/usr/lib/open-axiom/input/pdecomp0.as is in open-axiom-test 1.4.1+svn~2626-2ubuntu2.
This file is owned by root:root, with mode 0o644.
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#include "axiom.as"
-- Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
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--% Polynomial composition and decomposition functions
-- If f = g o h then g = leftFactor(f, h) & h = rightFactor(f, g)
-- SMW Dec 86
--% PolynomialComposition
--)abbrev package PCOMP PolynomialComposition
--)abbrev package PDECOMP PolynomialDecomposition
PolynomialComposition(UP: UnivariatePolynomialCategory(R), R: Ring): with
compose: (UP, UP) -> UP
== add
compose(g:UP, h:UP):UP ==
r: UP := 0
while g ~= 0 repeat
r := leadingCoefficient(g)*h**degree(g) + r
g := reductum g
r
-- Ref: Kozen and Landau, Cornell University TR 86-773
--% PolynomialDecomposition
PolynomialDecomposition(UP:UPC F, F:Field): PDcat == PDdef where
UPC ==> UnivariatePolynomialCategory
NNI ==> NonNegativeInteger
LR ==> Record(left: UP, right: UP)
PDcat ==> with
decompose: UP -> List UP
decompose: (UP, NNI, NNI) -> Union(value1:LR, failed:'failed')
leftFactor: (UP, UP) -> Union(value1:UP, failed:'failed')
rightFactorCandidate: (UP, NNI) -> UP
PDdef ==> add
import from F
import from LR
import from Union(value1:UP, failed:'failed')
import from Float
import from NNI
import from UniversalSegment NNI
import from Record(quotient:UP, remainder:UP);
leftFactor(f:UP, h:UP):Union(value1:UP, failed:'failed') ==
g: UP := 0
for i in 0.. while f ~= 0 repeat
fr := divide(f, h)
f := fr.quotient
r := fr.remainder
degree r > 0 => return [failed]
g := g + r * monomial(1, i)
[g]
decompose(f:UP, dg:NNI, dh:NNI):Union(value1:LR, failed:'failed') ==
df := degree f
dg*dh ~= df => [failed]
h := rightFactorCandidate(f, dh)
g:Union(value1:UP, failed:'failed') := leftFactor(f, h)
g case failed => [failed]
[[g.value1, h]]
decompose(f:UP):List UP ==
df := degree f
for dh in 2..df-1 | df rem dh = 0 repeat
h := rightFactorCandidate(f, dh)
g := leftFactor(f, h)
g case value1 => return
append(decompose(g.value1), decompose h)
[f]
rightFactorCandidate(f:UP, dh:NNI):UP ==
f := f / leadingCoefficient f
df := degree f
dg := df quo dh
h := monomial(1, dh)
for k in 1..dh repeat
hdg:= h**dg
c := (coefficient(f,df-k)-coefficient(hdg,df-k))/
(dg::Integer::F)
h := h + monomial(c, dh-k)
h - monomial(coefficient(h, 0), 0) -- drop constant term
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