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Works with ASCII text output formatting.
axiom
)set output tex off
 
axiom
)set output algebra on

axiom
solve([x^2 + y^2 - 2*(ax*x + ay*y) = l1, x^2 + y^2 - 2*(cx*x + cy*y) = l2],[x,y])
(1) [ (- 2cy + 2ay)y - l2 + l1 [x= ------------------------, 2cx - 2ax
2 2 2 2 2 (4cy - 8ay cy + 4cx - 8ax cx + 4ay + 4ax )y + 2 2 (4cy - 4ay)l2 + (- 4cy + 4ay)l1 + (8ax cx - 8ax )cy - 8ay cx + 8ax ay cx * y + 2 2 2 2 l2 + (- 2l1 + 4ax cx - 4ax )l2 + l1 + (- 4cx + 4ax cx)l1 = 0 ] ]
Type: List(List(Equation(Fraction(Polynomial(Integer)))))

But fails with LaTeX.

axiom
)set output tex on
 
axiom
)set output algebra off

The result of 0^0 depends on the type of '0':

axiom
(0::Float)^(0::Float)
>> Error detected within library code: 0^0 is undefined

The idea was, that defining 0^0 as 1 is ok whenever there is no notion of limit. However,

axiom
(0::EXPR INT)^(0::EXPR INT)

\label{eq1}1(1)
Type: Expression(Integer)

is not quite in line with this, I think. There has been some discussion on this subject on axiom-developer.

It is easy to change this behaviour, if we know better...

Let's see if the same happens here:
axiom
sinCosProducts := rule
  sin (x) * sin (y) == (cos(x-y) - cos(x+y))/2
  cos (x) * cos (y) == (cos(x-y) + cos(x+y))/2
  sin (x) * cos (y) == (sin(x-y) + sin(x+y))/2

\label{eq2}\begin{array}{@{}l}
\displaystyle
\left\{{{\%B \ {\sin \left({x}\right)}\ {\sin \left({y}\right)}}\mbox{\rm = =}{{-{\%B \ {\cos \left({y + x}\right)}}+{\%B \ {\cos \left({y - x}\right)}}}\over 2}}, \: \right.
\
\
\displaystyle
\left.{{\%C \ {\cos \left({x}\right)}\ {\cos \left({y}\right)}}\mbox{\rm = =}{{{\%C \ {\cos \left({y + x}\right)}}+{\%C \ {\cos \left({y - x}\right)}}}\over 2}}, \: \right.
\
\
\displaystyle
\left.{{\%D \ {\cos \left({y}\right)}\ {\sin \left({x}\right)}}\mbox{\rm = =}{{{\%D \ {\sin \left({y + x}\right)}}-{\%D \ {\sin \left({y - x}\right)}}}\over 2}}\right\} 
(2)
Type: Ruleset(Integer,Integer,Expression(Integer))

when typing --Bill Page, Mon, 30 Jan 2006 09:00:02 -0600 reply
When you are typing or when you cut-and-paste commands directly into the Axiom interpreter you must use an underscore character at the end of each incomplete line, and you must use the ( ) syntax instead of identation, like this:
  sinCosProducts := rule (_
  sin (x) * sin (y) == (cos(x-y) - cos(x+y))/2; _
  cos (x) * cos (y) == (cos(x-y) + cos(x+y))/2; _
  sin (x) * cos (y) == (sin(x-y) + sin(x+y))/2)

Alternatively, using a text editor you can enter the commands into a file called, for example sincos.input exactly as in MathActon? above and the use the command:

  )read sincos.input

axiom
guess([1, 5, 14, 34, 69, 135, 240, 416, 686, 1106], [guessRat], [guessSum, guessProduct, guessOne])

\label{eq3}\left[ \right](3)
Type: List(Expression(Integer))

conversion failed --Bill Page, Thu, 23 Mar 2006 22:21:41 -0600 reply
Unknown wrote:
z:=sum(myfn(x),x=1..10) -- This fails, why?

The reason this fails is because Axiom tries to evaluate myfn(x) first. But x is not yet an Integer so Axiom cannot compute myfn(x). I guess you were expecting Axiom to "wait" and not evaluate myfn(x) until after x has been assigned the value 1, right? But Axiom does not work this way.

The solution is to write myfn(x) so that is can be applied to something symbolic like x. For example something this:

axiom
myfn(i : Expression Integer) : Expression Integer == i
Function declaration myfn : Expression(Integer) -> Expression( Integer) has been added to workspace.
Type: Void
axiom
myfn(x)
axiom
Compiling function myfn with type Expression(Integer) -> Expression(
      Integer)

\label{eq4}x(4)
Type: Expression(Integer)
axiom
z:=sum(myfn(x),x=1..10)

\label{eq5}55(5)
Type: Expression(Integer)

Any hints for multivariate functions? --Bill (Name omitted), Fri, 24 Mar 2006 21:45:25 -0600 reply
Hi Bill:

Thanks for your quick response. I tried to respond to this earlier, but didn't see it in the sand box, please forgive me if you get multiple copies.

