 - v**2/x)*y(x) = 0 2 ,, , 2 4x y (x) + 4xy (x) + (x - v )y(x) (3) ----------------------------------= 0 2 4x Type: Equation Expression Integer Time: 0.35 (IN) + 0.03 (EV) + 0.08 (OT) = 0.47 sec solve(%, y, x) 2 2 v WARNING (genufact): No known algorithm to factor ? - --, trying square-free. 4 WARNING (genufact): No known algorithm to factor 3 2 2 2 ? - 3? + (- v + 3)? + v - 1, trying square-free. (4) [particular= 0,basis= []] Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) Time: 0.02 (IN) + 81.03 (EV) + 0.02 (OT) + 17.42 (GC) = 98.48 sec -- Delay (or mixed differential-difference) equation. See Daniel Zwillinger, -- _Handbook of Differential Equations_, Second Edition, Academic Press, Inc., -- 1992, p. 210 => y(t) = y0 sum((-a)^n (t - n + 1)^n/n!, n = 0..floor(t) + 1) D(y(t), t) + a*y(t - 1) = 0 , (5) y (t) + a y(t - 1)= 0 Type: Equation Expression Integer Time: 0.22 (IN) + 0.07 (OT) = 0.28 sec solve(%, y, t) t ++ (6) | a y(%V - 1)d%V + y(t) ++ Type: Union(Expression Integer,...) Time: 0.02 (IN) + 0.30 (EV) + 0.02 (OT) + 0.03 (GC) = 0.37 sec -- Discontinuous ODE [Zwillinger, p. 221] -- => y(t) = cosh t (0 <= t < T) -- (sin T cosh T + cos T sinh T) sin t -- + (cos T cosh T - sin T sinh T) cos t (T <= t) sgn(t) == if t < 0 then -1 else 1; Type: Void Time: 0 sec solve(D(y(t), t, 2) + sgn(t - TT)*y(t) = 0, y, t = 0, [1, 0]) Compiling function sgn with type Polynomial Integer -> Integer (8) cos(t) Type: Union(Expression Integer,...) Time: 0.15 (IN) + 0.13 (EV) + 0.03 (OT) = 0.32 sec )clear properties sgn Compiled code for sgn has been cleared. solve(D(y(t), t, 2) + sign(t - TT)*y(t) = 0, y, t = 0, [1, 0]) There are 31 exposed and 17 unexposed library operations named * having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op * to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named * with argument type(s) failed Expression Integer -- Integro-differential equation. See A. E. Fitzgerald, David E. Higginbotham -- and Arvin Grabel, _Basic Electrical Engineering_, Fourth Edition, -- McGraw-Hill Book Company, 1975, p. 117. -- => i(t) = 5/13 [-8 e^(-4 t) + e^(-t) (8 cos 2 t + sin 2 t)] i:= operator('i); Type: BasicOperator Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec eqn:= D(i(t), t) + 2*i(t) + 5*'integrate(i(tau), tau = 0..t) = 10*%e**(-4*t) , - 4t (10) i (t) + 2i(t) + 5integrate(i(tau),tau= 0,t )= 10%e Type: Equation OutputForm Time: 0.17 (IN) + 0.05 (EV) + 0.03 (OT) = 0.25 sec solve(eqn, i, t) There are 6 exposed and 1 unexposed library operations named solve having 3 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation OutputForm BasicOperator Variable t map(e +-> laplace(e, t, s), eqn) There are 1 exposed and 0 unexposed library operations named laplace having 3 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op laplace to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named laplace with argument type(s) OutputForm Variable t Variable s AXIOM will attempt to step through and interpret the code. Anonymous user functions created with +-> that are processed in interpret-code mode must have result target information available. This information is not present so AXIOM cannot proceed any further. This may be remedied by declaring the function. --subst(%, [t = 0, D(i(t), t) = 10]) )clear properties i eqn -- System of two linear, constant coefficient ODEs: -- x(t) = e^t [A cos(t) - B sin(t)], y(t) = e^t [A sin(t) + B cos(t)] x:= operator('x); Type: BasicOperator Time: 0 sec system:= [D(x(t), t) = x(t) - y(t), D(y(t), t) = x(t) + y(t)] , , (12) [x (t)= - y(t) + x(t),y (t)= y(t) + x(t)] Type: List Equation Expression Integer Time: 0.05 (IN) + 0.03 (EV) + 0.02 (OT) = 0.10 sec -- Check the answer ans:= [x(t) = %e^t*(A*cos(t) - B*sin(t)), y(t) = %e^t*(A*sin(t) + B*cos(t))] t t t t (13) [x(t)= - B %e sin(t) + A cos(t)%e ,y(t)= A %e sin(t) + B cos(t)%e ] Type: List Equation Expression Integer Time: 0.25 (IN) + 0.07 (EV) + 0.02 (OT) = 0.33 sec dans:= map(e +-> D(lhs(e), t) = D(rhs(e), t), ans) (14) , t t [x (t)= (- B - A)%e sin(t) + (- B + A)cos(t)%e , , t t y (t)= (- B + A)%e sin(t) + (B + A)cos(t)%e ] Type: List Equation Expression Integer Time: 0.02 (IN) + 0.07 (EV) + 0.02 (OT) = 0.10 sec map(e +-> subst(lhs(e), dans) = subst(rhs(e), ans), system) (15) [ t t (- B - A)%e sin(t) + (- B + A)cos(t)%e = t t (- B - A)%e sin(t) + (- B + A)cos(t)%e , t t t t (- B + A)%e sin(t) + (B + A)cos(t)%e = (- B + A)%e sin(t) + (B + A)cos(t)%e ] Type: List Equation Expression Integer Time: 0.02 (IN) + 0.03 (EV) + 0.03 (OT) = 0.08 sec )clear properties ans dans -- Triangular system of two ODEs: x(t) = A e^t [sin(t) + 2], -- y(t) = A e^t [5 - cos(t) + 2 sin(t)]/5 + B e^(-t) -- See Nicolas Robidoux, ``Does Axiom Solve Systems of O.D.E.'s Like -- Mathematica?'', LA-UR-93-2235, Los Alamos National Laboratory, Los Alamos, -- New Mexico. system:= [D(x(t), t) = x(t) * (1 + cos(t)/(2 + sin(t))), _ D(y(t), t) = x(t) - y(t)] , x(t)sin(t) + x(t)cos(t) + 2x(t) , (16) [x (t)= -------------------------------,y (t)= - y(t) + x(t)] sin(t) + 2 Type: List Equation Expression Integer Time: 0.08 (IN) + 0.03 (EV) + 0.03 (OT) = 0.15 sec -- Try solving this system one equation at a time solve(system.1, x, t) t t (17) [particular= 0,basis= [%e sin(t) + 2%e ]] Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) Time: 0.70 (EV) + 0.02 (OT) + 0.03 (GC) = 0.75 sec x(t) = C1 * %.basis.1 t t (18) x(t)= C1 %e sin(t) + 2C1 %e Type: Equation Expression Integer Time: 0.17 (IN) + 0.02 (OT) = 0.18 sec solve(subst(system.2, %), y, t) (19) - t t 2 - t t 2 2C1 %e (%e ) sin(t) + (- C1 cos(t) + 5C1)%e (%e ) [particular= ------------------------------------------------------, 5 - t basis= [%e ]] Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) Time: 0.03 (IN) + 2.0 (EV) + 0.02 (OT) + 0.05 (GC) = 2.10 sec y(t) = simplify(%.particular) + C2 * %.basis.1 t t - t 2C1 %e sin(t) + (- C1 cos(t) + 5C1)%e + 5C2 %e (20) y(t)= -------------------------------------------------- 5 Type: Equation Expression Integer Time: 0.08 (IN) + 0.23 (EV) + 0.03 (OT) = 0.35 sec )clear properties x y x:= operator('x); Type: BasicOperator Time: 0 sec y:= operator('y); Type: BasicOperator Time: 0.02 (IN) = 0.02 sec z:= operator('z); Type: BasicOperator Time: 0.03 (IN) = 0.03 sec -- 3 x 3 linear system with constant coefficients: -- (1) real distinct characteristic roots (= 2, 1, 3) [Roxin, p. 109] -- => x(t) = A e^(2 t), y(t) = B e^t + C e^(3 t), -- z(t) = -A e^(2 t) - C e^(3 t) system:= [D(x(t), t) = 2*x(t), _ D(y(t), t) = -2*x(t) + y(t) - 2*z(t), _ D(z(t), t) = x(t) + 3*z(t)] , , , (24) [x (t)= 2x(t),y (t)= - 2z(t) + y(t) - 2x(t),z (t)= 3z(t) + x(t)] Type: List Equation Expression Integer Time: 0.13 (IN) + 0.03 (OT) = 0.17 sec -- (2) complex characteristic roots (= 0, -1 +- sqrt(2) i) [Roxin, p. 111] -- => x(t) = A + e^(-t)/3 [-(B + sqrt(2) C) cos(sqrt(2) t) + -- (sqrt(2) B - C) sin(sqrt(2) t)], -- y(t) = e^(-t) [B cos(sqrt(2) t) + C sin(sqrt(2) t)], -- z(t) = e^(-t) [(-B + sqrt(2) C) cos(sqrt(2) t) -- -(sqrt(2) B + C) sin(sqrt(2) t)] system:= [D(x(t), t) = y(t), D(y(t), t) = z(t), _ D(z(t), t) = -3*y(t) - 2*z(t)] , , , (25) [x (t)= y(t),y (t)= z(t),z (t)= - 2z(t) - 3y(t)] Type: List Equation Expression Integer Time: 0.08 (IN) + 0.02 (EV) + 0.03 (OT) = 0.13 sec -- (3) multiple characteristic roots (= 2, 2, 2) [Roxin, p. 113] -- => x(t) = e^(2 t) [A + C (1 + t)], y(t) = B e^(2 t), -- z(t) = e^(2 t) [A + C t] system:= [D(x(t), t) = 3*x(t) - z(t), D(y(t), t) = 2*y(t), _ D(z(t), t) = x(t) + z(t)] , , , (26) [x (t)= - z(t) + 3x(t),y (t)= 2y(t),z (t)= z(t) + x(t)] Type: List Equation Expression Integer Time: 0.57 (IN) + 0.03 (EV) + 0.03 (OT) + 0.47 (GC) = 1.10 sec -- x(t) = x0 + [4 sin(w t)/w - 3 t] x0' [Rick Niles] -- + 6 [w t - sin(w t)] y0 + 2/w [1 - cos(w t)] y0', -- y(t) = -2/w [1 - cos(w t)] x0' + [4 - 3 cos(w t)] y0 + sin(w