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--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
)abbrev package INFSP InnerNumericFloatSolvePackage
++ Author: P. Gianni
++ Date Created: January 1990
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This is an internal package
++ for computing approximate solutions to systems of polynomial equations.
++ The parameter K specifies the coefficient field of the input polynomials
++ and must be either \spad{Fraction(Integer)} or \spad{Complex(Fraction Integer)}.
++ The parameter F specifies where the solutions must lie and can
++ be one of the following: \spad{Float}, \spad{Fraction(Integer)}, \spad{Complex(Float)},
++ \spad{Complex(Fraction Integer)}. The last parameter specifies the type
++ of the precision operand and must be either \spad{Fraction(Integer)} or \spad{Float}.
InnerNumericFloatSolvePackage(K,F,Par): Cat == Cap where
F : Field -- this is the field where the answer will be
K : GcdDomain -- type of the input
Par : Join(Field, OrderedRing ) -- it will be NF or RN
I ==> Integer
NNI ==> NonNegativeInteger
P ==> Polynomial
EQ ==> Equation
L ==> List
SUP ==> SparseUnivariatePolynomial
RN ==> Fraction Integer
NF ==> Float
CF ==> Complex Float
GI ==> Complex Integer
GRN ==> Complex RN
SE ==> Symbol
RFI ==> Fraction P I
Cat == with
innerSolve1 : (SUP K,Par) -> L F
++ innerSolve1(up,eps) returns the list of the zeros
++ of the univariate polynomial up with precision eps.
innerSolve1 : (P K,Par) -> L F
++ innerSolve1(p,eps) returns the list of the zeros
++ of the polynomial p with precision eps.
innerSolve : (L P K,L P K,L SE,Par) -> L L F
++ innerSolve(lnum,lden,lvar,eps) returns a list of
++ solutions of the system of polynomials lnum, with
++ the side condition that none of the members of lden
++ vanish identically on any solution. Each solution
++ is expressed as a list corresponding to the list of
++ variables in lvar and with precision specified by eps.
makeEq : (L F,L SE) -> L EQ P F
++ makeEq(lsol,lvar) returns a list of equations formed
++ by corresponding members of lvar and lsol.
Cap == add
------ Local Functions ------
isGeneric? : (L P K,L SE) -> Boolean
evaluate : (P K,SE,SE,F) -> F
numeric : K -> F
oldCoord : (L F,L I) -> L F
findGenZeros : (L P K,L SE,Par) -> L L F
failPolSolve : (L P K,L SE) -> Union(L L P K,"failed")
numeric(r:K):F ==
K is I =>
F is Float => r::I::Float
F is RN => r::I::RN
F is CF => r::I::CF
F is GRN => r::I::GRN
K is GI =>
gr:GI := r::GI
F is GRN => complex(real(gr)::RN,imag(gr)::RN)$GRN
F is CF => convert(gr)
error "case not handled"
-- construct the equation
makeEq(nres:L F,lv:L SE) : L EQ P F ==
[equation(x::(P F),r::(P F)) for x in lv for r in nres]
evaluate(pol:P K,xvar:SE,zvar:SE,z:F):F ==
rpp:=map(numeric,pol)$PolynomialFunctions2(K,F)
rpp := eval(rpp,zvar,z)
upol:=univariate(rpp,xvar)
retract(-coefficient(upol,0))/retract(leadingCoefficient upol)
myConvert(eps:Par) : RN ==
Par is RN => eps
Par is NF => retract(eps)$NF
innerSolve1(pol:P K,eps:Par) : L F == innerSolve1(univariate pol,eps)
innerSolve1(upol:SUP K,eps:Par) : L F ==
K is GI and (Par is RN or Par is NF) =>
(complexZeros(upol,
eps)$ComplexRootPackage(SUP K,Par)) pretend L(F)
K is I =>
F is Float =>
z:= realZeros(upol,myConvert eps)$RealZeroPackage(SUP I)
[convert((1/2)*(x.left+x.right))@Float for x in z] pretend L(F)
F is RN =>