I tried to simplify the code from my original program, and generated a univariate function, however my actual code has a multivariate function, and your excellent hint on the use of the Expression qualifier on the parameter and return type which works great for the univariate function case appears to fail for multivarite functions. Please consider the following example.

axiom
a(n : Expression Integer, k : Expression Integer, p : Expression Float) : Expression Float == binomial(n,k) * p^(k) * (1.0-p)^(n-k)
Function declaration a : (Expression(Integer),Expression(Integer), Expression(Float)) -> Expression(Float) has been added to workspace.
Type: Void
axiom
output(a(4,3,0.25)) -- see that the function actually evaluates for sensible values
axiom
Compiling function a with type (Expression(Integer),Expression(
      Integer),Expression(Float)) -> Expression(Float) 
   0.046875
Type: Void
axiom
z := sum(a(4,i,0.25), i=1..3) --- this fails
There are 6 exposed and 2 unexposed library operations named sum having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op sum 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 sum with argument type(s) Expression(Float) SegmentBinding(PositiveInteger)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. output(z)
55
Type: Void

I did notice in the Axiom online book, chapter 6.6, around page 241, the recommendation to use untyped functions, which appears to allow Axiom to do inference on parameter and result type.

axiom
b(n, k, p) == binomial(n,k) * p^(k) * (1.0-p)^(n-k)
Type: Void
axiom
output(b(4,3,0.25)) -- see that the function actually evaluates for sensible values
axiom
Compiling function b with type (PositiveInteger,PositiveInteger,
      Float) -> Float 
   0.046875
Type: Void
axiom
z := sum(b(4,i,0.25), i=1..3) --- this fails
axiom
Compiling function b with type (PositiveInteger,Variable(i),Float)
       -> Expression(Float) 
   There are 6 exposed and 2 unexposed library operations named sum 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                               )display op sum
      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 sum with argument type(s) Expression(Float) SegmentBinding(PositiveInteger)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. output(z)
55
Type: Void

For univariate functions the approach

axiom
c(k) == binomial(4,k) * 0.25^k * (1.0 - 0.25)^(4-k) -- This approach is only a test, but is not suitable for my program
Type: Void
axiom
output(c(3)) -- test to see if function can be evaluated for sensible arguments
axiom
Compiling function c with type PositiveInteger -> Float 
   0.046875
Type: Void
axiom
z := sum(c(i), i=1..3) -- still doesn't work
axiom
Compiling function c with type Variable(i) -> Expression(Float) 
   There are 6 exposed and 2 unexposed library operations named sum 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                               )display op sum
      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 sum with argument type(s) Expression(Float) SegmentBinding(PositiveInteger)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. output(z)
55
Type: Void

But interestingly something like

axiom
d(k) == 1.5 * k -- coerce output to be a Float
Type: Void
axiom
z := sum(d(i), i=1..3) -- This works!
axiom
Compiling function d with type Variable(i) -> Polynomial(Float)

\label{eq6}9.0(6)
Type: Fraction(Polynomial(Float))
axiom
output(z)
9.0
Type: Void

Bill, thanks again for your quick help, unforutnatly I lack a local Axiom expert, any ideas would really be welcome here.

reduce(+,[...]?) = sum(...) --billpage, Sat, 25 Mar 2006 16:01:07 -0600 reply
Try this
axiom
z := reduce(+,[b(4,i,0.25) for  i in 1..3])

\label{eq7}0.6796875(7)
Type: Float

Handling the result from functions returning a matrix --Bill (Name omitted), Mon, 27 Mar 2006 08:10:26 -0600 reply
Hi all:

Thanks Bill Page for your help, it is much appreciated (although I used a for loop and not reduce :-)).

I'm having a bit of difficulty getting a Function returning a matrix to work as expected, perhaps it is just cockpit error, but I don't see the error of my ways.

axiom
CFM(Q : Matrix(Float)): Matrix(Float) ==
   x := nrows(Q)
   MyIdentityMatrix : Matrix(Float) := new(x, x, 0)
   for i in 1..nrows(MyIdentityMatrix) repeat
      MyIdnetityMatrix(i,i) := 1.0
   Ninv := MyIdnetityMatrix - Q
   N := inverse(Ninv)
   N
Function declaration CFM : Matrix(Float) -> Matrix(Float) has been added to workspace.
Type: Void
axiom
--test ComputeFundamentalMatrix
X := matrix[[0, 0.5, 0],[0.5, 0, 0.5],[0, 0.5, 0]]

\label{eq8}\left[ 
\begin{array}{ccc}
{0.0}&{0.5}&{0.0}
\
{0.5}&{0.0}&{0.5}
\
{0.0}&{0.5}&{0.0}
(8)
Type: Matrix(Float)
axiom
output(X)
+0.0 0.5 0.0+ | | |0.5 0.0 0.5| | | +0.0 0.5 0.0+
Type: Void
axiom
N := CFM(X)
The form on the left hand side of an assignment must be a single variable, a Tuple of variables or a reference to an entry in an object supporting the setelt operation. output(N)
N
Type: Void