t)/w y0' system:= [D(x(t), t, 2) = 2*w*D(y(t), t), _ D(y(t), t, 2) = -2*w*D(x(t), t) + 3*w**2*y(t)] ,, , ,, , 2 (27) [x (t)= 2wy (t),y (t)= - 2wx (t) + 3w y(t)] Type: List Equation Expression Integer Time: 0.13 (IN) + 0.03 (EV) + 0.02 (OT) = 0.18 sec )clear properties x y z system -- ---------- Quit ---------- )quit real 267.4 user 92.4 sys 1.8 \end{verbatim} \section{Partial Differential Equations} \begin{verbatim} -- ---------- Partial Differential Equations ---------- -- A very simple PDE => g(x) + h(y) for arbitrary functions g and h f:= operator('f); Type: BasicOperator Time: 0.07 (IN) + 0.10 (OT) + 0.05 (GC) = 0.22 sec D(f(x, y), [x, y]) = 0 (2) f (x,y)= 0 ,1,2 Type: Equation Expression Integer Time: 0.65 (IN) + 0.27 (EV) + 0.27 (OT) + 0.15 (GC) = 1.33 sec )clear properties f -- Heat equation: the fundamental solution is 1/sqrt(4 pi t) exp(-x^2/[4 t]). -- If f(x, t) and a(x, t) are solutions, the most general solution obtainable -- from f(x, t) by group transformations is of the form u(x, t) = a(x, t) -- + 1/sqrt(1 + 4 e6 t) exp(e3 - [e5 x + e6 x^2 - e5^2 t]/[1 + 4 e6 t]) -- f([e^(-e4) (x - 2 e5 t)]/[1 + 4 e6 t] - e1, [e^(-2 e4) t]/[1 + 4 e6 t] - e2) -- See Peter J. Olver, _Applications of Lie Groups to Differential Equations_, -- Second Edition, Springer Verlag, 1993, p. 120 (an excellent book). See also -- Heat.input u:= operator('u); Type: BasicOperator Time: 0.03 (IN) + 0.02 (OT) = 0.05 sec D(u(x, t), t) = D(u(x, t), x, 2) (4) u (x,t)= u (x,t) ,2 ,1,1 Type: Equation Expression Integer Time: 0.07 (IN) + 0.05 (EV) + 0.03 (OT) = 0.15 sec )clear properties u -- Potential equation on a circular disk---a separable PDE -- => v(r, theta) = a[0] + sum(a[n] r^n cos(n theta), n = 1..infinity) -- + sum(b[n] r^n sin(n theta), n = 1..infinity) v:= operator('v); Type: BasicOperator Time: 0.03 (IN) = 0.03 sec 1/r * D(r * D(v(r, theta), r), r) _ + 1/r**2 * D(v(r, theta), theta, 2) = 0 2 v (r,theta) + r v (r,theta) + rv (r,theta) ,2,2 ,1,1 ,1 (6) -------------------------------------------------= 0 2 r Type: Equation Expression Integer Time: 0.63 (IN) + 0.15 (EV) + 0.10 (OT) = 0.88 sec )clear properties v -- ---------- Quit ---------- )quit real 9.5 user 3.3 sys 0.3 \end{verbatim} \section{Operators} \begin{verbatim} -- ---------- Operators ---------- f(x) == exp(x) Type: Void Time: 0.02 (IN) = 0.02 sec g(x) == x**2 Type: Void Time: 0.02 (IN) = 0.02 sec -- (f + 2 g)(y) => e^y + 2 y^2 (f + 2*g)(y) There are 31 exposed and 17 unexposed library operations named * having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op * to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named * with argument type(s) PositiveInteger FunctionCalled g -- (f o g)(y) => e^(y^2) (f * g)(y) There are 31 exposed and 17 unexposed library operations named * having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op * to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named * with argument type(s) FunctionCalled f FunctionCalled g )clear properties f g -- Linear differential operator DD : LODO(Expression Integer, e +-> D(e, x)) := D() (3) D Type: LinearOrdinaryDifferentialOperator(Expression Integer,theMap("NIL",1)) Time: 0.08 (IN) + 0.02 (EV) + 0.15 (OT) + 0.07 (GC) = 0.32 sec L:= (DD - 1) * (DD + 2) 2 (4) D + D - 2 Type: LinearOrdinaryDifferentialOperator(Expression Integer,theMap("NIL",1)) Time: 0.13 (IN) + 0.10 (EV) + 0.03 (GC) = 0.27 sec f:= operator('f); Type: BasicOperator Time: 0.07 (IN) + 0.02 (OT) = 0.08 sec g:= operator('g); Type: BasicOperator Time: 0.05 (IN) = 0.05 sec -- => f'' + f' - 2 f L(f(x)) ,, , (7) f (x) + f (x) - 2f(x) Type: Expression Integer Time: 0.53 (IN) + 0.22 (EV) + 0.13 (OT) + 0.07 (GC) = 0.95 sec -- => g''(y) + g'(y) - 2 g(y) subst(L(subst(g(y), y = x)), x = y) ,, , (8) g (y) + g (y) - 2g(y) Type: Expression Integer Time: 0.54 (IN) + 0.08 (EV) + 0.10 (OT) = 0.72 sec -- => 2 A [(1 + z) cos(z^2) - (1 + 2 z^2) sin(z^2)] subst(L(subst(A * sin(z**2), z = x)), x = z) 2 2 2 (9) (- 4A z - 2A)sin(z ) + (2A z + 2A)cos(z ) Type: Expression Integer Time: 0.13 (IN) + 0.20 (EV) + 0.07 (OT) + 0.02 (GC) = 0.42 sec -- Truncated Taylor series operator T:= (f, xx, a) +-> subst((DD**0)(f(x)), x = a)/factorial(0) * (xx - a)**0 + _ subst((DD**1)(f(x)), x = a)/factorial(1) * (xx - a)**1 + _ subst((DD**2)(f(x)), x = a)/factorial(2) * (xx - a)**2 (10) (f,xx,a) +-> 0 1 subst(DD (f(x)),x= a) 0 subst(DD (f(x)),x= a) 1 --------------------- (xx - a) + --------------------- (xx - a) factorial(0) factorial(1) + 2 subst(DD (f(x)),x= a) 2 --------------------- (xx - a) factorial(2) Type: AnonymousFunction Time: 0.02 (OT) = 0.02 sec -- => f(a) + f'(a) (x - a) + f''(a) (x - a)^2/2 T(f, x, a) 2 2 ,, , (x - 2a x + a )f (a) + (2x - 2a)f (a) + 2f(a) (11) ----------------------------------------------- 2 Type: Expression Integer Time: 0.50 (IN) + 0.13 (EV) + 0.10 (OT) = 0.73 sec -- => g(b) + g'(b) (y - b) + g''(b) (y - b)^2/2 T(g, y, b) 2 2 ,, , (y - 2b y + b )g (b) + (2y - 2b)g (b) + 2g(b) (12) ----------------------------------------------- 2 Type: Expression Integer Time: 0.18 (IN) + 0.12 (EV) + 0.07 (OT) = 0.37 sec -- => sin(c) + cos(c) (z - c) - sin(c) (z - c)^2/2 T(sin, z, c) 2 2 (- z + 2c z - c + 2)sin(c) + (2z - 2c)cos(c) (13) ---------------------------------------------- 2 Type: Expression Integer Time: 0.18 (IN) + 0.07 (EV) + 0.05 (OT) = 0.30 sec )clear properties DD L f g T -- Define the binary infix operator ~ so that x ~ y => sqrt(x^2 + y^2) -- Make it associative: 3 ~ 4 ~ 12 => 13 -- Define the matchfix pair of operators | and | so that | x | => abs(x) -- ---------- Quit ---------- )quit real 6.8 user 5.8 sys 0.4 \end{verbatim} \section{Programming and Miscellaneous} \begin{verbatim} -- ---------- Programming and Miscellaneous ---------- -- How easy is it to substitute x for a + b in the following expression? -- => (x + c)^2 + (d - x)^2 expr:= (a + b + c)**2 + (d - a - b)**2 2 2 2 2 (1) d + (- 2b - 2a)d + c + (2b + 2a)c + 2b + 4a b + 2a Type: Polynomial Integer Time: 0.23 (IN) + 0.02 (EV) + 0.18 (OT) + 0.10 (GC) = 0.53 sec subst(expr, a + b = x) >> Error detected within library code: left hand side must be a single kernel initial (2) -> real 3.2 user 1.8 sys 0.2 ------------------------------------------------------------------------------- Mon Jan 5 00:26:00 MET 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected )set streams calculate 7 -- ---------- Programming and Miscellaneous ---------- -- How easy is it to substitute x for a + b in the following expression? -- => (x + c)^2 + (d - x)^2 expr:= (a + b + c)**2 + (d - a - b)**2 2 2 2 2 (1) d + (- 2b - 2a)d + c + (2b + 2a)c + 2b + 4a b + 2a Type: Polynomial Integer Time: 0.25 (IN) + 0.03 (EV) + 0.15 (OT) + 0.07 (GC) = 0.50 sec subst(expr, b = x - a) 2 2 2 (2) 2x + (- 2d + 2c)x + d + c Type: Expression Integer Time: 0.38 (IN) + 0.05 (EV) + 0.17 (OT) + 0.12 (GC) = 0.72 sec )clear properties expr -- How easy is it to substitute r for sqrt(x^2 + y^2) in the following -- expression? => x/r x/sqrt(x**2 + y**2) x (3) ---------- +-------+ | 2 2 \|y + x Type: Expression Integer Time: 0.30 (IN) + 0.07 (EV) + 0.12 (OT) = 0.48 sec subst(%, sqrt(x**2 + y**2) = r) x (4) - r Type: Expression Integer Time: 0.15 (IN) + 0.02 (EV) = 0.17 sec -- Change variables so that the following transcendental expression is -- converted into a rational expression [Vernor Vinge] -- => (r - 1)^4 (u^4 - r u^3 - r^3 u + r u + r^4)/[u^4 (2 r - 1)^2] q:= (1/r**4 + 1/(r**2 - 2*r*cos(t) + 1)**2 _ - 2*(r - cos(t))/(r**2 * (r**2 - 2*r*cos(t) + 1)**(3/2))) / _ (1/r**4 + 1/(r - 1)**4 - 2*(r - 1)/(r**2 * (r**2 - 2*r + 1)**(3/2))) (5) 6 5 4 3 2 2 (4r - 16r + 24r - 16r + 4r )cos(t) + 7 6 5 4 3 2 8 7 (- 4r + 16r - 28r + 32r - 28r + 16r - 4r)cos(t) + 2r - 8r + 6 5 4 3 2 14r - 16r + 15r - 12r + 8r - 4r + 1 * +-----------+ +--------------------+ | 2 | 2 \|r - 2r + 1 \|- 2r