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intpar.spad
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intpar.spad
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)abbrev package RNORM RealNormalizationUtilities
++ Description: This package tries to rewrite an expression in
++ real form preserving its value.
RealNormalizationUtilities(R, F) : Exports == Implementation where
R : Join(GcdDomain, Comparable, CharacteristicZero,
PolynomialFactorizationExplicit,
RetractableTo Integer, LinearlyExplicitOver Integer)
F : Join(TranscendentalFunctionCategory, PrimitiveFunctionCategory,
AlgebraicallyClosedFunctionSpace R)
K ==> Kernel F
SY ==> Symbol
Exports ==> with
try_real : (F, SY) -> F
Implementation ==> add
if F has LiouvillianFunctionCategory then
imf := sqrt(-1)$F
imk := retract(imf)@Kernel(F)
split_expr(e : F) : List(F) ==
ue := univariate(e, imk)
u_num := numer(ue)
degree(u_num) > 1 => error "impossible"
u_den := denom(ue)
degree(u_den) > 1 => error "impossible"
r_num := coefficient(u_num, 0)
i_num := coefficient(u_num, 1)
cd := r_den := coefficient(u_den, 0)
if degree(u_den) > 0 then
i_den := leadingCoefficient(u_den)
cd := r_den*r_den + i_den*i_den
nrn := r_num*r_den + i_num*i_den
i_num := i_num*r_den - r_num*i_den
r_num := nrn
r_num := r_num/cd
i_num := i_num/cd
[r_num, i_num]
do_liou1(f1 : F, op : BasicOperator, a : F) : F ==
ll := split_expr(a)
ra := ll(1)
ia := ll(2)
is?(op, 'Ei) and ra = 0 =>
Ci(ia) + imf*Si(ia)
is?(op, 'erf) and ra = 0 => imf*erfi(ia)
is?(op, 'erf) and ra = ia =>
a1 := 2*ra/sqrt(pi())
(1 + imf)*(fresnelC(a1) - imf*fresnelS(a1))
is?(op, 'erf) and ra = -ia =>
a1 := 2*ra/sqrt(pi())
(1 - imf)*(fresnelC(a1) + imf*fresnelS(a1))
f1
real_k(k : K) : F ==
kf := k::F
op := operator(k)
arg := argument(k)
#arg = 1 and has?(op, 'prim) => do_liou1(kf, op, first(arg))
kf
do_c(c : F) : F ==
lk0 := kernels(c)
lk : List(K) := []
lv : List(F) := []
for k in lk0 repeat
op := operator(k)
arg := argument(k)
if #arg = 1 and is?(k, 'exp) then
a1 := first(arg)
ll := split_expr(a1)
ra := ll(1)
ia := ll(2)
if ra = 0 then
v := (cos(ia) + imf*sin(ia))
lk := cons(k, lk)
lv := cons(v, lv)
not(empty?(lk)) => subst(c, lk, lv)
c
try_real(f : F, x : SY) : F ==
lk := kernels(f)
res_p : F := 0
for k in lk repeat
fu := univariate(f, k)
num_f := numer(fu)
den_f := denom(fu)
degree(den_f) > 0 or degree(num_f) ~= 1 => iterate
c_den := leadingCoefficient(den_f)
c := leadingCoefficient(num_f)/c_den
D(c, x) ~= 0 => iterate
res_p := do_c(c)*real_k(k) + res_p
f := coefficient(num_f, 0)/c_den
lk := kernels(res_p)
lck : List(K) := []
rpp := res_p
for k in lk repeat
op := operator(k)
arg := argument(k)
not(is?(op, 'Ci) and #arg = 1) => iterate
fu := univariate(rpp, k)
num_f := numer(fu)
den_f := denom(fu)
degree(den_f) > 0 or degree(num_f) ~= 1 => iterate
c_den := leadingCoefficient(den_f)
c := leadingCoefficient(num_f)/c_den
D(c, x) ~= 0 => iterate
lck := cons(k, lck)
rpp := coefficient(num_f, 0)/c_den
lck := reverse(removeDuplicates(lck))
lck1 : List(F) := []
for k in lck repeat
a := first(argument(k))
for k1 in lck repeat
k = k1 => break
a1 := first(argument(k1))
a = -a1 =>
k := k1
break
lck1 := cons(k::F, lck1)
res_p := subst(res_p, lck, reverse(lck1))
f + res_p
else
try_real(f : F, x : SY) : F == f
)abbrev package LINCOMB LinearCombinationUtilities
++ Description: This package implements frequently used operations
++ on linear combinations.