Any ideas where I'm blowing it here? I tried explicitly setting N to be a Matrix type but that failed too.

axiom
CFM(Q : Matrix(Float)): Matrix(Float) ==
   x := nrows(Q)
   MyIdentityMatrix : Matrix(Float) := new(x, x, 0)
   for i in 1..nrows(MyIdentityMatrix) repeat
      MyIdnetityMatrix(i,i) := 1.0
   Ninv := MyIdnetityMatrix - Q
   N := inverse(Ninv)
   N
Function declaration CFM : Matrix(Float) -> Matrix(Float) has been added to workspace. Compiled code for CFM has been cleared. 1 old definition(s) deleted for function or rule CFM
Type: Void
axiom
--test ComputeFundamentalMatrix
X := matrix[[0, 0.5, 0],[0.5, 0, 0.5],[0, 0.5, 0]]

\label{eq9}\left[ 
\begin{array}{ccc}
{0.0}&{0.5}&{0.0}
\
{0.5}&{0.0}&{0.5}
\
{0.0}&{0.5}&{0.0}
(9)
Type: Matrix(Float)
axiom
output(X)
+0.0 0.5 0.0+ | | |0.5 0.0 0.5| | | +0.0 0.5 0.0+
Type: Void
axiom
N : Matrix(Float) := CFM(X)
The form on the left hand side of an assignment must be a single variable, a Tuple of variables or a reference to an entry in an object supporting the setelt operation. output(N)
N is declared as being in Matrix(Float) but has not been given a value.

Thanks again for all your help.

Regards:

Bill M. (Sorry, my unique last name attracts too much spam).

typo and identity --billpage, Mon, 27 Mar 2006 09:34:35 -0600 reply
although I used a for loop and not reduce :-)

Good thinking. ;)

You have a simple typographical error. You have written both:

  MyIdentityMatrix

and :

  MyIdnetityMatrix

BTW, instead of the complicated construction of the identify matrix you should just write:

  Ninv := 1 - Q

For matrices 1 denotes the identity.

axiom
)set output tex off
 
axiom
)set output algebra on
FunFun := x^4 - 6* x^3 + 11* x^2 + 2* x + 1
4 3 2 (28) x - 6x + 11x + 2x + 1
Type: Polynomial(Integer)
axiom
radicalSolve(FunFun)
(29) [ x = - ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 - 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 30 |--------------------- - 169 3| +-+ \| 27\|3 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 - 144 |--------------------- 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 |------------------------------------------------------------- + 3 | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 / 2 ,
x = ROOT +---------------------+2 +---------------------+ | +-+ +----+ | +-+ +----+ |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 - 9 |--------------------- + 30 |--------------------- 3| +-+ 3| +-+ \| 27\|3 \| 27\|3 + - 169 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 - 144 |--------------------- 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 * +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 |------------------------------------------------------------- | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 + +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 |------------------------------------------------------------- + 3 | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 / 2 ,
x = - ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 - 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 30 |--------------------- - 169 3| +-+ \| 27\|3 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 144 |--------------------- 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 - |------------------------------------------------------------- + 3 | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 / 2 ,
x = ROOT +---------------------+2 +---------------------+ | +-+ +----+ | +-+ +----+ |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 - 9 |--------------------- + 30 |--------------------- 3| +-+ 3| +-+ \| 27\|3 \| 27\|3 + - 169 * ROOT +---------------------+2 | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 15 |--------------------- + 169 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 + +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 144 |--------------------- 3| +-+ \| 27\|3 / +---------------------+ | +-+ +----+ |2069\|3 + 144\|- 79 9 |--------------------- 3| +-+ \| 27\|3 * +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 |------------------------------------------------------------- | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 + +-------------------------------------------------------------+ | +---------------------+2 +---------------------+ | | +-+ +----+ | +-+ +----+ | |2069\|3 + 144\|- 79 |2069\|3 + 144\|- 79 |9 |--------------------- + 15 |--------------------- + 169 | 3| +-+ 3| +-+ | \| 27\|3 \| 27\|3 - |------------------------------------------------------------- + 3 | +---------------------+ | | +-+ +----+ | |2069\|3 + 144\|- 79 | 9 |--------------------- | 3| +-+ \| \| 27\|3 / 2 ]
Type: List(Equation(Expression(Integer)))
axiom
)set output tex on
 
axiom
)set output algebra off

Matthias

axiom
t:=matrix ([[0,1,1],[1,-2,2],[1,2,-1]])

\label{eq10}\left[ 
\begin{array}{ccc}
0 & 1 & 1 
\
1 & - 2 & 2 
\
1 & 2 & - 1 
(10)
Type: Matrix(Integer)

We can diagonalise t by finding it's eigenvalues.