cos(t) + r + 1 + 7 6 5 4 3 2 (- 4r + 16r - 24r + 16r - 4r )cos(t) + 8 7 6 5 4 3 2 9 8 (6r - 24r + 38r - 32r + 18r - 8r + 2r )cos(t) - 2r + 8r + 7 6 5 4 3 - 14r + 16r - 14r + 8r - 2r * +-----------+ | 2 \|r - 2r + 1 / 6 5 4 3 2 2 (8r - 16r + 24r - 16r + 4r )cos(t) + 7 6 5 4 3 2 8 7 (- 8r + 16r - 32r + 32r - 28r + 16r - 4r)cos(t) + 2r - 4r + 6 5 4 3 2 10r - 12r + 15r - 12r + 8r - 4r + 1 * +-----------+ | 2 \|r - 2r + 1 + 7 6 5 4 2 (- 8r + 24r - 24r + 8r )cos(t) + 8 7 6 5 4 3 9 8 7 6 (8r - 24r + 32r - 32r + 24r - 8r )cos(t) - 2r + 6r - 10r + 14r + 5 4 3 2 - 14r + 10r - 6r + 2r * +--------------------+ | 2 \|- 2r cos(t) + r + 1 Type: Expression Integer Time: 0.61 (IN) + 0.28 (EV) + 0.28 (OT) + 0.07 (GC) = 1.25 sec subst(q, cos(t) = (r**2 - u**2 + 1)/(2*r)) (6) +-----------+ 4 3 2 4 8 7 6 5 4 | 2 ((r - 4r + 6r - 4r + 1)u + r - 4r + 6r - 4r + r )\|r - 2r + 1 * +--+ | 2 \|u + 5 4 3 2 4 7 6 5 3 2 2 ((- r + 4r - 6r + 4r - r)u + (- r + 4r - 5r + 5r - 4r + r)u ) * +-----------+ | 2 \|r - 2r + 1 / +-----------+ 4 3 2 4 | 2 (2r - 4r + 6r - 4r + 1)u \|r - 2r + 1 + 5 4 3 2 4 (- 2r + 6r - 6r + 2r )u * +--+ | 2 \|u Type: Expression Integer Time: 0.15 (IN) + 0.17 (EV) + 0.12 (OT) = 0.43 sec (rule sqrt(x**2) == x)(%) (7) 4 3 2 4 5 4 3 2 3 (r - 4r + 6r - 4r + 1)u + (- r + 4r - 6r + 4r - r)u + 7 6 5 3 2 8 7 6 5 4 (- r + 4r - 5r + 5r - 4r + r)u + r - 4r + 6r - 4r + r * +-----------+ | 2 \|r - 2r + 1 / +-----------+ 4 3 2 4 | 2 5 4 3 2 4 (2r - 4r + 6r - 4r + 1)u \|r - 2r + 1 + (- 2r + 6r - 6r + 2r )u Type: Expression Integer Time: 0.07 (IN) + 0.93 (EV) + 0.12 (OT) = 1.12 sec (rule sqrt(r**2 - 2*r + 1) == r - 1)(%) (8) 4 3 2 4 5 4 3 2 3 (r - 4r + 6r - 4r + 1)u + (- r + 4r - 6r + 4r - r)u + 7 6 5 3 2 8 7 6 5 4 (- r + 4r - 5r + 5r - 4r + r)u + r - 4r + 6r - 4r + r / 2 4 (4r - 4r + 1)u Type: Expression Integer Time: 0.03 (IN) + 0.47 (EV) + 0.07 (OT) = 0.57 sec map(factor, % :: Fraction Polynomial Integer) 4 4 3 3 4 (r - 1) (u - r u + (- r + r)u + r ) (9) -------------------------------------- 2 4 (2r - 1) u Type: Fraction Factored Polynomial Integer Time: 0.17 (IN) + 0.25 (EV) + 0.03 (OT) + 0.02 (GC) = 0.47 sec -- Establish a rule to symmetrize a differential operator: [Stanly Steinberg] -- f g'' + f' g' -> (f g')' f:= operator('f); Type: BasicOperator Time: 0.05 (IN) + 0.02 (OT) = 0.07 sec g:= operator('g); Type: BasicOperator Time: 0.03 (IN) = 0.03 sec symmetrize:= rule _ f(x)*D(g(x), x, 2) + D(f(x), x)*D(g(x), x) == D(f(x)*D(g(x), x), x) ,, , , (12) f(x)g (x) + f (x)g (x) + %B == 'D('f(x)'D('g(x),x),x) + %B Type: RewriteRule(Integer,Integer,Expression Integer) Time: 0.62 (IN) + 0.05 (EV) + 0.07 (OT) = 0.73 sec q:= f(x)*D(g(x), x, 2) + D(f(x), x)*D(g(x), x) ,, , , (13) f(x)g (x) + f (x)g (x) Type: Expression Integer Time: 0.03 (IN) + 0.03 (EV) + 0.05 (OT) = 0.12 sec symmetrize q There are no library operations named %diff Use HyperDoc Browse or issue )what op %diff to learn if there is any operation containing " %diff " in its name. Cannot find a definition or applicable library operation named %diff with argument type(s) Expression Integer Variable %%01 Variable x -- => 2 (f g')' + f g symmetrize 2*q + f(x)*g(x) ,, , , (14) 2f(x)g (x) + 2f (x)g (x) + f(x)g(x) Type: Expression Integer Time: 0.05 (IN) + 0.02 (OT) = 0.07 sec )clear properties f g q -- Infinite lists: [1 2 3 4 5 ...] * [1 3 5 7 9 ...] -- => [1 6 15 28 45 66 91 ...] l1:= [i for i in 1..]; Type: Stream PositiveInteger Time: 0.02 (IN) + 0.03 (OT) = 0.05 sec l2:= [2*i-1 for i in 1..]; Type: Stream Integer Time: 0.03 (IN) + 0.02 (EV) = 0.05 sec [l1.i*l2.i for i in 1..] (17) [1,6,15,28,45,66,91,...] Type: Stream Integer Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec )clear properties l1 l2 -- Write a simple program to compute Legendre polynomials p(n, x) == 1/(2**n*factorial(n)) * D((x**2 - 1)**n, x, n) Type: Void Time: 0 sec -- p[0](x) = 1, p[1](x) = x, p[2](x) = (3 x^2 - 1)/2, -- p[3](x) = (5 