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weier.spad
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weier.spad
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)abbrev package WEIER WeierstrassPreparation
++ Author: William H. Burge
++ Date Created: Sept 1988
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description: This package implements the Weierstrass preparation
++ theorem f for multivariate power series.
++ weierstrass(v, p) where v is a variable, and p is a
++ TaylorSeries(R) in which there is term of form c*v^s where c is
++ a constant. Let s be minimal as above. The result
++ is a list of TaylorSeries coefficients A[i] of the
++ equivalent polynomial
++ A = A[0] + A[1]*v + A[2]*v^2 + ... + A[s-1]*v^(s-1) + v^s
++ such that p=A*B , B being a TaylorSeries of minimum degree 0
WeierstrassPreparation(R) : Defn == Impl where
R : Field
VarSet==>Symbol
SMP ==> Polynomial R
PS ==> InnerTaylorSeries SMP
NNI ==> NonNegativeInteger
ST ==> Stream
StS ==> Stream SMP
STPS==>StreamTaylorSeriesOperations
STTAYLOR==>StreamTaylorSeriesOperations
SUP==> SparseUnivariatePolynomial(SMP)
ST2==>StreamFunctions2
SMPS==> TaylorSeries(R)
L==>List
coef ==> coefficient$SUP
Defn ==> with
weierstrass : (VarSet, SMPS)->L SMPS
++\spad{weierstrass(v, ts)} where v is a variable and ts is
++ a TaylorSeries, implements the Weierstrass Preparation
++ Theorem. The result is a list of TaylorSeries that
++ are the coefficients of the equivalent series.
Impl ==> add
import from TaylorSeries(R)
import from StreamTaylorSeriesOperations SMP
import from StreamTaylorSeriesOperations SMPS
map1==>map$(ST2(SMP, SUP))
map2==>map$(ST2(StS, SMP))
map3==>map$(ST2(StS, StS))
qqq : (NNI, SMPS, ST SMPS) -> ((ST SMPS) -> ST SMPS)
transback(smps : ST SMPS) : L SMPS ==
if empty?(smps)
then empty()$(L SMPS)
else
if empty?(first (smps pretend (ST StS)))
then empty()$(L SMPS)
else
cons(map2((ss : StS) : SMP +->
if empty?(ss) then 0 else first ss,
smps pretend ST StS)pretend SMPS,
transback(map3((ss : StS) : StS +-> rest(ss, 1),
smps pretend ST StS) pretend (ST SMPS))
)$(L SMPS)
streamlikeUniv(p : SUP, n : NNI) : StS ==
if n = 0
then cons(coef(p, 0), empty()$StS)
else cons(coef(p, n), streamlikeUniv(p, (n-1)::NNI))
transpose(s : ST StS) : ST StS ==delay(
if empty?(s)
then empty()$(ST StS)
else cons(map2(first, s), transpose(map3(rest, rst s))))
zp==>map$StreamFunctions3(SUP, NNI, StS)
NO_COEFF ==> "can not find power of variable with constant coefficient"
sts2stst(var : VarSet, sts : StS) : ST StS ==
si0 : ST NNI := (integers 0) pretend ST NNI
zp((x, y) +-> streamlikeUniv(x, y),
map1((p : SMP) : SUP +-> univariate(p, var), sts), si0)
tp(v : VarSet, sts : StS) : ST StS == transpose sts2stst(v, sts)
map4==>map$(ST2 (StS, StS))
maptake(n : NNI, p : ST StS) : ST SMPS ==
map4((ss : StS) : StS +-> first(ss, n), p) pretend ST SMPS
mapdrop(n : NNI, p : ST StS) : ST SMPS ==
map4((ss : StS) : StS +-> rest(ss, n), p) pretend ST SMPS
YSS==>Y$ParadoxicalCombinatorsForStreams(SMPS)
weier(v : VarSet, sts : StS) : ST SMPS ==
p := tp(v, sts) pretend (ST SMPS)
b : StS := first p pretend StS
a : NNI
for a0 in 0.. repeat
a0 = 1000 => error NO_COEFF
empty?(b) => error NO_COEFF
(b0 := first b) = 0 =>
b := rest b
iterate
c := retractIfCan(b0)@Union(R, "failed")
c case R =>
a := a0
break
b := rest b
e := recip b
f := if e case "failed"
then error "no reciprocal"
else e@StS
q := (YSS qqq(a, f pretend SMPS, rest p))
maptake(a, (p*q) pretend ST StS)
qq : (NNI, SMPS, ST SMPS, ST SMPS)->ST SMPS
qq(a, e, p, c)==
cons(e, (-e)*mapdrop(a, (p*c)pretend(ST StS)))
qqq(a, e, p) == s +-> qq(a, e, p, s)
wei(v : VarSet, s : SMPS) : ST SMPS == weier(v, s pretend StS)
weierstrass(v, smps) ==
coefficient(smps, 0) ~= 0 => empty()
transback wei (v, smps)
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
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