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spad
)abbrev domain RR Real
++ ===================================
++ THE REAL LINE MODELLED AS EXPR(INT)
++ ===================================
++ Modified version of Expression R 
++ Original header:
++ Author: Manuel Bronstein
++ Date Created: 19 July 1988
++ Date Last Updated: October 1993 (P.Gianni), February 1995 (MB)
++ Description: Expressions involving symbolic functions.
++ Keywords: operator, kernel, function.
Real : Exports == Implementation where
  R   ==> Integer
  Q   ==> Fraction Integer
  K   ==> Kernel %
  MP  ==> SparseMultivariatePolynomial(R, K)
  AF  ==> AlgebraicFunction(R, %)
  EF  ==> ElementaryFunction(R, %)
  CF  ==> CombinatorialFunction(R, %)
  LF  ==> LiouvillianFunction(R, %)
  AN  ==> AlgebraicNumber
  KAN ==> Kernel AN
  FSF ==> FunctionalSpecialFunction(R, %)
  ESD ==> ExpressionSpace_&(%)
  FSD ==> FunctionSpace_&(%, R)
  POWER  ==> '%power
  SUP    ==> SparseUnivariatePolynomial
Exports ==> FunctionSpace R with if R has IntegralDomain then AlgebraicallyClosedFunctionSpace R TranscendentalFunctionCategory CombinatorialOpsCategory LiouvillianFunctionCategory SpecialFunctionCategory reduce : % -> % ++ reduce(f) simplifies all the unreduced algebraic quantities ++ present in f by applying their defining relations. number? : % -> Boolean ++ number?(f) tests if f is rational simplifyPower : (%, Integer) -> % ++ simplifyPower(f, n) \undocumented{} if R has GcdDomain then factorPolynomial : SUP % -> Factored SUP % ++ factorPolynomial(p) \undocumented{} squareFreePolynomial : SUP % -> Factored SUP % ++ squareFreePolynomial(p) \undocumented{} if R has RetractableTo Integer then RetractableTo AN setSimplifyDenomsFlag : Boolean -> Boolean ++ setSimplifyDenomsFlag(x) sets flag affecting simplification ++ of denominators. If true irrational algebraics are removed from ++ denominators. If false they are kept. getSimplifyDenomsFlag : () -> Boolean ++ getSimplifyDenomsFlag() gets values of flag affecting ++ simplification of denominators. See setSimplifyDenomsFlag.
Implementation ==> add import from KernelFunctions2(R, %)
SYMBOL := '%symbol ALGOP := '%alg
retNotUnit : % -> R retNotUnitIfCan : % -> Union(R, "failed")
belong? op == true
retNotUnit x == (u := constantIfCan(k := retract(x)@K)) case R => u::R error "Not retractable"
retNotUnitIfCan x == (r := retractIfCan(x)@Union(K,"failed")) case "failed" => "failed" constantIfCan(r::K)
SPCH ==> SparsePolynomialCoercionHelpers(R, Symbol, K)
if R has Ring then poly_to_MP(p : Polynomial(R)) : MP == ps := p pretend SparseMultivariatePolynomial(R, Symbol) vl1 : List Symbol := variables(ps) vl2 : List K := [kernel(z)$K for z in vl1] remap_variables(ps, vl1, vl2)$SPCH
if R has IntegralDomain then reduc : (%, List Kernel %) -> % algreduc : (%, List Kernel %) -> % commonk : (%, %) -> List K commonk0 : (List K, List K) -> List K toprat : % -> % algkernels : List K -> List K algtower : % -> List K evl : (MP, K, SparseUnivariatePolynomial %) -> Fraction MP evl0 : (MP, K) -> SparseUnivariatePolynomial Fraction MP
Rep := Fraction MP 0 == 0$Rep 1 == 1$Rep one? x == (x = 1)$Rep zero? x == zero?(x)$Rep - x : % == -$Rep x n : Integer * x : % == n *$Rep x coerce(n : Integer) == coerce(n)$Rep@Rep::% x : % * y : % == algreduc(x *$Rep y, commonk(x, y)) x : % + y : % == algreduc(x +$Rep y, commonk(x, y)) (x : % - y : %) : % == algreduc(x -$Rep y, commonk(x, y)) x : % / y : % == algreduc(x /$Rep y, commonk(x, y))
number?(x : %) : Boolean == if R has RetractableTo(Integer) then ground?(x) or ((retractIfCan(x)@Union(Q,"failed")) case Q) else ground?(x)
simplifyPower(x : %, n : Integer) : % == k : List K := kernels x is?(x, POWER) => -- Look for a power of a number in case we can do -- a simplification args : List % := argument first k not(#args = 2) => error "Too many arguments to ^" number?(args.1) => reduc((args.1) ^$Rep n, algtower(args.1))^(args.2) (first args)^(n*second(args)) reduc(x ^$Rep n, algtower(x))
x : % ^ n : NonNegativeInteger == n = 0 => 1$% n = 1 => x simplifyPower(numerator x, n::Integer) / simplifyPower(denominator x, n::Integer)
x : % ^ n : Integer == n = 0 => 1$% n = 1 => x n = -1 => 1/x simplifyPower(numerator x, n) / simplifyPower(denominator x, n)
x : % ^ n : PositiveInteger == n = 1 => x simplifyPower(numerator x, n::Integer) / simplifyPower(denominator x, n::Integer)
smaller?(x : %, y : %) == smaller?(x, y)$Rep x : % = y : % == (x - y) =$Rep 0$Rep numer x == numer(x)$Rep denom x == denom(x)$Rep
EREP := Record(num : MP, den : MP)
coerce(p : MP) : % == [p, 1]$EREP pretend %
coerce(p : Polynomial(R)) : % == en := poly_to_MP(p) [en, 1]$EREP pretend %
coerce(pq : Fraction(Polynomial(R))) : % == en := poly_to_MP(numer(pq)) ed := poly_to_MP(denom(pq)) [en, ed]$EREP pretend %
reduce x == reduc(x, algtower x) commonk(x, y) == commonk0(algtower x, algtower y) algkernels l == select!(x +-> has?(operator x, ALGOP), l) toprat f == ratDenom(f, algtower f )$AlgebraicManipulations(R, %)
alg_ker_set(x : %) : List(K) == resl : List(K) := [] ak1 : List(K) := [] for k in kernels x repeat not(is?(k, 'nthRoot) or is?(k, 'rootOf)) => "iterate" ak1 := cons(k, ak1) while not(empty?(ak1)) repeat ak := ak1 ak1 := [] for k in ak repeat needed := true for k1 in resl while needed repeat if EQ(k1, k)$Lisp then needed := false for k1 in resl while needed repeat if k1 = k then needed := false not(needed) => "iterate" resl := cons(k, resl) ak1 := cons(k, ak1) arg := argument(k) for k1 in kernels(arg.1) repeat if (is?(k1, 'nthRoot) or is?(k1, 'rootOf)) then ak1 := cons(k1, ak1) resl