x^3 - 3 x)/2, p[4](x) = (35 x^4 - 30 x^2 + 3)/8 for i in 0..4 repeat _ (output(""); _ output(concat(["p(", string(i), ", x) = "])); _ output(p(i, x)) ) Compiling function p with type (NonNegativeInteger,Variable x) -> Polynomial Fraction Integer p(0, x) = 1 p(1, x) = x p(2, x) = 3 2 1 - x - - 2 2 p(3, x) = 5 3 3 - x - - x 2 2 p(4, x) = 35 4 15 2 3 -- x - -- x + - 8 4 8 Type: Void Time: 0.20 (IN) + 0.08 (EV) + 0.12 (OT) = 0.40 sec -- p[4](1) = 1 eval(p(4, x), x = 1) Compiling function p with type (PositiveInteger,Variable x) -> Polynomial Fraction Integer (20) 1 Type: Polynomial Fraction Integer Time: 0.63 (IN) + 0.02 (EV) + 0.12 (OT) = 0.77 sec -- Now, perform the same computation using a recursive definition pp(0, x) == 1 Type: Void Time: 0.02 (OT) = 0.02 sec pp(1, x) == x Type: Void Time: 0 sec pp(n, x) == ((2*n - 1)*x*pp(n - 1, x) - (n - 1)*pp(n - 2, x))/n Type: Void Time: 0 sec for i in 0..4 repeat _ (output(""); _ output(concat(["pp(", string(i), ", x) = "])); _ output(pp(i, x)) ) Compiling function pp with type (Integer,Variable x) -> Polynomial Fraction Integer pp(0, x) = 1 pp(1, x) = x pp(2, x) = 3 2 1 - x - - 2 2 pp(3, x) = 5 3 3 - x - - x 2 2 pp(4, x) = 35 4 15 2 3 -- x - -- x + - 8 4 8 Type: Void Time: 0.33 (IN) + 0.03 (EV) + 0.05 (OT) = 0.42 sec pp(4, 1) Compiling function pp with type (Integer,Integer) -> Fraction Integer +++ |*2;pp;1;initial| redefined (25) 1 Type: Fraction Integer Time: 0.08 (IN) + 0.05 (OT) = 0.13 sec )clear properties p pp Compiled code for p has been cleared. Compiled code for pp has been cleared. -- Iterative computation of Fibonacci numbers myfib(n) == ( _ local i, j, k, f; _ if n < 0 then _ error("undefined") _ else if n < 2 then _ n _ else _ (j:= 0, k:= 1, _ for i in 2..n repeat _ (f:= j + k, j:= k, k:= f), _ return(f))); Type: Void Time: 0.02 (IN) = 0.02 sec -- Convert the function into FORTRAN syntax outputAsFortran(myfib) There are 4 exposed and 0 unexposed library operations named outputAsFortran having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op outputAsFortran to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named outputAsFortran with argument type(s) FunctionCalled myfib -- Create a list of the first 11 values of the function. [myfib(i) for i in 0..10] Compiling function myfib with type NonNegativeInteger -> Any (27) [0,1,1,2,3,5,8,13,21,34,55] Type: List Any Time: 0.22 (IN) + 0.07 (OT) = 0.28 sec )clear properties myfib Compiled code for myfib has been cleared. -- Define the function p(x) = x^2 - 4 x + 7 such that p(lambda) = 0 for -- lambda = 2 +- i sqrt(3) and p(A) = @[[0 0], [0 0]] for A = @[[1 -2], [2 3]] -- (the lambda are the eigenvalues and p(x) is the characteristic polynomial of -- A) [Johnson and Reiss, p. 184] p(x) == x**2 - 4*x + 7 Type: Void Time: 0 sec p(2 + %i*sqrt(3)) Compiling function p with type Expression Complex Integer -> Expression Complex Integer (29) 0 Type: Expression Complex Integer Time: 0.43 (IN) + 0.08 (EV) + 0.12 (OT) + 0.03 (GC) = 0.67 sec p(matrix(@[[1, -2], [2, 3]])) There are 11 exposed and 5 unexposed library operations named + having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op + to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named + with argument type(s) Matrix Integer PositiveInteger AXIOM will attempt to step through and interpret the code. +0 0+ (30) | | +0 0+ Type: SquareMatrix(2,Integer) Time: 0.22 (IN) + 0.03 (EV) + 0.03 (OT) + 0.05 (GC) = 0.33 sec )clear properties p Compiled code for p has been cleared. -- Define a function to be the result of a calculation -log(x**2 - 2**(1/3)*x + 2**(2/3))/(6 * 2**(2/3)) _ + atan((2*x - 2**(1/3))/(2**(1/3) * sqrt(3))) / (2**(2/3) * sqrt(3)) _ + log(x + 2**(1/3))/(3 * 2**(2/3)) (31) 3+-+2 +-+ +-+ 3+-+2 3+-+ 2 +-+ 3+-+ (x \|2 - 1)\|3 - \|3 log(\|2 - x\|2 + x ) + 2\|3 log(\|2 + x) + 6atan(-----------------) 3 ----------------------------------------------------------------------------- 