/usr/share/axiom-20170501/src/algebra/INTTOOLS.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 | )abbrev package INTTOOLS IntegrationTools
++ Author: Manuel Bronstein
++ Date Created: 25 April 1990
++ Date Last Updated: 9 June 1993
++ Description:
++ Tools for the integrator
IntegrationTools(R,F) : SIG == CODE where
R : OrderedSet
F : FunctionSpace R
K ==> Kernel F
SE ==> Symbol
P ==> SparseMultivariatePolynomial(R, K)
UP ==> SparseUnivariatePolynomial F
IR ==> IntegrationResult F
ANS ==> Record(special:F, integrand:F)
U ==> Union(ANS, "failed")
ALGOP ==> "%alg"
SIG ==> with
varselect : (List K, SE) -> List K
++ varselect([k1,...,kn], x) returns the ki which involve x.
kmax : List K -> K
++ kmax([k1,...,kn]) returns the top-level ki for integration.
ksec : (K, List K, SE) -> K
++ ksec(k, [k1,...,kn], x) returns the second top-level ki
++ after k involving x.
union : (List K, List K) -> List K
++ union(l1, l2) returns set-theoretic union of l1 and l2.
vark : (List F, SE) -> List K
++ vark([f1,...,fn],x) returns the set-theoretic union of
++ \spad{(varselect(f1,x),...,varselect(fn,x))}.
if R has IntegralDomain then
removeConstantTerm : (F, SE) -> F
++ removeConstantTerm(f, x) returns f minus any additive constant
++ with respect to x.
if R has GcdDomain and F has ElementaryFunctionCategory then
mkPrim : (F, SE) -> F
++ mkPrim(f, x) makes the logs in f which are linear in x
++ primitive with respect to x.
if R has ConvertibleTo Pattern Integer and R has PatternMatchable Integer
and F has LiouvillianFunctionCategory and F has RetractableTo SE then
intPatternMatch : (F, SE, (F, SE) -> IR, (F, SE) -> U) -> IR
++ intPatternMatch(f, x, int, pmint) tries to integrate \spad{f}
++ first by using the integration function \spad{int}, and then
++ by using the pattern match intetgration function \spad{pmint}
++ on any remaining unintegrable part.
CODE ==> add
better?: (K, K) -> Boolean
union(l1, l2) == setUnion(l1, l2)
varselect(l, x) == [k for k in l | member?(x, variables(k::F))]
ksec(k, l, x) == kmax setUnion(remove(k, l), vark(argument k, x))
vark(l, x) ==
varselect(reduce("setUnion",[kernels f for f in l],empty()$List(K)), x)
kmax l ==
ans := first l
for k in rest l repeat
if better?(k, ans) then ans := k
ans
-- true if x should be considered before y in the tower
better?(x, y) ==
height(y) ^= height(x) => height(y) < height(x)
has?(operator y, ALGOP) or
(is?(y, "exp"::SE) and not is?(x, "exp"::SE)
and not has?(operator x, ALGOP))
if R has IntegralDomain then
removeConstantTerm(f, x) ==
not freeOf?((den := denom f)::F, x) => f
(u := isPlus(num := numer f)) case "failed" =>
freeOf?(num::F, x) => 0
f
ans:P := 0
for term in u::List(P) repeat
if not freeOf?(term::F, x) then ans := ans + term
ans / den
if R has GcdDomain and F has ElementaryFunctionCategory then
psimp : (P, SE) -> Record(coef:Integer, logand:F)
cont : (P, List K) -> P
logsimp : (F, SE) -> F
linearLog?: (K, F, SE) -> Boolean
logsimp(f, x) ==
r1 := psimp(numer f, x)
r2 := psimp(denom f, x)
g := gcd(r1.coef, r2.coef)
g * log(r1.logand ** (r1.coef quo g) / r2.logand ** (r2.coef quo g))
cont(p, l) ==
empty? l => p
q := univariate(p, first l)
cont(unitNormal(leadingCoefficient q).unit * content q, rest l)
linearLog?(k, f, x) ==
is?(k, "log"::SE) and
((u := retractIfCan(univariate(f,k))@Union(UP,"failed")) case UP)
and (degree(u::UP) = 1)
and not member?(x, variables leadingCoefficient(u::UP))
mkPrim(f, x) ==
lg := [k for k in kernels f | linearLog?(k, f, x)]
eval(f, lg, [logsimp(first argument k, x) for k in lg])
psimp(p, x) ==
(u := isExpt(p := ((p exquo cont(p, varselect(variables p, x)))::P)))
case "failed" => [1, p::F]
[u.exponent, u.var::F]
if R has Join(ConvertibleTo Pattern Integer, PatternMatchable Integer)
and F has Join(LiouvillianFunctionCategory, RetractableTo SE) then
intPatternMatch(f, x, int, pmint) ==
ir := int(f, x)
empty?(l := notelem ir) => ir
ans := ratpart ir
nl:List(Record(integrand:F, intvar:F)) := empty()
lg := logpart ir
for rec in l repeat
u := pmint(rec.integrand, retract(rec.intvar))
if u case ANS then
rc := u::ANS
ans := ans + rc.special
if rc.integrand ^= 0 then
ir0 := intPatternMatch(rc.integrand, x, int, pmint)
ans := ans + ratpart ir0
lg := concat(logpart ir0, lg)
nl := concat(notelem ir0, nl)
else nl := concat(rec, nl)
mkAnswer(ans, lg, nl)
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