axiom
)set output tex off
 
axiom
)set output algebra on
e:=radicalEigenvectors(t)
(31) [ +-----------------+2 +-----------------+ | +-+ +------+ | +-+ +------+ |3\|3 + \|- 1345 |3\|3 + \|- 1345 3 |----------------- - 3 |----------------- + 7 3| +-+ 3| +-+ \| 6\|3 \| 6\|3 [radval= --------------------------------------------------, radmult= 1, +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 3 |----------------- 3| +-+ \| 6\|3
radvect = [ [ [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 12\|3 |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +-+ +------+ |3\|3 + \|- 1345 +-+ +------+ (60\|3 + 6\|- 1345 ) |----------------- + 205\|3 + 3\|- 1345 3| +-+ \| 6\|3 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 126\|3 |----------------- 3| +-+ \| 6\|3 ] ,
[ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 6\|3 |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +-+ +------+ |3\|3 + \|- 1345 +-+ +------+ (117\|3 - 3\|- 1345 ) |----------------- - 71\|3 + 9\|- 1345 3| +-+ \| 6\|3 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 126\|3 |----------------- 3| +-+ \| 6\|3 ] , [1]] ] ] ,
[ radval = +-----------------+2 | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 - 3) |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 + 3) |----------------- + 14 3| +-+ \| 6\|3 / +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (3\|- 3 - 3) |----------------- 3| +-+ \| 6\|3 , radmult= 1,
radvect = [ [ [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 24\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((- 60\|- 3 - 60)\|3 - 6\|- 1345 \|- 3 - 6\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (205\|- 3 - 205)\|3 + 3\|- 1345 \|- 3 - 3\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] ,
[ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 12\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((- 117\|- 3 - 117)\|3 + 3\|- 1345 \|- 3 + 3\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (- 71\|- 3 + 71)\|3 + 9\|- 1345 \|- 3 - 9\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [1]] ] ] ,
[ radval = +-----------------+2 | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 + 3) |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 - 3) |----------------- - 14 3| +-+ \| 6\|3 / +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (3\|- 3 + 3) |----------------- 3| +-+ \| 6\|3 , radmult= 1,
radvect = [ [ [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 24\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((60\|- 3 - 60)\|3 + 6\|- 1345 \|- 3 - 6\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (- 205\|- 3 - 205)\|3 - 3\|- 1345 \|- 3 - 3\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] ,
[ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 12\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((117\|- 3 - 117)\|3 - 3\|- 1345 \|- 3 + 3\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (71\|- 3 + 71)\|3 - 9\|- 1345 \|- 3 - 9\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [1]] ] ] ]
Type: List(Record(radval: Expression(Integer),radmult: Integer,radvect: List(Matrix(Expression(Integer)))))
axiom
d:=diagonalMatrix([e.1.radval,e.2.radval,e.3.radval])
Function definition for d is being overwritten. Compiled code for d has been cleared.
(32) +-----------------+2 +-----------------+ | +-+ +------+ | +-+ +------+ |3\|3 + \|- 1345 |3\|3 + \|- 1345 3 |----------------- - 3 |----------------- + 7 3| +-+ 3| +-+ \| 6\|3 \| 6\|3 [[--------------------------------------------------,0,0], +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 3 |----------------- 3| +-+ \| 6\|3
[0,
+-----------------+2 | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 - 3) |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 + 3) |----------------- + 14 3| +-+ \| 6\|3 / +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (3\|- 3 - 3) |----------------- 3| +-+ \| 6\|3 , 0] ,
[0, 0,
+-----------------+2 | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 + 3) |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (- 3\|- 3 - 3) |----------------- - 14 3| +-+ \| 6\|3 / +-----------------+ | +-+ +------+ +---+ |3\|3 + \|- 1345 (3\|- 3 + 3) |----------------- 3| +-+ \| 6\|3 ] ]
Type: Matrix(Expression(Integer))

Now prove it by constructing the simularity transformation from the eigenvectors:

axiom
p:=horizConcat(horizConcat(e.1.radvect.1,e.2.radvect.1),e.3.radvect.1)
(33) [ [ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 12\|3 |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +-+ +------+ |3\|3 + \|- 1345 +-+ +------+ (60\|3 + 6\|- 1345 ) |----------------- + 205\|3 + 3\|- 1345 3| +-+ \| 6\|3 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 126\|3 |----------------- 3| +-+ \| 6\|3 ,
+-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 24\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((- 60\|- 3 - 60)\|3 - 6\|- 1345 \|- 3 - 6\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (205\|- 3 - 205)\|3 + 3\|- 1345 \|- 3 - 3\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ,
+-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 - 24\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((60\|- 3 - 60)\|3 + 6\|- 1345 \|- 3 - 6\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (- 205\|- 3 - 205)\|3 - 3\|- 1345 \|- 3 - 3\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] ,
[ +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 6\|3 |----------------- 3| +-+ \| 6\|3 + +-----------------+ | +-+ +------+ +-+ +------+ |3\|3 + \|- 1345 +-+ +------+ (117\|3 - 3\|- 1345 ) |----------------- - 71\|3 + 9\|- 1345 3| +-+ \| 6\|3 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 126\|3 |----------------- 3| +-+ \| 6\|3 ,
+-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 12\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((- 117\|- 3 - 117)\|3 + 3\|- 1345 \|- 3 + 3\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (- 71\|- 3 + 71)\|3 + 9\|- 1345 \|- 3 - 9\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ,
+-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 12\|3 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ ((117\|- 3 - 117)\|3 - 3\|- 1345 \|- 3 + 3\|- 1345 ) * +-----------------+ | +-+ +------+ |3\|3 + \|- 1345 |----------------- 3| +-+ \| 6\|3 + +---+ +-+ +------+ +---+ +------+ (71\|- 3 + 71)\|3 - 9\|- 1345 \|- 3 - 9\|- 1345 / +-----------------+2 | +-+ +------+ +-+ |3\|3 + \|- 1345 252\|3 |----------------- 3| +-+ \| 6\|3 ] , [1,1,1]]
Type: Matrix(Expression(Integer))
axiom
p*d*inverse(p)
+0 1 1 + | | (34) |1 - 2 2 | | | +1 2 - 1+
Type: Matrix(Expression(Integer))
axiom
)set output tex on
 