algtower(x : %) : List K == reverse!(sort! alg_ker_set(x))
simple_root(r : K) : Boolean == is?(r, 'nthRoot) => al := argument(r) al.2 ~= 2::% => false a := al.1 #algkernels(kernels(a)) > 0 => false true false
root_reduce(x : %, r : K) : % == a := argument(r).1 an := numer(a) dn := denom(a) dp := univariate(denom x, r) n0 := numer x c1 := leadingCoefficient(dp) c0 := leadingCoefficient(reductum(dp)) n1 := dn*(c0*n0 - monomial(1, r, 1)$MP*c1*n0) d1 := c0*c0*dn - an*c1*c1 reduc(n1 /$Rep d1, [r])
DEFVAR(algreduc_flag$Lisp, false$Boolean)$Lisp
getSimplifyDenomsFlag() == algreduc_flag$Lisp
setSimplifyDenomsFlag(x) == res := getSimplifyDenomsFlag() SETF(algreduc_flag$Lisp, x)$Lisp res
algreduc(x, ckl) == x1 := reduc(x, ckl) not(getSimplifyDenomsFlag()) => x1 akl := algtower(1$MP /$Rep denom x1) #akl = 0 => x1 if #akl = 1 then r := akl.1 simple_root(r) => return root_reduce(x, r) sas := create()$SingletonAsOrderedSet for k in akl repeat q := univariate(x1, k, minPoly k )$PolynomialCategoryQuotientFunctions(IndexedExponents K, K, R, MP, %) x1 := retract(eval(q, sas, k::%))@% reduc(x1, akl)
x : MP / y : MP == reduc(x /$Rep y, commonk(x /$Rep 1$MP, y /$Rep 1$MP))
-- since we use the reduction from FRAC SMP which asssumes -- that the variables are independent, we must remove algebraic -- from the denominators
reducedSystem(m : Matrix %) : Matrix(R) == mm : Matrix(MP) := reducedSystem(map(toprat, m))$Rep reducedSystem(mm)$MP
reducedSystem(m : Matrix %, v : Vector %): Record(mat : Matrix R, vec : Vector R) == r : Record(mat : Matrix MP, vec : Vector MP) := reducedSystem(map(toprat, m), map(toprat, v))$Rep reducedSystem(r.mat, r.vec)$MP
-- The result MUST be left sorted deepest first MB 3/90 commonk0(x, y) == ans := empty()$List(K) for k in reverse! x repeat if member?(k, y) then ans := concat(k, ans) ans
rootOf(x : SparseUnivariatePolynomial %, v : Symbol) == rootOf(x, v)$AF rootSum(x : %, p : SparseUnivariatePolynomial %, v : Symbol) : % == rootSum(x, p, v)$AF pi() == pi()$EF exp x == exp(x)$EF log x == log(x)$EF sin x == sin(x)$EF cos x == cos(x)$EF tan x == tan(x)$EF cot x == cot(x)$EF sec x == sec(x)$EF csc x == csc(x)$EF asin x == asin(x)$EF acos x == acos(x)$EF atan x == atan(x)$EF acot x == acot(x)$EF asec x == asec(x)$EF acsc x == acsc(x)$EF sinh x == sinh(x)$EF cosh x == cosh(x)$EF tanh x == tanh(x)$EF coth x == coth(x)$EF sech x == sech(x)$EF csch x == csch(x)$EF asinh x == asinh(x)$EF acosh x == acosh(x)$EF atanh x == atanh(x)$EF acoth x == acoth(x)$EF asech x == asech(x)$EF acsch x == acsch(x)$EF
abs x == abs(x)$FSF Gamma x == Gamma(x)$FSF Gamma(a, x) == Gamma(a, x)$FSF Beta(x, y) == Beta(x, y)$FSF digamma x == digamma(x)$FSF polygamma(k, x) == polygamma(k, x)$FSF besselJ(v, x) == besselJ(v, x)$FSF besselY(v, x) == besselY(v, x)$FSF besselI(v, x) == besselI(v, x)$FSF besselK(v, x) == besselK(v, x)$FSF airyAi x == airyAi(x)$FSF airyAiPrime(x) == airyAiPrime(x)$FSF airyBi x == airyBi(x)$FSF airyBiPrime(x) == airyBiPrime(x)$FSF lambertW(x) == lambertW(x)$FSF polylog(s, x) == polylog(s, x)$FSF weierstrassP(g2, g3, x) == weierstrassP(g2, g3, x)$FSF weierstrassPPrime(g2, g3, x) == weierstrassPPrime(g2, g3, x)$FSF weierstrassSigma(g2, g3, x) == weierstrassSigma(g2, g3, x)$FSF weierstrassZeta(g2, g3, x) == weierstrassZeta(g2, g3, x)$FSF -- weierstrassPInverse(g2, g3, z) == weierstrassPInverse(g2, g3, z)$FSF whittakerM(k, m, z) == whittakerM(k, m, z)$FSF whittakerW(k, m, z) == whittakerW(k, m, z)$FSF angerJ(v, z) == angerJ(v, z)$FSF weberE(v, z) == weberE(v, z)$FSF struveH(v, z) == struveH(v, z)$FSF struveL(v, z) == struveL(v, z)$FSF hankelH1(v, z) == hankelH1(v, z)$FSF hankelH2(v, z) == hankelH2(v, z)$FSF lommelS1(mu, nu, z) == lommelS1(mu, nu, z)$FSF lommelS2(mu, nu, z) == lommelS2(mu, nu, z)$FSF kummerM(mu, nu, z) == kummerM(mu, nu, z)$FSF kummerU(mu, nu, z) == kummerU(mu, nu, z)$FSF legendreP(nu, mu, z) == legendreP(nu, mu, z)$FSF legendreQ(nu, mu, z) == legendreQ(nu, mu, z)$FSF kelvinBei(v, z) == kelvinBei(v, z)$FSF kelvinBer(v, z) == kelvinBer(v, z)$FSF kelvinKei(v, z) == kelvinKei(v, z)$FSF kelvinKer(v, z) == kelvinKer(v, z)$FSF ellipticK(m) == ellipticK(m)$FSF ellipticE(m) == ellipticE(m)$FSF ellipticE(z, m) == ellipticE(z, m)$FSF ellipticF(z, m) == ellipticF(z, m)$FSF ellipticPi(z, n, m) == ellipticPi(z, n, m)$FSF jacobiSn(z, m) == jacobiSn(z, m)$FSF jacobiCn(z, m) == jacobiCn(z, m)$FSF jacobiDn(z, m) == jacobiDn(z, m)$FSF jacobiZeta(z, m) == jacobiZeta(z, m)$FSF jacobiTheta(q, z) == jacobiTheta(q, z)$FSF lerchPhi(z, s, a) == lerchPhi(z, s, a)$FSF riemannZeta(z) == riemannZeta(z)$FSF charlierC(n, a, z) == charlierC(n, a, z)$FSF hermiteH(n, z) == hermiteH(n, z)$FSF jacobiP(n, a, b, z) == jacobiP(n, a, b, z)$FSF laguerreL(n, a, z) == laguerreL(n, a, z)$FSF meixnerM(n, b, c, z) == meixnerM(n, b, c, z)$FSF
if % has RetractableTo(Integer) then hypergeometricF(la, lb, x) == hypergeometricF(la, lb, x)$FSF meijerG(la, lb, lc, ld, x) == meijerG(la, lb, lc, ld, x)$FSF
x : % ^ y : % == x ^$CF y factorial x == factorial(x)$CF binomial(n, m) == binomial(n, m)$CF permutation(n, m) == permutation(n, m)$CF factorials x == factorials(x)$CF factorials(x, n) == factorials(x, n)$CF summation(x : %, n : Symbol) == summation(x, n)$CF summation(x : %, s : SegmentBinding %) == summation(x, s)$CF product(x : %, n : Symbol) == product(x, n)$CF product(x : %, s : SegmentBinding %) == product(x, s)$CF