z:= realZeros(upol,myConvert eps)$RealZeroPackage(SUP I)
[(1/2)*(x.left + x.right) for x in z] pretend L(F)
error "improper arguments to INFSP"
error "improper arguments to INFSP"
-- find the zeros of components in "generic" position --
findGenZeros(lp:L P K,rlvar:L SE,eps:Par) : L L F ==
rlp:=reverse lp
f:=rlp.first
zvar:= rlvar.first
rlp:=rlp.rest
lz:=innerSolve1(f,eps)
[reverse cons(z,[evaluate(pol,xvar,zvar,z) for pol in rlp
for xvar in rlvar.rest]) for z in lz]
-- convert to the old coordinates --
oldCoord(numres:L F,lval:L I) : L F ==
rnumres:=reverse numres
rnumres.first:= rnumres.first +
(+/[n*nr for n in lval for nr in rnumres.rest])
reverse rnumres
-- real zeros of a system of 2 polynomials lp (incomplete)
innerSolve2(lp:L P K,lv:L SE,eps: Par):L L F ==
mainvar := first lv
up1:=univariate(lp.1, mainvar)
up2:=univariate(lp.2, mainvar)
vec := subresultantVector(up1,up2)$SubResultantPackage(P K,SUP P K)
p0 := primitivePart multivariate(vec.0, mainvar)
p1 := primitivePart(multivariate(vec.1, mainvar),mainvar)
zero? p1 or
not one? gcd(p0, leadingCoefficient(univariate(p1,mainvar))) =>
innerSolve(cons(0,lp),empty(),lv,eps)
findGenZeros([p1, p0], reverse lv, eps)
-- real zeros of the system of polynomial lp --
innerSolve(lp:L P K,ld:L P K,lv:L SE,eps: Par) : L L F ==
-- empty?(ld) and (#lv = 2) and (# lp = 2) => innerSolve2(lp, lv, eps)
lnp:= [pToDmp(p)$PolToPol(lv,K) for p in lp]
OV:=OrderedVariableList(lv)
lvv:L OV:= [variable(vv)::OV for vv in lv]
DP:=DirectProduct(#lv,NonNegativeInteger)
dmp:=DistributedMultivariatePolynomial(lv,K)
lq:L dmp:=[]
if ld~=[] then
lq:= [(pToDmp(q1)$PolToPol(lv,K)) pretend dmp for q1 in ld]
partRes:=groebSolve(lnp,lvv)$GroebnerSolve(lv,K,K) pretend (L L dmp)
partRes=list [] => []
-- remove components where denominators vanish
if lq~=[] then
gb:=GroebnerInternalPackage(K,DirectProduct(#lv,NNI),OV,dmp)
partRes:=[pr for pr in partRes|
and/[not zero?(redPol(fq,pr pretend List(dmp))$gb)
for fq in lq]]
-- select the components in "generic" form
rlv:=reverse lv
rrlvv:= rest reverse lvv
listGen:L L dmp:=[]
for res in partRes repeat
res1:=rest reverse res
"and"/[("max"/degree(f,rrlvv))=1 for f in res1] =>
listGen:=concat(res pretend (L dmp),listGen)
result:L L F := []
if listGen~=[] then
listG :L L P K:=
[[dmpToP(pf)$PolToPol(lv,K) for pf in pr] for pr in listGen]
result:=
"append"/[findGenZeros(res,rlv,eps) for res in listG]
for gres in listGen repeat
partRes:=delete(partRes,position(gres,partRes))
-- adjust the non-generic components
for gres in partRes repeat
genRecord := genericPosition(gres,lvv)$GroebnerSolve(lv,K,K)
lgen := genRecord.dpolys
lval := genRecord.coords
lgen1:=[dmpToP(pf)$PolToPol(lv,K) for pf in lgen]
lris:=findGenZeros(lgen1,rlv,eps)
result:= append([oldCoord(r,lval) for r in lris],result)
result
)abbrev package FLOATRP FloatingRealPackage
++ Author: P. Gianni
++ Date Created: January 1990
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors: SystemSolvePackage, RadicalSolvePackage,
++ FloatingComplexPackage
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This is a package for the approximation of real solutions for
++ systems of polynomial equations over the rational numbers.
++ The results are expressed as either rational numbers or floats
++ depending on the type of the precision parameter which can be
++ either a rational number or a floating point number.