axiom
)set output algebra off

\end{axiom}

Axiom can't integrame exp(x^4) ;( --unknown, Fri, 28 Apr 2006 14:03:28 -0500 reply
Axiom can't integrame exp(x^4) ;(

axiom
integrate(exp(x^4),x)

\label{eq11}\int^{
\displaystyle
x}{{{e}^{{\%E}^{4}}}\ {d \%E}}(11)
Type: Union(Expression(Integer),...)

But Maple can...

axiom
f(x) == (1/4)*x*(-Gamma(1/4,-x^4)*Gamma(3/4)+%pi*sqrt(2))/((-x^4)^(1/4)*Gamma(3/4))
Type: Void
axiom
D(f(x),x)
axiom
Compiling function f with type Variable(x) -> Expression(Integer)

\label{eq12}{e}^{{x}^{4}}(12)
Type: Expression(Integer)

Axiom cannot integrate e^(4*x) --kratt6, Fri, 28 Apr 2006 16:34:16 -0500 reply
This is not a big surprise: note that Gamma(x,y) is not an elementary function.

Martin

This was wrong:
axiom
integrate(1/(1+x^4),x=%minusInfinity..%plusInfinity)

\label{eq13}{\pi \ {\sqrt{2}}}\over 2(13)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)
axiom
numeric(integrate(1/(1+x^4),x=0..1))

\label{eq14}0.8669729873_3991103758(14)
Type: Float

axiom
)clear co
All user variables and function definitions have been cleared. All )browse facility databases have been cleared. Internally cached functions and constructors have been cleared. )clear completely is finished. n := 32

\label{eq15}32(15)
axiom
y : FARRAY INT := new(n,1)

\label{eq16}\begin{array}{@{}l}
\displaystyle
\left[ 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \right.
\
\
\displaystyle
\left.\: 1, \: 1, \: 1, \: 1, \: 1, \: 1 \right] 
(16)
Type: FlexibleArray?(Integer)
axiom
n0 := n

\label{eq17}32(17)
axiom
n1 := sum(x^1, x=0..n-1)

\label{eq18}496(18)
Type: Fraction(Polynomial(Integer))
axiom
n2 := sum(x^2, x=0..n-1)

\label{eq19}10416(19)
Type: Fraction(Polynomial(Integer))
axiom
n3 := sum(x^3, x=0..n-1)

\label{eq20}246016(20)
Type: Fraction(Polynomial(Integer))
axiom
n4 := sum(x^4, x=0..n-1)

\label{eq21}6197520(21)
Type: Fraction(Polynomial(Integer))
axiom
A := matrix([[n4, n3, n2],_
            [n3, n2, n1],_
            [n2, n1, n0]])

\label{eq22}\left[ 
\begin{array}{ccc}
{6197520}&{246016}&{10416}
\
{246016}&{10416}&{496}
\
{10416}&{496}&{32}
(22)
Type: Matrix(Fraction(Polynomial(Integer)))
axiom
X := vector([x1, x2, x3])

\label{eq23}\left[ x 1, \: x 2, \: x 3 \right](23)
Type: Vector(OrderedVariableList([x1,x2,x3]))
axiom
B := vector([sum(x^2* u, x=0..n-1),_
       sum(x*   v, x=0..n-1),_
       sum(     w, x=0..n-1)])

\label{eq24}\left[{{10416}\  u}, \:{{496}\  v}, \:{{32}\  w}\right](24)
Type: Vector(Fraction(Polynomial(Integer)))
axiom
solve([A * X = B], [x1, x2, x3])
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(Equation(Vector(Fraction(Polynomial(Integer))))) List(OrderedVariableList([x1,x2,x3]))
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

can this be correct? --unknown, Tue, 30 May 2006 23:51:26 -0500 reply
axiom
integrate(1/((x+t)*sqrt(1+(x*t)^2)),t=0..%plusInfinity,"noPole")