erf x == erf(x)$LF erfi x == erfi(x)$LF Ei x == Ei(x)$LF Si x == Si(x)$LF Ci x == Ci(x)$LF Shi x == Shi(x)$LF Chi x == Chi(x)$LF li x == li(x)$LF dilog x == dilog(x)$LF fresnelS x == fresnelS(x)$LF fresnelC x == fresnelC(x)$LF integral(x : %, n : Symbol) == integral(x, n)$LF integral(x : %, s : SegmentBinding %) == integral(x, s)$LF
operator op == belong?(op)$AF => operator(op)$AF belong?(op)$EF => operator(op)$EF belong?(op)$CF => operator(op)$CF belong?(op)$LF => operator(op)$LF belong?(op)$FSF => operator(op)$FSF belong?(op)$FSD => operator(op)$FSD belong?(op)$ESD => operator(op)$ESD nullary? op and has?(op, SYMBOL) => operator(kernel(name op)$K) (n := arity op) case "failed" => operator name op operator(name op, n::NonNegativeInteger)
reduc(x, l) == for k in l repeat p := minPoly k x := evl(numer x, k, p) /$Rep evl(denom x, k, p) x
evl0(p, k) == numer univariate(p::Fraction(MP), k)$PolynomialCategoryQuotientFunctions(IndexedExponents K, K, R, MP, Fraction MP)
-- uses some operations from Rep instead of % in order not to -- reduce recursively during those operations. evl(p, k, m) == degree(p, k) < degree m => p::Fraction(MP) (((evl0(p, k) pretend SparseUnivariatePolynomial(%)) rem m) pretend SparseUnivariatePolynomial Fraction MP) (k::MP::Fraction(MP))
if R has GcdDomain then noalg? : SUP % -> Boolean
noalg? p == while p ~= 0 repeat not empty? algkernels kernels leadingCoefficient p => return false p := reductum p true
gcdPolynomial(p : SUP %, q : SUP %) == noalg? p and noalg? q => gcdPolynomial(p, q)$Rep gcdPolynomial(p, q)$GcdDomain_&(%)
factorPolynomial(x : SUP %) : Factored SUP % == uf := factor(x pretend SUP(Rep))$SupFractionFactorizer( IndexedExponents K, K, R, MP) uf pretend Factored SUP %
squareFreePolynomial(x : SUP %) : Factored SUP % == uf := squareFree(x pretend SUP(Rep))$SupFractionFactorizer( IndexedExponents K, K, R, MP) uf pretend Factored SUP %
if (R has RetractableTo Integer) then x : % ^ r : Q == x ^$AF r minPoly k == minPoly(k)$AF definingPolynomial x == definingPolynomial(x)$AF retract(x : %) : Q == retract(x)$Rep retractIfCan(x : %) : Union(Q, "failed") == retractIfCan(x)$Rep
if not(R is AN) then k2expr : KAN -> % smp2expr : SparseMultivariatePolynomial(Integer, KAN) -> % R2AN : R -> Union(AN, "failed") k2an : K -> Union(AN, "failed") smp2an : MP -> Union(AN, "failed")
coerce(x : AN) : % == smp2expr(numer x) / smp2expr(denom x) k2expr k == map(x +-> x::%, k)$ExpressionSpaceFunctions2(AN, %)
smp2expr p == map(k2expr, x +-> x::%, p )$PolynomialCategoryLifting(IndexedExponents KAN, KAN, Integer, SparseMultivariatePolynomial( Integer, KAN), %)
retractIfCan(x : %) : Union(AN, "failed") == ((n := smp2an numer x) case AN) and ((d := smp2an denom x) case AN) => (n::AN) / (d::AN) "failed"
R2AN r == (u := retractIfCan(r::%)@Union(Q, "failed")) case Q => u::Q::AN "failed"
k2an k == not(belong?(op := operator k)$AN) => "failed" is?(op, 'rootOf) => args := argument(k) a2 := args.2 k1u := retractIfCan(a2)@Union(K, "failed") k1u case "failed" => "failed" k1 := k1u::K s1u := retractIfCan(a2)@Union(Symbol, "failed") s1u case "failed" => "failed" s1 := s1u::Symbol a1 := args.1 denom(a1) ~= 1 => error "Bad argument to rootOf" eq := univariate(numer(a1), k1) eqa : SUP(AN) := 0 while eq ~= 0 repeat cc := leadingCoefficient(eq)::% ccu := retractIfCan(cc)@Union(AN, "failed") ccu case "failed" => return "failed" eqa := eqa + monomial(ccu::AN, degree eq) eq := reductum eq rootOf(eqa, s1)$AN arg : List(AN) := empty() for x in argument k repeat if (a := retractIfCan(x)@Union(AN, "failed")) case "failed" then return "failed" else arg := concat(a::AN, arg) (operator(op)$AN) reverse!(arg)
smp2an p == (x1 := mainVariable p) case "failed" => R2AN leadingCoefficient p up := univariate(p, k := x1::K) (t := k2an k) case "failed" => "failed" ans : AN := 0 while not ground? up repeat (c := smp2an leadingCoefficient up) case "failed" => return "failed" ans := ans + (c::AN) * (t::AN) ^ (degree up) up := reductum up (c := smp2an leadingCoefficient up) case "failed" => "failed" ans + c::AN
if R has ConvertibleTo InputForm then convert(x : %) : InputForm == convert(x)$Rep import from MakeUnaryCompiledFunction(%, %, %) eval(f : %, op : BasicOperator, g : %, x : Symbol) : % == eval(f, [op], [g], x) eval(f : %, ls : List BasicOperator, lg : List %, x : Symbol) == -- handle subscripted symbols by renaming -> eval -- -> renaming back llsym : List List Symbol := [variables g for g in lg] lsym : List Symbol := removeDuplicates concat llsym lsd : List Symbol := select (scripted?, lsym) empty? lsd => eval(f, ls, [compiledFunction(g, x) for g in lg]) ns : List Symbol := [new()$Symbol for i in lsd] lforwardSubs : List Equation % := [(i::%)= (j::%) for i in lsd for j in ns] lbackwardSubs : List Equation % := [(j::%)= (i::%) for i in lsd for j in ns] nlg : List % := [subst(g, lforwardSubs) for g in lg] res : % := eval(f, ls, [compiledFunction(g, x) for g in nlg]) subst(res, lbackwardSubs)
if R has PatternMatchable Integer then patternMatch(x : %, p : Pattern Integer, l : PatternMatchResult(Integer, %)) == patternMatch(x, p, l)$PatternMatchFunctionSpace(Integer, R, %)