3+-+2 +-+ 6\|2 \|3 Type: Expression Integer Time: 0.65 (IN) + 0.55 (EV) + 0.13 (OT) = 1.33 sec function(%, f, x) (32) f Type: Symbol Time: 0.08 (IN) + 0.03 (EV) = 0.12 sec expr:= f(y) Compiling function f with type Variable y -> Expression Integer (33) 3+-+2 +-+ +-+ 3+-+2 3+-+ 2 +-+ 3+-+ (y \|2 - 1)\|3 - \|3 log(\|2 - y\|2 + y ) + 2\|3 log(\|2 + y) + 6atan(-----------------) 3 ----------------------------------------------------------------------------- 3+-+2 +-+ 6\|2 \|3 Type: Expression Integer Time: 0.23 (IN) + 0.22 (EV) + 0.17 (OT) = 0.62 sec -- Display the top-level structure of a nasty expression, hiding the -- lower-level details. name(mainKernel(expr)) (34) log Type: Symbol Time: 0 sec )clear properties expr f Compiled code for f has been cleared. -- Convert the following expression into TeX or LaTeX y = sqrt((exp(x**2) + exp(-x**2))/(sqrt(3)*x - sqrt(2))) +-------------+ | 2 2 | x - x |%e + %e (35) y= |------------- | +-+ +-+ \| x\|3 - \|2 Type: Equation Expression Integer Time: 0.20 (IN) + 0.07 (EV) + 0.05 (OT) = 0.32 sec outputAsTex(%) $$ y={\sqrt {{{{e \sp {x \sp 2}}+{e \sp {\left( -{x \sp 2} \right)}}} \over {{x \ {\sqrt {3}}} -{\sqrt {2}}}}}} \leqno(36) $$ Type: Void Time: 0.07 (IN) + 0.03 (EV) = 0.10 sec -- ---------- Quit ---------- )quit real 29.9 user 14.9 sys 0.7 \end{verbatim} \section{Makefile} <<*>>= TANGLE=/usr/local/bin/NOTANGLE WEAVE=/usr/local/bin/NOWEAVE LATEX=/usr/bin/latex LISP=/sei/lisp MAKEINDEX=/usr/bin/makeindex all: code doc run code: wester.pamphlet ${TANGLE} -Rcode wester.pamphlet >wester.lisp doc: ${WEAVE} -t8 -delay wester.pamphlet >wester.tex ${LATEX} wester.tex 2>/dev/null 1>/dev/null @{MAKEINDEX} wester.idx ${LATEX} wester.tex 2>/dev/null 1>/dev/null run: remake: ${TANGLE} -t8 wester.pamphlet >Makefile.wester @ \eject \begin{thebibliography}{99} \bibitem{1} Wester, Michael J. (ed) "Computer Algebra Systems" Wiley 1999 ISBN 0-471-98353-5 \end{thebibliography} \printindex \end{document} From unknown Thu Feb 2 14:23:56 -0600 2006 From: unknown Date: Thu, 02 Feb 2006 14:23:56 -0600 Subject: document still unavailable Message-ID: <20060202142356-0600@wiki.axiom-developer.org> The exact links I'm pointed to are http://wiki.axiom-developer.org/images/WesterProblemSet.pdf http://wiki.axiom-developer.org/images/WesterProblemSet.dvi and so forth From BillPage Thu Feb 2 15:20:22 -0600 2006 From: Bill Page Date: Thu, 02 Feb 2006 15:20:22 -0600 Subject: confusion ... re: document still unavailable. Message-ID: <20060202152022-0600@wiki.axiom-developer.org> I don't know exactly what's happening here. When I tried the links just now, they failed as you said. But I am sure that I did a 'refresh' (see bottom right corner of page) and tested this yesterday. 'refresh' recreates the files. So I did a 'refresh' again. And now they work again. Please try again. If the links don't work. Hit 'refresh' and see if that helps. From unknown Sat Feb 4 06:23:17 -0600 2006 From: unknown Date: Sat, 04 Feb 2006 06:23:17 -0600 Subject: Even more confusion! Message-ID: <20060204062317-0600@wiki.axiom-developer.org> When I click on any of those buttons, the same page indicating the non-existence of the document under consideration appears. But on hitting refresh, and then clicking on the button seem to work for me. What's refresh anyway? And what does tangle button, which makes it appearance on the rhs of the greyish download strip, does? From BillPage Sat Feb 4 11:02:28 -0600 2006 From: Bill Page Date: Sat, 04 Feb 2006 11:02:28 -0600 Subject: something seems remove the files Message-ID: <20060204110228-0600@wiki.axiom-developer.org> Must be some subtle coding error I guess, but something seems cause these files to disappear after some period of time. Right now I do not have any idea of the possible cause. The 'refresh' button causes the entire contents of the page to be re-generated from the underlying source (i.e. the source is what you see whey you click 'edit') and is the same as the contents of the 'src' link - it is the pamphlet source file itself. 