\label{eq25}{\left(
\begin{array}{@{}l}
\displaystyle
- 
\
\
\displaystyle
{\log{{\left({{{{\left({2 \ {{x}^{10}}}-{4 \ {{x}^{8}}}+{6 \ {{x}^{6}}}-{6 \ {{x}^{4}}}+{4 \ {{x}^{2}}}- 2 \right)}\ {\sqrt{{{x}^{4}}+ 1}}}+{2 \ {{x}^{12}}}-{4 \ {{x}^{10}}}+{7 \ {{x}^{8}}}-{8 \ {{x}^{6}}}+{7 \ {{x}^{4}}}-{4 \ {{x}^{2}}}+ 2}\over{{x}^{2}}}\right)}}}+ 
\
\
\displaystyle
{\log \left({{{x}^{6}}+{{x}^{2}}}\right)}
(25)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)
axiom
subst(%,x=1)

\label{eq26}0(26)
Type: Expression(Integer)
axiom
integrate(1/((1+t)*sqrt(1+(1*t)^2)),t=0..%plusInfinity,"noPole")

\label{eq27}{{\sqrt{2}}\ {\log \left({{{12}\ {\sqrt{2}}}+{17}}\right)}}\over 4(27)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)
axiom
simplify(%-subst((asinh(x^2)+asinh(1/x^2))/sqrt(1+x^4),x=1))

\label{eq28}{{\log \left({{{12}\ {\sqrt{2}}}+{17}}\right)}-{4 \ {asinh \left({1}\right)}}}\over{2 \ {\sqrt{2}}}(28)
Type: Expression(Integer)
axiom
%::Expression Float

\label{eq29}0.0(29)
Type: Expression(Float)

axiom
a := matrix([ [-1,0,0,0,1,0], [0,1,0,0,0,0], [0,0,2,0,0,-2], [0,0,0,4,0,0], [0,0,0,0,3,0], [0,0,-3,0,0,3]])

\label{eq30}\left[ 
\begin{array}{cccccc}
- 1 & 0 & 0 & 0 & 1 & 0 
\
0 & 1 & 0 & 0 & 0 & 0 
\
0 & 0 & 2 & 0 & 0 & - 2 
\
0 & 0 & 0 & 4 & 0 & 0 
\
0 & 0 & 0 & 0 & 3 & 0 
\
0 & 0 & - 3 & 0 & 0 & 3 
(30)
Type: Matrix(Integer)
axiom
determinant(a)

\label{eq31}0(31)
axiom
inverse(a)

\label{eq32}\mbox{\tt "failed"}(32)
Type: Union("failed",...)

axiom
As := matrix([ [-3,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]])

\label{eq33}\left[ 
\begin{array}{cccc}
- 3 & 1 & 1 & 1 
\
1 & 1 & 1 & 1 
\
1 & 1 & 1 & 1 
\
1 & 1 & 1 & 1 
(33)
Type: Matrix(Integer)
axiom
A := subMatrix(As, 2,4,2,4)

\label{eq34}\left[ 
\begin{array}{ccc}
1 & 1 & 1 
\
1 & 1 & 1 
\
1 & 1 & 1 
(34)
Type: Matrix(Integer)
axiom
ob := orthonormalBasis(A)

\label{eq35}\begin{array}{@{}l}
\displaystyle
\left[{\left[ 
\begin{array}{c}
-{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}
\
{{\sqrt{2}}\over{\sqrt{3}}}
\
-{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}
(35)
Type: List(Matrix(Expression(Integer)))
axiom
P : Matrix(Expression Integer) := new(3,3,0)

\label{eq36}\left[ 
\begin{array}{ccc}
0 & 0 & 0 
\
0 & 0 & 0 
\
0 & 0 & 0 
(36)
Type: Matrix(Expression(Integer))
axiom
setsubMatrix!(P,1,1,ob.3)

\label{eq37}\left[ 
\begin{array}{ccc}
{1 \over{\sqrt{3}}}& 0 & 0 
\
{1 \over{\sqrt{3}}}& 0 & 0 
\
{1 \over{\sqrt{3}}}& 0 & 0 
(37)
Type: Matrix(Expression(Integer))
axiom
setsubMatrix!(P,1,2,ob.1)

\label{eq38}\left[ 
\begin{array}{ccc}
{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& 0 
\
{1 \over{\sqrt{3}}}&{{\sqrt{2}}\over{\sqrt{3}}}& 0 
\
{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& 0 
(38)
Type: Matrix(Expression(Integer))
axiom
setsubMatrix!(P,1,3,ob.2)

\label{eq39}\left[ 
\begin{array}{ccc}
{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& -{1 \over{\sqrt{2}}}
\
{1 \over{\sqrt{3}}}&{{\sqrt{2}}\over{\sqrt{3}}}& 0 
\
{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{1 \over{\sqrt{2}}}
(39)
Type: Matrix(Expression(Integer))
axiom
Pt := transpose(P)