)abbrev domain RSPACE RealSpace ++ Author: kfp ++ Date Created: Thu Nov 06 03:51:56 CET 2014 ++ License: BSD (same as Axiom) ++ Date Last Updated: ++ Basic Operations: ++ Related Domains: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ Examples: ++ References: ++ ++ Description: ++ ++ RealSpace(n) : Exports == Implementation where
NNI ==> NonNegativeInteger PI ==> PositiveInteger I ==> Integer R ==> Real n:NNI
Exports == Join(CoercibleTo OutputForm, ConvertibleTo String) with
coerce : % -> OutputForm coerce : List R -> % coerce : R -> %
dot : (%,%) -> R dim : % -> NNI
"+" : (%,%) -> % "-" : (%,%) -> % "*" : (R,%) -> % "*" : (NNI,%) -> % "*" : (PI,%) -> % "*" : (I,%) -> % "/" : (%,R) -> %
"#" : % -> NNI "-" : % -> % "=" : (%,%) -> Boolean "~=": (%,%) -> Boolean
unitVector : PI -> % elt : (%,I) -> R
eval: (%, Equation R) -> % eval: (%, List Equation R) -> %
every?: (R -> Boolean, %) -> Boolean map: (R -> R, %) -> % -- strange, to check member?: (R, %) -> Boolean any?: (R -> Boolean, %) -> Boolean copy: % -> % D: (%, Symbol) -> % D: (%, Symbol, NonNegativeInteger) -> % D: (%, List Symbol) -> % D: (%, List Symbol, List NonNegativeInteger) -> %
dist : (%,%) -> R
Implementation == DirectProduct(n,R) add
Rep := DirectProduct(n,R)
coerce(l:List R):% == #l=n => directProduct((vector l)::Vector(R))$Rep
coerce(x:R):% == x*unitVector(1)$Rep
if n=1 then coerce(x:%):OutputForm == elt(x,1)::OutputForm
dim(x:%):NNI == n dist(x:%,y:%):R == sqrt dot(x-y,x-y)
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5840138844114655499-25px001.spad
      using old system compiler.
   RR abbreviates domain Real 
------------------------------------------------------------------------
   initializing NRLIB RR for Real 
   compiling into NRLIB RR 
   importing KernelFunctions2(Integer,$)
   compiling exported belong? : BasicOperator -> Boolean
      RR;belong?;BoB;1 is replaced by QUOTET 
Time: 0.06 SEC.
compiling local retNotUnit : $ -> Integer Time: 0 SEC.
compiling local retNotUnitIfCan : $ -> Union(Integer,failed) Time: 0 SEC.
processing macro definition SPCH ==> SparsePolynomialCoercionHelpers(Integer,Symbol,Kernel $) compiling local poly_to_MP : Polynomial Integer -> SparseMultivariatePolynomial(Integer,Kernel $) Time: 0.01 SEC.
compiling exported Zero : () -> $ Time: 0.01 SEC.
compiling exported One : () -> $ Time: 0 SEC.
compiling exported one? : $ -> Boolean Time: 0.01 SEC.
compiling exported zero? : $ -> Boolean Time: 0 SEC.
compiling exported - : $ -> $ Time: 0 SEC.
compiling exported * : (Integer,$) -> $ Time: 0 SEC.
compiling exported coerce : Integer -> $ Time: 0 SEC.
compiling exported * : ($,$) -> $ Time: 0 SEC.
compiling exported + : ($,$) -> $ Time: 0 SEC.
compiling exported - : ($,$) -> $ Time: 0 SEC.
compiling exported / : ($,$) -> $ Time: 0.01 SEC.
compiling exported number? : $ -> Boolean Time: 0 SEC.
compiling exported simplifyPower : ($,Integer) -> $ Time: 0.06 SEC.
compiling exported ^ : ($,NonNegativeInteger) -> $ Time: 0 SEC.
compiling exported ^ : ($,Integer) -> $ Time: 0.01 SEC.
compiling exported ^ : ($,PositiveInteger) -> $ Time: 0 SEC.
compiling exported smaller? : ($,$) -> Boolean Time: 0 SEC.
compiling exported = : ($,$) -> Boolean Time: 0.01 SEC.
compiling exported numer : $ -> SparseMultivariatePolynomial(Integer,Kernel $) Time: 0 SEC.
compiling exported denom : $ -> SparseMultivariatePolynomial(Integer,Kernel $) Time: 0 SEC.
compiling exported coerce : SparseMultivariatePolynomial(Integer,Kernel $) -> $ Time: 0 SEC.
compiling exported coerce : Polynomial Integer -> $ Time: 0.01 SEC.
compiling exported coerce : Fraction Polynomial Integer -> $ Time: 0 SEC.
compiling exported reduce : $ -> $ Time: 0 SEC.
compiling local commonk : ($,$) -> List Kernel $ Time: 0 SEC.
compiling local algkernels : List Kernel $ -> List Kernel $ Time: 0.01 SEC.
compiling local toprat : $ -> $ Time: 0 SEC.
compiling local alg_ker_set : $ -> List Kernel $ Time: 0.01 SEC.
compiling local algtower : $ -> List Kernel $ Time: 0 SEC.
compiling local simple_root : Kernel $ -> Boolean Time: 0.01 SEC.
compiling local root_reduce : ($,Kernel $) -> $ Time: 0.84 SEC.
compiling exported getSimplifyDenomsFlag : () -> Boolean RR;getSimplifyDenomsFlag;B;36 is replaced by algreduc_flag Time: 0 SEC.
compiling exported setSimplifyDenomsFlag : Boolean -> Boolean Time: 0 SEC.
compiling local algreduc : ($,List Kernel $) -> $ Time: 0.02 SEC.
compiling exported / : (SparseMultivariatePolynomial(Integer,Kernel $),SparseMultivariatePolynomial(Integer,Kernel $)) -> $ Time: 0 SEC.
compiling exported reducedSystem : Matrix $ -> Matrix Integer Time: 0.02 SEC.
compiling exported reducedSystem : (Matrix $,Vector $) -> Record(mat: Matrix Integer,vec: Vector Integer) Time: 0.01 SEC.
compiling local commonk0 : (List Kernel $,List Kernel $) -> List Kernel $ Time: 0 SEC.
compiling exported rootOf : (SparseUnivariatePolynomial $,Symbol) -> $ Time: 0 SEC.
compiling exported rootSum : ($,SparseUnivariatePolynomial $,Symbol) -> $ Time: 0 SEC.
compiling exported pi : () -> $ Time: 0 SEC.
compiling exported exp : $ -> $ Time: 0 SEC.
compiling exported log : $ -> $ Time: 0 SEC.
compiling exported sin : $ -> $ Time: 0 SEC.
compiling exported cos : $ -> $ Time: 0 SEC.
compiling exported tan : $ -> $ Time: 0 SEC.
compiling exported cot : $ -> $ Time: 0 SEC.
compiling exported sec : $ -> $ Time: 0 SEC.
compiling exported csc : $ -> $ Time: 0 SEC.
compiling exported asin : $ -> $ Time: 0 SEC.
compiling exported acos : $ -> $ Time: 0 SEC.