FloatingRealPackage(Par): Cat == Cap where
I ==> Integer
NNI ==> NonNegativeInteger
P ==> Polynomial
EQ ==> Equation
L ==> List
SUP ==> SparseUnivariatePolynomial
RN ==> Fraction Integer
NF ==> Float
CF ==> Complex Float
GI ==> Complex Integer
GRN ==> Complex RN
SE ==> Symbol
RFI ==> Fraction P I
INFSP ==> InnerNumericFloatSolvePackage
Par : Join(OrderedRing, Field) -- RN or NewFloat
Cat == with
solve: (L RFI,Par) -> L L EQ P Par
++ solve(lp,eps) finds all of the real solutions of the
++ system lp of rational functions over the rational numbers
++ with respect to all the variables appearing in lp,
++ with precision eps.
solve: (L EQ RFI,Par) -> L L EQ P Par
++ solve(leq,eps) finds all of the real solutions of the
++ system leq of equationas of rational functions
++ with respect to all the variables appearing in lp,
++ with precision eps.
solve: (RFI,Par) -> L EQ P Par
++ solve(p,eps) finds all of the real solutions of the
++ univariate rational function p with rational coefficients
++ with respect to the unique variable appearing in p,
++ with precision eps.
solve: (EQ RFI,Par) -> L EQ P Par
++ solve(eq,eps) finds all of the real solutions of the
++ univariate equation eq of rational functions
++ with respect to the unique variables appearing in eq,
++ with precision eps.
realRoots: (L RFI,L SE,Par) -> L L Par
++ realRoots(lp,lv,eps) computes the list of the real
++ solutions of the list lp of rational functions with rational
++ coefficients with respect to the variables in lv,
++ with precision eps. Each solution is expressed as a list
++ of numbers in order corresponding to the variables in lv.
realRoots : (RFI,Par) -> L Par
++ realRoots(rf, eps) finds the real zeros of a univariate
++ rational function with precision given by eps.
Cap == add
makeEq(nres:L Par,lv:L SE) : L EQ P Par ==
[equation(x::(P Par),r::(P Par)) for x in lv for r in nres]
-- find the real zeros of an univariate rational polynomial --
realRoots(p:RFI,eps:Par) : L Par ==
innerSolve1(numer p,eps)$INFSP(I,Par,Par)
-- real zeros of the system of polynomial lp --
realRoots(lp:L RFI,lv:L SE,eps: Par) : L L Par ==
lnum:=[numer p for p in lp]
lden:=[dp for p in lp |not one?(dp:=denom p)]
innerSolve(lnum,lden,lv,eps)$INFSP(I,Par,Par)
solve(lp:L RFI,eps : Par) : L L EQ P Par ==
lnum:=[numer p for p in lp]
lden:=[dp for p in lp |not one?(dp:=denom p)]
lv:="setUnion"/[variables np for np in lnum]
if lden~=[] then
lv:=setUnion(lv,"setUnion"/[variables dp for dp in lden])
[makeEq(numres,lv) for numres
in innerSolve(lnum,lden,lv,eps)$INFSP(I,Par,Par)]
solve(le:L EQ RFI,eps : Par) : L L EQ P Par ==
lp:=[lhs ep - rhs ep for ep in le]
lnum:=[numer p for p in lp]
lden:=[dp for p in lp |not one?(dp:=denom p)]
lv:="setUnion"/[variables np for np in lnum]
if lden~=[] then
lv:=setUnion(lv,"setUnion"/[variables dp for dp in lden])
[makeEq(numres,lv) for numres
in innerSolve(lnum,lden,lv,eps)$INFSP(I,Par,Par)]
solve(p : RFI,eps : Par) : L EQ P Par ==
(mvar := mainVariable numer p ) case "failed" =>
error "no variable found"
x:P Par:=mvar::SE::(P Par)
[equation(x,val::(P Par)) for val in realRoots(p,eps)]
solve(eq : EQ RFI,eps : Par) : L EQ P Par ==
solve(lhs eq - rhs eq,eps)
)abbrev package FLOATCP FloatingComplexPackage
++ Author: P. Gianni
++ Date Created: January 1990
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors: SystemSolvePackage, RadicalSolvePackage,
++ FloatingRealPackage
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This is a package for the approximation of complex solutions for
++ systems of equations of rational functions with complex rational
++ coefficients. The results are expressed as either complex rational
++ numbers or complex floats depending on the type of the precision
++ parameter which can be either a rational number or a floating point number.