\label{eq40}\left[ 
\begin{array}{ccc}
{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}}
\
-{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{{\sqrt{2}}\over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}
\
-{1 \over{\sqrt{2}}}& 0 &{1 \over{\sqrt{2}}}
(40)
Type: Matrix(Expression(Integer))
axiom
Ps : Matrix(Expression Integer) := new(4,4,0)

\label{eq41}\left[ 
\begin{array}{cccc}
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
(41)
Type: Matrix(Expression(Integer))
axiom
Ps(1,1) := 1

\label{eq42}1(42)
Type: Expression(Integer)
axiom
setsubMatrix!(Ps,2,2,P)

\label{eq43}\left[ 
\begin{array}{cccc}
1 & 0 & 0 & 0 
\
0 &{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& -{1 \over{\sqrt{2}}}
\
0 &{1 \over{\sqrt{3}}}&{{\sqrt{2}}\over{\sqrt{3}}}& 0 
\
0 &{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{1 \over{\sqrt{2}}}
(43)
Type: Matrix(Expression(Integer))
axiom
PsT := transpose(Ps)

\label{eq44}\left[ 
\begin{array}{cccc}
1 & 0 & 0 & 0 
\
0 &{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}}
\
0 & -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{{\sqrt{2}}\over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}
\
0 & -{1 \over{\sqrt{2}}}& 0 &{1 \over{\sqrt{2}}}
(44)
Type: Matrix(Expression(Integer))
axiom
PsTAsPs := PsT * As * Ps

\label{eq45}\left[ 
\begin{array}{cccc}
- 3 &{3 \over{\sqrt{3}}}& 0 & 0 
\
{3 \over{\sqrt{3}}}& 3 & 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
(45)
Type: Matrix(Expression(Integer))
axiom
b1 := PsTAsPs(2,1)

\label{eq46}3 \over{\sqrt{3}}(46)
Type: Expression(Integer)
axiom
l1 := PsTAsPs(2,2)

\label{eq47}3(47)
Type: Expression(Integer)
axiom
Us : Matrix(Expression Integer) := new(4,4,0)

\label{eq48}\left[ 
\begin{array}{cccc}
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
(48)
Type: Matrix(Expression(Integer))
axiom
Us(1,1) := 1

\label{eq49}1(49)
Type: Expression(Integer)
axiom
Us(2,2) := 1

\label{eq50}1(50)
Type: Expression(Integer)
axiom
Us(3,3) := 1

\label{eq51}1(51)
Type: Expression(Integer)
axiom
Us(4,4) := 1

\label{eq52}1(52)
Type: Expression(Integer)
axiom
Us(2,1) := -b1 / l1

\label{eq53}-{1 \over{\sqrt{3}}}(53)
Type: Expression(Integer)
axiom
PsUs := Ps * Us

\label{eq54}\left[ 
\begin{array}{cccc}
1 & 0 & 0 & 0 
\
-{1 \over 3}&{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& -{1 \over{\sqrt{2}}}
\
-{1 \over 3}&{1 \over{\sqrt{3}}}&{{\sqrt{2}}\over{\sqrt{3}}}& 0 
\
-{1 \over 3}&{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{1 \over{\sqrt{2}}}
(54)
Type: Matrix(Expression(Integer))
axiom
PsUsT := transpose(PsUs)

\label{eq55}\left[ 
\begin{array}{cccc}
1 & -{1 \over 3}& -{1 \over 3}& -{1 \over 3}
\
0 &{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}}
\
0 & -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{{\sqrt{2}}\over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}
\
0 & -{1 \over{\sqrt{2}}}& 0 &{1 \over{\sqrt{2}}}
(55)
Type: Matrix(Expression(Integer))
axiom
PsUsTAsPsUs := PsUsT * As * PsUs

\label{eq56}\left[ 
\begin{array}{cccc}
- 4 & 0 & 0 & 0 
\
0 & 3 & 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
(56)
Type: Matrix(Expression(Integer))
axiom
C := inverse(PsUs)

\label{eq57}\left[ 
\begin{array}{cccc}
1 & 0 & 0 & 0 
\
{{\sqrt{3}}\over 3}&{{\sqrt{3}}\over 3}&{{\sqrt{3}}\over 3}&{{\sqrt{3}}\over 3}
\
0 & -{{\sqrt{3}}\over{3 \ {\sqrt{2}}}}&{{2 \ {\sqrt{3}}}\over{3 \ {\sqrt{2}}}}& -{{{\sqrt{2}}\ {\sqrt{3}}}\over 6}
\
0 & -{{\sqrt{2}}\over 2}& 0 &{{\sqrt{2}}\over 2}
(57)
Type: Union(Matrix(Expression(Integer)),...)
axiom
c := PsUsTAsPsUs(1,1)

\label{eq58}- 4(58)
Type: Expression(Integer)
axiom
gQ := PsUsTAsPsUs / c

\label{eq59}\left[ 
\begin{array}{cccc}
1 & 0 & 0 & 0 
\
0 & -{3 \over 4}& 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
(59)
Type: Matrix(Expression(Integer))
axiom
x1 := transpose(matrix([[1,2,3,4]]))