compiling exported atan : $ -> $ Time: 0 SEC.
compiling exported acot : $ -> $ Time: 0 SEC.
compiling exported asec : $ -> $ Time: 0 SEC.
compiling exported acsc : $ -> $ Time: 0.01 SEC.
compiling exported sinh : $ -> $ Time: 0 SEC.
compiling exported cosh : $ -> $ Time: 0 SEC.
compiling exported tanh : $ -> $ Time: 0 SEC.
compiling exported coth : $ -> $ Time: 0 SEC.
compiling exported sech : $ -> $ Time: 0 SEC.
compiling exported csch : $ -> $ Time: 0 SEC.
compiling exported asinh : $ -> $ Time: 0 SEC.
compiling exported acosh : $ -> $ Time: 0 SEC.
compiling exported atanh : $ -> $ Time: 0 SEC.
compiling exported acoth : $ -> $ Time: 0 SEC.
compiling exported asech : $ -> $ Time: 0 SEC.
compiling exported acsch : $ -> $ Time: 0 SEC.
compiling exported abs : $ -> $ Time: 0.01 SEC.
compiling exported Gamma : $ -> $ Time: 0 SEC.
compiling exported Gamma : ($,$) -> $ Time: 0 SEC.
compiling exported Beta : ($,$) -> $ Time: 0 SEC.
compiling exported digamma : $ -> $ Time: 0 SEC.
compiling exported polygamma : ($,$) -> $ Time: 0 SEC.
compiling exported besselJ : ($,$) -> $ Time: 0 SEC.
compiling exported besselY : ($,$) -> $ Time: 0.01 SEC.
compiling exported besselI : ($,$) -> $ Time: 0 SEC.
compiling exported besselK : ($,$) -> $ Time: 0 SEC.
compiling exported airyAi : $ -> $ Time: 0 SEC.
compiling exported airyAiPrime : $ -> $ Time: 0 SEC.
compiling exported airyBi : $ -> $ Time: 0 SEC.
compiling exported airyBiPrime : $ -> $ Time: 0 SEC.
compiling exported lambertW : $ -> $ Time: 0 SEC.
compiling exported polylog : ($,$) -> $ Time: 0 SEC.
compiling exported weierstrassP : ($,$,$) -> $ Time: 0.01 SEC.
compiling exported weierstrassPPrime : ($,$,$) -> $ Time: 0 SEC.
compiling exported weierstrassSigma : ($,$,$) -> $ Time: 0 SEC.
compiling exported weierstrassZeta : ($,$,$) -> $ Time: 0 SEC.
compiling exported whittakerM : ($,$,$) -> $ Time: 0 SEC.
compiling exported whittakerW : ($,$,$) -> $ Time: 0 SEC.
compiling exported angerJ : ($,$) -> $ Time: 0 SEC.
compiling exported weberE : ($,$) -> $ Time: 0 SEC.
compiling exported struveH : ($,$) -> $ Time: 0 SEC.
compiling exported struveL : ($,$) -> $ Time: 0 SEC.
compiling exported hankelH1 : ($,$) -> $ Time: 0.01 SEC.
compiling exported hankelH2 : ($,$) -> $ Time: 0 SEC.
compiling exported lommelS1 : ($,$,$) -> $ Time: 0 SEC.
compiling exported lommelS2 : ($,$,$) -> $ Time: 0 SEC.
compiling exported kummerM : ($,$,$) -> $ Time: 0 SEC.
compiling exported kummerU : ($,$,$) -> $ Time: 0 SEC.
compiling exported legendreP : ($,$,$) -> $ Time: 0 SEC.
compiling exported legendreQ : ($,$,$) -> $ Time: 0 SEC.
compiling exported kelvinBei : ($,$) -> $ Time: 0.01 SEC.
compiling exported kelvinBer : ($,$) -> $ Time: 0 SEC.
compiling exported kelvinKei : ($,$) -> $ Time: 0 SEC.
compiling exported kelvinKer : ($,$) -> $ Time: 0 SEC.
compiling exported ellipticK : $ -> $ Time: 0 SEC.
compiling exported ellipticE : $ -> $ Time: 0 SEC.
compiling exported ellipticE : ($,$) -> $ Time: 0 SEC.
compiling exported ellipticF : ($,$) -> $ Time: 0 SEC.
compiling exported ellipticPi : ($,$,$) -> $ Time: 0.01 SEC.
compiling exported jacobiSn : ($,$) -> $ Time: 0 SEC.
compiling exported jacobiCn : ($,$) -> $ Time: 0 SEC.
compiling exported jacobiDn : ($,$) -> $ Time: 0 SEC.
compiling exported jacobiZeta : ($,$) -> $ Time: 0 SEC.
compiling exported jacobiTheta : ($,$) -> $ Time: 0 SEC.
compiling exported lerchPhi : ($,$,$) -> $ Time: 0 SEC.
compiling exported riemannZeta : $ -> $ Time: 0 SEC.
compiling exported charlierC : ($,$,$) -> $ Time: 0.01 SEC.
compiling exported hermiteH : ($,$) -> $ Time: 0 SEC.
compiling exported jacobiP : ($,$,$,$) -> $ Time: 0 SEC.
compiling exported laguerreL : ($,$,$) -> $ Time: 0 SEC.
compiling exported meixnerM : ($,$,$,$) -> $ Time: 0 SEC.
compiling exported hypergeometricF : (List $,List $,$) -> $ Time: 0.03 SEC.
compiling exported meijerG : (List $,List $,List $,List $,$) -> $ Time: 0.01 SEC.
compiling exported ^ : ($,$) -> $ Time: 0 SEC.
compiling exported factorial : $ -> $ Time: 0 SEC.
compiling exported binomial : ($,$) -> $ Time: 0 SEC.
compiling exported permutation : ($,$) -> $ Time: 0 SEC.
compiling exported factorials : $ -> $ Time: 0 SEC.
compiling exported factorials : ($,Symbol) -> $ Time: 0 SEC.
compiling exported summation : ($,Symbol) -> $ Time: 0 SEC.
compiling exported summation : ($,SegmentBinding $) -> $ Time: 0.01 SEC.
compiling exported product : ($,Symbol) -> $ Time: 0 SEC.
compiling exported product : ($,SegmentBinding $) -> $ Time: 0 SEC.
compiling exported erf : $ -> $ Time: 0 SEC.
compiling exported erfi : $ -> $ Time: 0 SEC.
compiling exported Ei : $ -> $ Time: 0 SEC.
compiling exported Si : $ -> $ Time: 0 SEC.
compiling exported Ci : $ -> $ Time: 0 SEC.
compiling exported Shi : $ -> $ Time: 0 SEC.
compiling exported Chi : $ -> $ Time: 0 SEC.
compiling exported li : $ -> $ Time: 0 SEC.
compiling exported dilog : $ -> $ Time: 0 SEC.
compiling exported fresnelS : $ -> $ Time: 0 SEC.
compiling exported fresnelC : $ -> $ Time: 0.01 SEC.
compiling exported integral : ($,Symbol) -> $ Time: 0 SEC.
compiling exported integral : ($,SegmentBinding $) -> $ Time: 0 SEC.
compiling exported operator : BasicOperator -> BasicOperator Time: 0.01 SEC.
compiling local reduc : ($,List Kernel $) -> $ Time: 0.02 SEC.
compiling local evl0 : (SparseMultivariatePolynomial(Integer,Kernel $),Kernel $) -> SparseUnivariatePolynomial Fraction SparseMultivariatePolynomial(Integer,Kernel $) Time: 0.01 SEC.