FloatingComplexPackage(Par): Cat == Cap where
Par : Join(Field, OrderedRing)
K ==> GI
FPK ==> Fraction P K
C ==> Complex
I ==> Integer
NNI ==> NonNegativeInteger
P ==> Polynomial
EQ ==> Equation
L ==> List
SUP ==> SparseUnivariatePolynomial
RN ==> Fraction Integer
NF ==> Float
CF ==> Complex Float
GI ==> Complex Integer
GRN ==> Complex RN
SE ==> Symbol
RFI ==> Fraction P I
INFSP ==> InnerNumericFloatSolvePackage
Cat == with
complexSolve: (L FPK,Par) -> L L EQ P C Par
++ complexSolve(lp,eps) finds all the complex solutions to
++ precision eps of the system lp of rational functions
++ over the complex rationals with respect to all the
++ variables appearing in lp.
complexSolve: (L EQ FPK,Par) -> L L EQ P C Par
++ complexSolve(leq,eps) finds all the complex solutions
++ to precision eps of the system leq of equations
++ of rational functions over complex rationals
++ with respect to all the variables appearing in lp.
complexSolve: (FPK,Par) -> L EQ P C Par
++ complexSolve(p,eps) find all the complex solutions of the
++ rational function p with complex rational coefficients
++ with respect to all the variables appearing in p,
++ with precision eps.
complexSolve: (EQ FPK,Par) -> L EQ P C Par
++ complexSolve(eq,eps) finds all the complex solutions of the
++ equation eq of rational functions with rational rational coefficients
++ with respect to all the variables appearing in eq,
++ with precision eps.
complexRoots : (FPK,Par) -> L C Par
++ complexRoots(rf, eps) finds all the complex solutions of a
++ univariate rational function with rational number coefficients.
++ The solutions are computed to precision eps.
complexRoots : (L FPK,L SE,Par) -> L L C Par
++ complexRoots(lrf, lv, eps) finds all the complex solutions of a
++ list of rational functions with rational number coefficients
++ with respect the the variables appearing in lv.
++ Each solution is computed to precision eps and returned as
++ list corresponding to the order of variables in lv.
Cap == add
-- find the complex zeros of an univariate polynomial --
complexRoots(q:FPK,eps:Par) : L C Par ==
p:=numer q
complexZeros(univariate p,eps)$ComplexRootPackage(SUP GI, Par)
-- find the complex zeros of an univariate polynomial --
complexRoots(lp:L FPK,lv:L SE,eps:Par) : L L C Par ==
lnum:=[numer p for p in lp]
lden:=[dp for p in lp |not one?(dp:=denom p)]
innerSolve(lnum,lden,lv,eps)$INFSP(K,C Par,Par)
complexSolve(lp:L FPK,eps : Par) : L L EQ P C Par ==
lnum:=[numer p for p in lp]
lden:=[dp for p in lp |not one?(dp:=denom p)]
lv:="setUnion"/[variables np for np in lnum]
if lden~=[] then
lv:=setUnion(lv,"setUnion"/[variables dp for dp in lden])
[[equation(x::(P C Par),r::(P C Par)) for x in lv for r in nres]
for nres in innerSolve(lnum,lden,lv,eps)$INFSP(K,C Par,Par)]
complexSolve(le:L EQ FPK,eps : Par) : L L EQ P C Par ==
lp:=[lhs ep - rhs ep for ep in le]
lnum:=[numer p for p in lp]
lden:=[dp for p in lp |not one?(dp:=denom p)]
lv:="setUnion"/[variables np for np in lnum]
if lden~=[] then
lv:=setUnion(lv,"setUnion"/[variables dp for dp in lden])
[[equation(x::(P C Par),r::(P C Par)) for x in lv for r in nres]
for nres in innerSolve(lnum,lden,lv,eps)$INFSP(K,C Par,Par)]
complexSolve(p : FPK,eps : Par) : L EQ P C Par ==
(mvar := mainVariable numer p ) case "failed" =>
error "no variable found"
x:P C Par:=mvar::SE::(P C Par)
[equation(x,val::(P C Par)) for val in complexRoots(p,eps)]
complexSolve(eq : EQ FPK,eps : Par) : L EQ P C Par ==
complexSolve(lhs eq - rhs eq,eps)
|