\label{eq60}\left[ 
\begin{array}{c}
1 
\
2 
\
3 
\
4 
(60)
Type: Matrix(Integer)
axiom
v1 := transpose(x1) * As * x1

\label{eq61}\left[ 
\begin{array}{c}
{96}
(61)
Type: Matrix(Integer)
axiom
x2 := C * x1

\label{eq62}\left[ 
\begin{array}{c}
1 
\
{{{10}\ {\sqrt{3}}}\over 3}
\
0 
\
{\sqrt{2}}
(62)
Type: Matrix(Expression(Integer))
axiom
v2 := transpose(x2) * PsUsTAsPsUs * x2

\label{eq63}\left[ 
\begin{array}{c}
{96}
(63)
Type: Matrix(Expression(Integer))

axiom
)clear all
All user variables and function definitions have been cleared. draw(y^2/2+(x^2-1)^2/4-1=0, x,y, range ==[-2..2, -1..1])
Graph data being transmitted to the viewport manager... FriCAS2D data being transmitted to the viewport manager...

\label{eq64}\mbox{\rm \hbox{\axiomType{TwoDimensionalViewport}\ } :}\mbox{\tt "FriCAS2D"}(64)
Type: TwoDimensionalViewport?

axiom
f1 := taylor(1 - x^2,x = 0)

\label{eq65}1 -{{x}^{2}}(65)
Type: UnivariateTaylorSeries?(Expression(Integer),x,0)
axiom
asin f1

\label{eq66}\begin{array}{@{}l}
\displaystyle
{\pi \over 2}-{{2 \over{\sqrt{2}}}\  x}-{{1 \over{6 \ {\sqrt{2}}}}\ {{x}^{3}}}-{{3 \over{{80}\ {\sqrt{2}}}}\ {{x}^{5}}}-{{5 \over{{448}\ {\sqrt{2}}}}\ {{x}^{7}}}- 
\
\
\displaystyle
{{{35}\over{{9216}\ {\sqrt{2}}}}\ {{x}^{9}}}+{O \left({{x}^{1
1}}\right)}
(66)
Type: UnivariateTaylorSeries?(Expression(Integer),x,0)
axiom
sin %

\label{eq67}1 -{{x}^{2}}+{O \left({{x}^{11}}\right)}(67)
Type: UnivariateTaylorSeries?(Expression(Integer),x,0)

SandboxMSkuce?

axiom
1+1

\label{eq68}2(68)

axiom
integrate((x-1)/log(x), x)

\label{eq69}{li \left({{x}^{2}}\right)}-{li \left({x}\right)}(69)
Type: Union(Expression(Integer),...)
axiom
integrate(x*exp(x)*sin(x),x)

\label{eq70}{{x \ {{e}^{x}}\ {\sin \left({x}\right)}}+{{\left(- x + 1 \right)}\ {\cos \left({x}\right)}\ {{e}^{x}}}}\over 2(70)
Type: Union(Expression(Integer),...)

Working With Lists --daneshpajouh, Sat, 16 Jun 2007 07:00:00 -0500 reply
axiom
[p for p in primes(2,1000)|(p rem 16)=1]

\label{eq71}\begin{array}{@{}l}
\displaystyle
\left[{17}, \:{97}, \:{113}, \:{193}, \:{241}, \:{257}, \:{33
7}, \:{353}, \:{401}, \:{433}, \:{449}, \:{577}, \:{593}, \: \right.
\
\
\displaystyle
\left.{641}, \:{673}, \:{769}, \:{881}, \:{929}, \:{977}\right] (71)
Type: List(Integer)
axiom
[p^2+1 for p in primes(2,100)]

\label{eq72}\begin{array}{@{}l}
\displaystyle
\left[ 5, \:{10}, \:{26}, \:{50}, \:{122}, \:{170}, \:{290}, \:{362}, \:{530}, \:{842}, \:{962}, \:{1370}, \:{1682}, \: \right.
\
\
\displaystyle
\left.{1850}, \:{2210}, \:{2810}, \:{3482}, \:{3722}, \:{4490}, \:{5042}, \:{5330}, \:{6242}, \:{6890}, \: \right.
\
\
\displaystyle
\left.{7922}, \:{9410}\right] 
(72)
Type: List(Integer)

axiom
integrate (2*x^2 + 2*x, x)

\label{eq73}{{2 \over 3}\ {{x}^{3}}}+{{x}^{2}}(73)
Type: Polynomial(Fraction(Integer))

axiom
radix(36,37)

\label{eq74}36(74)
Type: RadixExpansion?(37)

Is it error?

(better) example (with axiom markers this time) ;-) --pbwagner, Mon, 10 Sep 2007 13:01:48 -0500 reply
axiom
integrate(log(log(x)),x)

\label{eq75}{x \ {\log \left({\log \left({x}\right)}\right)}}-{li \left({x}\right)}(75)
Type: Union(Expression(Integer),...)




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