compiling local evl : (SparseMultivariatePolynomial(Integer,Kernel $),Kernel $,SparseUnivariatePolynomial $) -> Fraction SparseMultivariatePolynomial(Integer,Kernel $) Time: 0.07 SEC.
compiling local noalg? : SparseUnivariatePolynomial $ -> Boolean Time: 0 SEC.
compiling exported gcdPolynomial : (SparseUnivariatePolynomial $,SparseUnivariatePolynomial $) -> SparseUnivariatePolynomial $ Time: 0 SEC.
compiling exported factorPolynomial : SparseUnivariatePolynomial $ -> Factored SparseUnivariatePolynomial $ Time: 0.02 SEC.
compiling exported squareFreePolynomial : SparseUnivariatePolynomial $ -> Factored SparseUnivariatePolynomial $ Time: 0 SEC.
compiling exported ^ : ($,Fraction Integer) -> $ Time: 0.01 SEC.
compiling exported minPoly : Kernel $ -> SparseUnivariatePolynomial $ Time: 0 SEC.
compiling exported definingPolynomial : $ -> $ Time: 0 SEC.
compiling exported retract : $ -> Fraction Integer Time: 0 SEC.
compiling exported retractIfCan : $ -> Union(Fraction Integer,failed) Time: 0 SEC.
compiling exported coerce : AlgebraicNumber -> $ Time: 0.01 SEC.
compiling local k2expr : Kernel AlgebraicNumber -> $ Time: 0 SEC.
compiling local smp2expr : SparseMultivariatePolynomial(Integer,Kernel AlgebraicNumber) -> $ Time: 0 SEC.
compiling exported retractIfCan : $ -> Union(AlgebraicNumber,failed) Time: 0.01 SEC.
compiling local R2AN : Integer -> Union(AlgebraicNumber,failed) Time: 0 SEC.
compiling local k2an : Kernel $ -> Union(AlgebraicNumber,failed) Time: 0.03 SEC.
compiling local smp2an : SparseMultivariatePolynomial(Integer,Kernel $) -> Union(AlgebraicNumber,failed) Time: 0.01 SEC.
compiling exported convert : $ -> InputForm Time: 0.01 SEC.
importing MakeUnaryCompiledFunction($,$,$) compiling exported eval : ($,BasicOperator,$,Symbol) -> $ Time: 0.01 SEC.
compiling exported eval : ($,List BasicOperator,List $,Symbol) -> $ Time: 0.02 SEC.
compiling exported patternMatch : ($,Pattern Integer,PatternMatchResult(Integer,$)) -> PatternMatchResult(Integer,$) Time: 0 SEC.
****** Domain: (Integer) already in scope augmenting (Integer): (CharacteristicNonZero) ****** Domain: (Integer) already in scope augmenting (Integer): (ConvertibleTo (Pattern (Float))) ****** Domain: (Integer) already in scope augmenting (Integer): (Group) ****** Domain: (Integer) already in scope augmenting (Integer): (PatternMatchable (Float)) (time taken in buildFunctor: 210)
;;; *** |Real| REDEFINED
;;; *** |Real| REDEFINED Time: 0.27 SEC.
Warnings: [1] retNotUnit: $$ has no value [2] poly_to_MP: $$ has no value [3] simplifyPower: $$ has no value [4] alg_ker_set: ak1 has no value [5] alg_ker_set: resl has no value [6] algreduc: $$ has no value [7] reducedSystem: $$ has no value [8] reducedSystem: mat has no value [9] reducedSystem: vec has no value [10] pi: not known that (RadicalCategory) is of mode (CATEGORY domain (IF (has (Integer) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Integer))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Integer) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Integer) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch)) [11] summation: $$ has no value [12] erf: not known that (RadicalCategory) is of mode (CATEGORY domain (IF (has (Integer) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Integer))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Integer) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Integer) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch)) [13] erf: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has (Integer) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Integer))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Integer) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Integer) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch)) [14] evl0: $$ has no value [15] factorPolynomial: $$ has no value [16] patternMatch: $$ has no value [17] unknown Functor code (DEFVAR algreduc_flag (call (XLAM ignore (QUOTE ))))
Cumulative Statistics for Constructor Real Time: 1.76 seconds
finalizing NRLIB RR Processing Real for Browser database: --------constructor--------- --------(reduce (% %))--------- --------(number? ((Boolean) %))--------- --------(simplifyPower (% % (Integer)))--------- --------(factorPolynomial ((Factored (SparseUnivariatePolynomial %)) (SparseUnivariatePolynomial %)))--------- --------(squareFreePolynomial ((Factored (SparseUnivariatePolynomial %)) (SparseUnivariatePolynomial %)))--------- --------(setSimplifyDenomsFlag ((Boolean) (Boolean)))--------- --------(getSimplifyDenomsFlag ((Boolean)))--------- ; compiling file "/var/aw/var/LatexWiki/RR.NRLIB/RR.lsp" (written 01 DEC 2014 05:54:49 PM):
; /var/aw/var/LatexWiki/RR.NRLIB/RR.fasl written ; compilation finished in 0:00:00.777 ------------------------------------------------------------------------ Real is now explicitly exposed in frame initial Real will be automatically loaded when needed from /var/aw/var/LatexWiki/RR.NRLIB/RR