'refresh' runs 'weave' on the pamphlet file to generate the LaTeX ('tex') file and then runs latex to generate the 'dvi', and dvips to generate the 'ps' file and ps2pdf to generate the 'pdf' and finally dvipng to display the graphic image of the first page. Pamphlet files are intended to be used for *both* documentation and the program code itself. 'tangle' is used to extract the code chunks (programs) from the pamphlet file. In the case of the Wester problem set, this pamphlet file does not contain any separately identified code chunks (but it should!). For more information about pamphlet files see [Literate Programming]. From BillPage Sat Feb 4 13:46:10 -0600 2006 From: Bill Page Date: Sat, 04 Feb 2006 13:46:10 -0600 Subject: bug fixed Message-ID: <20060204134610-0600@wiki.axiom-developer.org> I think I found what was causing the 'pdf' and related files to be deleted. It turned out that the LaTeX output generation in another part of AxiomWiki (actually part of LatexWiki - latexWrapper.py) was being too agressive about cleaning up what it assumed were just unneeded temporary files. The result was that if someone clicked 'Save' on some other page that contained some new LaTeX or Axiom code, then as a side-effect of generating the required png images, it also deleted files that it should not have. Please let me know if the problem of disappearing files happens again.
| Download: pdf dvi ps src tex log |
document still unavailable --unknown, Thu, 02 Feb 2006 14:23:56 -0600 reply
The exact links I'm pointed to are
http://wiki.axiom-developer.org/images/WesterProblemSet.pdf
http://wiki.axiom-developer.org/images/WesterProblemSet.dvi
and so forth
I don't know exactly what's happening here. When I tried
the links just now, they failed as you said. But I am sure
that I did a refresh (see bottom right corner of page)
and tested this yesterday. refresh recreates the files.
So I did a refresh again. And now they work again.
Please try again. If the links don't work. Hit refresh
and see if that helps.
What's refresh anyway? And what does tangle button, which makes it appearance on the rhs of the greyish download strip, does?
Must be some subtle coding error I guess, but something seems cause these files to disappear after some period of time. Right now I do not have any idea of the possible cause.The refresh button causes the entire contents of the page to
be re-generated from the underlying source (i.e. the source is
what you see whey you click edit) and is the same as the contents
of the src link - it is the pamphlet source file itself. refresh
runs weave on the pamphlet file to generate the LaTeX? (tex)
file and then runs latex to generate the dvi, and dvips to
generate the ps file and ps2pdf to generate the pdf and
finally dvipng to display the graphic image of the first page.
Pamphlet files are intended to be used for both documentation and the program code itself.
tangle is used to extract the code chunks (programs) from the
pamphlet file. In the case of the Wester problem set, this
pamphlet file does not contain any separately identified code
chunks (but it should!).
For more information about pamphlet files see [Literate Programming]?.
I think I found what was causing thepdf and related files to be
deleted. It turned out that the LaTeX? output generation in another
part of AxiomWiki? (actually part of LatexWiki? - latexWrapper.py) was
being too agressive about cleaning up what it assumed were just unneeded
temporary files. The result was that if someone clicked Save on some
other page that contained some new LaTeX? or Axiom code, then as a
side-effect of generating the required png images, it also deleted
files that it should not have.
Please let me know if the problem of disappearing files happens again.