RSPACE abbreviates domain RealSpace ------------------------------------------------------------------------ initializing NRLIB RSPACE for RealSpace compiling into NRLIB RSPACE Local variable Rep type redefined: (DirectProductCategory n (Real)) to (Join (QuotientFieldCategory (SparseMultivariatePolynomial (Integer) (Kernel $))) (CATEGORY package (IF (has (SparseMultivariatePolynomial (Integer) (Kernel $)) (IntegerNumberSystem)) (IF (has (SparseMultivariatePolynomial (Integer) (Kernel $)) (OpenMath)) (ATTRIBUTE (OpenMath)) noBranch) noBranch) (IF (has (SparseMultivariatePolynomial (Integer) (Kernel $)) (Canonical)) (IF (has (SparseMultivariatePolynomial (Integer) (Kernel $)) (GcdDomain)) (IF (has (SparseMultivariatePolynomial (Integer) (Kernel $)) (canonicalUnitNormal)) (ATTRIBUTE (Canonical)) noBranch) noBranch) noBranch))) compiling exported coerce : List Real -> $ Time: 0.01 SEC.
compiling exported coerce : Real -> $ Time: 0.01 SEC.
compiling exported coerce : $ -> OutputForm Time: 0 SEC.
compiling exported dim : $ -> NonNegativeInteger Time: 0 SEC.
compiling exported dist : ($,$) -> Real Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |RealSpace| REDEFINED
;;; *** |RealSpace| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor RealSpace Time: 0.02 seconds
--------------non extending category---------------------- .. RealSpace(#1) of cat (|Join| (|CoercibleTo| (|OutputForm|)) (|ConvertibleTo| (|String|)) (CATEGORY |domain| (SIGNATURE |coerce| ((|OutputForm|) $)) (SIGNATURE |coerce| ($ (|List| (|Real|)))) (SIGNATURE |coerce| ($ (|Real|))) (SIGNATURE |dot| ((|Real|) $ $)) (SIGNATURE |dim| ((|NonNegativeInteger|) $)) (SIGNATURE + ($ $ $)) (SIGNATURE - ($ $ $)) (SIGNATURE * ($ (|Real|) $)) (SIGNATURE * ($ (|NonNegativeInteger|) $)) (SIGNATURE * ($ (|PositiveInteger|) $)) (SIGNATURE * ($ (|Integer|) $)) (SIGNATURE / ($ $ (|Real|))) (SIGNATURE |#| ((|NonNegativeInteger|) $)) (SIGNATURE - ($ $)) (SIGNATURE = ((|Boolean|) $ $)) (SIGNATURE ~= ((|Boolean|) $ $)) (SIGNATURE |unitVector| ($ (|PositiveInteger|))) (SIGNATURE |elt| ((|Real|) $ (|Integer|))) (SIGNATURE |eval| ($ $ (|Equation| (|Real|)))) (SIGNATURE |eval| ($ $ (|List| (|Equation| (|Real|))))) (SIGNATURE |every?| ((|Boolean|) (|Mapping| (|Boolean|) (|Real|)) $)) (SIGNATURE |map| ($ (|Mapping| (|Real|) (|Real|)) $)) (SIGNATURE |member?| ((|Boolean|) (|Real|) $)) (SIGNATURE |any?| ((|Boolean|) (|Mapping| (|Boolean|) (|Real|)) $)) (SIGNATURE |copy| ($ $)) (SIGNATURE D ($ $ (|Symbol|))) (SIGNATURE D ($ $ (|Symbol|) (|NonNegativeInteger|))) (SIGNATURE D ($ $ (|List| (|Symbol|)))) (SIGNATURE D ($ $ (|List| (|Symbol|)) (|List| (|NonNegativeInteger|)))) (SIGNATURE |dist| ((|Real|) $ $)))) has no (|DirectProductCategory| |#1| (|Real|)) finalizing NRLIB RSPACE Processing RealSpace for Browser database: --------constructor--------- --------(reduce (% %))--------- --------(number? ((Boolean) %))--------- --------(simplifyPower (% % (Integer)))--------- --------(factorPolynomial ((Factored (SparseUnivariatePolynomial %)) (SparseUnivariatePolynomial %)))--------- --------(squareFreePolynomial ((Factored (SparseUnivariatePolynomial %)) (SparseUnivariatePolynomial %)))--------- --------(setSimplifyDenomsFlag ((Boolean) (Boolean)))--------- --------(getSimplifyDenomsFlag ((Boolean)))--------- --------constructor--------- --->-->RealSpace((coerce ((OutputForm) %))): Not documented!!!! --->-->RealSpace((coerce (% (List (Real))))): Not documented!!!! --->-->RealSpace((coerce (% (Real)))): Not documented!!!! --->-->RealSpace((dot ((Real) % %))): Not documented!!!! --->-->RealSpace((dim ((NonNegativeInteger) %))): Not documented!!!! --->-->RealSpace((+ (% % %))): Not documented!!!! --->-->RealSpace((- (% % %))): Not documented!!!! --->-->RealSpace((* (% (Real) %))): Not documented!!!! --->-->RealSpace((* (% (NonNegativeInteger) %))): Not documented!!!! --->-->RealSpace((* (% (PositiveInteger) %))): Not documented!!!! --->-->RealSpace((* (% (Integer) %))): Not documented!!!! --->-->RealSpace((/ (% % (Real)))): Not documented!!!! --->-->RealSpace((# ((NonNegativeInteger) %))): Not documented!!!! --->-->RealSpace((- (% %))): Not documented!!!! --->-->RealSpace((= ((Boolean) % %))): Not documented!!!! --->-->RealSpace((~= ((Boolean) % %))): Not documented!!!! --->-->RealSpace((unitVector (% (PositiveInteger)))): Not documented!!!! --->-->RealSpace((elt ((Real) % (Integer)))): Not documented!!!! --->-->RealSpace((eval (% % (Equation (Real))))): Not documented!!!! --->-->RealSpace((eval (% % (List (Equation (Real)))))): Not documented!!!! --->-->RealSpace((every? ((Boolean) (Mapping (Boolean) (Real)) %))): Not documented!!!! --->-->RealSpace((map (% (Mapping (Real) (Real)) %))): Not documented!!!! --->-->RealSpace((member? ((Boolean) (Real) %))): Not documented!!!! --->-->RealSpace((any? ((Boolean) (Mapping (Boolean) (Real)) %))): Not documented!!!! --->-->RealSpace((copy (% %))): Not documented!!!! --->-->RealSpace((D (% % (Symbol)))): Not documented!!!! --->-->RealSpace((D (% % (Symbol) (NonNegativeInteger)))): Not documented!!!! --->-->RealSpace((D (% % (List (Symbol))))): Not documented!!!! --->-->RealSpace((D (% % (List (Symbol)) (List (NonNegativeInteger))))): Not documented!!!! --->-->RealSpace((dist ((Real) % %))): Not documented!!!! --->-->RealSpace(): Spurious comments: \blankline \blankline ; compiling file "/var/aw/var/LatexWiki/RSPACE.NRLIB/RSPACE.lsp" (written 01 DEC 2014 05:54:50 PM):
; /var/aw/var/LatexWiki/RSPACE.NRLIB/RSPACE.fasl written ; compilation finished in 0:00:00.017 ------------------------------------------------------------------------ RealSpace is now explicitly exposed in frame initial RealSpace will be automatically loaded when needed from /var/aw/var/LatexWiki/RSPACE.NRLIB/RSPACE

fricas
P:=[x,y,z::RR]::RealSpace(3)

\label{eq1}\left[ x , \: y , \: z \right](1)
Type: RealSpace?(3)
fricas
Q:=[u,v,w::RR]::RealSpace(3)

\label{eq2}\left[ u , \: v , \: w \right](2)
Type: RealSpace?(3)

fricas
s:=s::RR

\label{eq3}s(3)
Type: Real
fricas
t:=t::RR

\label{eq4}t(4)
Type: Real

fricas
P+Q

\label{eq5}\left[{x + u}, \:{y + v}, \:{z + w}\right](5)
Type: RealSpace?(3)
fricas
P-Q

\label{eq6}\left[{x - u}, \:{y - v}, \:{z - w}\right](6)
Type: RealSpace?(3)
fricas
-P

\label{eq7}\left[ - x , \: - y , \: - z \right](7)
Type: RealSpace?(3)
fricas
dot(P,Q)

\label{eq8}{w \  z}+{v \  y}+{u \  x}(8)
Type: Real
fricas
dist(P,Q)

\label{eq9}\sqrt{{{z}^{2}}-{2 \  w \  z}+{{y}^{2}}-{2 \  v \  y}+{{x}^{2}}-{2 \  u \  x}+{{w}^{2}}+{{v}^{2}}+{{u}^{2}}}(9)
Type: Real
fricas
t*P+s*Q

\label{eq10}\left[{{t \  x}+{s \  u}}, \:{{t \  y}+{s \  v}}, \:{{t \  z}+{s \  w}}\right](10)
Type: RealSpace?(3)
fricas
unitVector(1)$RSPACE(3)

\label{eq11}\left[ 1, \: 0, \: 0 \right](11)
Type: RealSpace?(3)
fricas
P.1

\label{eq12}x(12)
Type: Real
fricas
Q.2

\label{eq13}v(13)
Type: Real

Functions

fricas
f:Real -> Real
Type: Void
fricas
f(t) == t^2*sin(t)
Type: Void
fricas
X:Real -> RealSpace(4)
Type: Void
fricas
X(t) == [f(t),f(2*t),t^2,cos(t::RR)]::RealSpace(4)
Type: Void
fricas
X(t) -- curve in R^4
fricas
Compiling function f with type Real -> Real
fricas
Compiling function X with type Real -> RealSpace(4)

\label{eq14}\left[{{{t}^{2}}\ {\sin \left({t}\right)}}, \:{4 \ {{t}^{2}}\ {\sin \left({2 \  t}\right)}}, \:{{t}^{2}}, \:{\cos \left({t}\right)}\right](14)
Type: RealSpace?(4)
fricas
D(X(t),'t)

\label{eq15}\begin{array}{@{}l}
\displaystyle
\left[{{2 \  t \ {\sin \left({t}\right)}}+{{{t}^{2}}\ {\cos \left({t}\right)}}}, \:{{8 \  t \ {\sin \left({2 \  t}\right)}}+{8 \ {{t}^{2}}\ {\cos \left({2 \  t}\right)}}}, \:{2 \  t}, \: -{\sin \left({t}\right)}\right] (15)
Type: RealSpace?(4)
fricas
D(X(t),'t,2)

\label{eq16}\begin{array}{@{}l}
\displaystyle
\left[{{{\left(-{{t}^{2}}+ 2 \right)}\ {\sin \left({t}\right)}}+{4 \  t \ {\cos \left({t}\right)}}}, \: \right.
\
\
\displaystyle
\left.{{{\left(-{{16}\ {{t}^{2}}}+ 8 \right)}\ {\sin \left({2 \  t}\right)}}+{{32}\  t \ {\cos \left({2 \  t}\right)}}}, \: 2, \: -{\cos \left({t}\right)}\right] 
(16)
Type: RealSpace?(4)
fricas
eval(X(t),t=1)

\label{eq17}\left[{\sin \left({1}\right)}, \:{4 \ {\sin \left({2}\right)}}, \: 1, \:{\cos \left({1}\right)}\right](17)
Type: RealSpace?(4)
fricas
Z:RealSpace(2)->RealSpace(3)
Type: Void
fricas
z:=[x,y::RR]::RSPACE(2)

\label{eq18}\left[ x , \: y \right](18)
Type: RealSpace?(2)
fricas
Z(q) == [q.1^2,exp(-q.1*q.2),(q.1+q.2)^n]
Type: Void
fricas
Z(z)
fricas
Compiling function Z with type RealSpace(2) -> RealSpace(3)

\label{eq19}\left[{{x}^{2}}, \:{{e}^{-{x \  y}}}, \:{{\left(y + x \right)}^{n}}\right](19)
Type: RealSpace?(3)
fricas
D(Z(z),['x,'y])

\label{eq20}\left[ 0, \:{{\left({x \  y}- 1 \right)}\ {{e}^{-{x \  y}}}}, \:{{\left({{n}^{2}}- n \right)}\ {{\left(y + x \right)}^{n - 2}}}\right](20)
Type: RealSpace?(3)
fricas
D(Z(z),['x,'y],[2,3])

\label{eq21}\begin{array}{@{}l}
\displaystyle
\left[ 0, \:{{\left(-{{{x}^{3}}\ {{y}^{2}}}+{6 \ {{x}^{2}}\  y}-{6 \  x}\right)}\ {{e}^{-{x \  y}}}}, \: \right.
\
\
\displaystyle
\left.{{\left({{n}^{5}}-{{10}\ {{n}^{4}}}+{{35}\ {{n}^{3}}}-{{5
0}\ {{n}^{2}}}+{{24}\  n}\right)}\ {{\left(y + x \right)}^{n - 5}}}\right] 
(21)
Type: RealSpace?(3)
fricas
D(Z(z),'x,3)

\label{eq22}\left[ 0, \: -{{{y}^{3}}\ {{e}^{-{x \  y}}}}, \:{{\left({{n}^{3}}-{3 \ {{n}^{2}}}+{2 \  n}\right)}\ {{\left(y + x \right)}^{n - 3}}}\right](22)
Type: RealSpace?(3)

fricas
limit(1/n^2::RR,n=%plusInfinity)

\label{eq23}0(23)
Type: Union(OrderedCompletion?(Real),...)
fricas
limit(X(t).4,'t=0)

\label{eq24}1(24)
Type: Union(OrderedCompletion?(Real),...)

fricas
integrate((sin(y::RR))^2,y)

\label{eq25}{-{{\cos \left({y}\right)}\ {\sin \left({y}\right)}}+ y}\over 2(25)
Type: Union(Real,...)
fricas
integrate(sin(s)^2,'s)

\label{eq26}{-{{\cos \left({s}\right)}\ {\sin \left({s}\right)}}+ s}\over 2(26)
Type: Union(Real,...)
fricas
integrate(dist(X(u),X(v)),u)
>> Error detected within library code: integrate: implementation incomplete (constant residues)

-- continue with PROP domain




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