LinearCombinationUtilities(F, UP) : Exports == Implementation where
F : Field
UP : UnivariatePolynomialCategory F
N ==> NonNegativeInteger
RF ==> Fraction UP
GP ==> LaurentPolynomial(F, UP)
Param_Rec_F ==> Record(ratpart : F, coeffs : Vector F)
L_Param_F ==> List Param_Rec_F
Partial_F ==> Union(Param_Rec_F, "failed")
Both_F ==> Record(particular : Partial_F, basis : L_Param_F)
Q ==> Fraction(Integer)
Param_Rec_Q2 ==> Record(ratpart : F, coeffs : Vector Q)
L_Param_Q2 ==> List Param_Rec_Q2
Partial_Q2 ==> Union(Param_Rec_Q2, "failed")
Both_Q2 ==> Record(particular : Partial_Q2, basis : L_Param_Q2)
Exports ==> with
dehomogenize : L_Param_F -> Both_F
++ dehomogenize(ls) converts list of solutions
++ (a, [c0, c1, ..., cn]) to homogeneous equation
++ L(a) + c0 f + c1 g1 + ... + cn gn = 0 into list of
++ solutions of inhomogeneous equation
++ L(a) + f + c1 g1 + ... + cn gn = 0. This transformation
++ works the same for all equations, so we only
++ need list of solutions as argument
lin_comb : (Vector F, List(F)) -> F
++ lin_comb(v, [f1, ..., fn]) computes linear combination
++ v(1) f1 + ... v(n) fn. Vector v and list [f1, ..., fn]
++ must be of equal length.
lin_comb : (Vector Q, List(F)) -> F
++ lin_comb(v, [f1, ..., fn]) computes linear combination
++ v(1) f1 + ... v(n) fn. Vector v and list [f1, ..., fn]
++ must be of equal length.
lin_comb! : (Vector F, Vector F, List(Vector F)) -> Vector F
++ lin_comb!(v, w, [f1, ..., fn]) computes linear combination
++ w + v(1) f1 + ... v(n) fn by modifying w in place.
++ Vector v and list [f1, ..., fn] must be of equal length.
lin_comb : (Vector F, List(Vector F)) -> Vector F
++ lin_comb(v, [f1, ..., fn]) computes linear combination
++ v(1) f1 + ... v(n) fn. Vector v and list [f1, ..., fn]
++ must be of equal positive length.
lin_comb : (Vector F, List(RF)) -> RF
++ lin_comb(v, [f1, ..., fn]) computes linear combination
++ v(1) f1 + ... v(n) fn. Vector v and list [f1, ..., fn]
++ must be of equal length.
lin_comb : (Vector Q, List(RF)) -> RF
++ lin_comb(v, [f1, ..., fn]) computes linear combination
++ v(1) f1 + ... v(n) fn. Vector v and list [f1, ..., fn]
++ must be of equal length.
lin_comb : (Vector F, List(GP)) -> GP
++ lin_comb(v, [f1, ..., fn]) computes linear combination
++ v(1) f1 + ... v(n) fn. Vector v and list [f1, ..., fn]
++ must be of equal length.
lin_comb : (Vector F, List(UP)) -> UP
++ lin_comb(v, [f1, ..., fn]) computes linear combination
++ v(1) f1 + ... v(n) fn. Vector v and list [f1, ..., fn]
++ must be of equal length.
Implementation ==> add
dehomogenize_body(Lpar, Par) ==>
empty?(ls) => ["failed", []]
nn := #(ls(1).coeffs)
be1 : Par
found : Boolean := false
res2 : Lpar := []
for be in ls repeat
c := (be.coeffs)(1)
c = 0 or found => res2 := cons(be, res2)
be1 := be
found := true
not(found) => ["failed", []]
c1inv := 1/(be1.coeffs)(1)
ppa := c1inv*be1.ratpart
ppv := (c1inv*be1.coeffs)(2..nn)
res3 : Lpar := []
for be in res2 repeat
c := (be.coeffs)(1)
bv := (be.coeffs)(2..nn)
rp := be.ratpart
if c ~= 0 then
rp := rp - c*ppa
bv := bv - c*ppv
res3 := cons([rp, bv], res3)
[[ppa, ppv], res3]
dehomogenize(ls : L_Param_F) : Both_F ==
dehomogenize_body(L_Param_F, Param_Rec_F)
lin_comb(v : Vector F, lf : List(F)) : F ==
res : F := 0
for i in 1..#v for f in lf repeat
res := res + v(i)*f
res
lin_comb(v : Vector Q, lf : List(F)) : F ==
res : F := 0
for i in 1..#v for f in lf repeat
res := res + v(i)::F*f
res
lin_comb!(u : Vector F, v : Vector F,
lw : List(Vector F)) : Vector F ==
res := v
n := #res
for i in 1..#u for w in lw repeat
c := u(i)
for j in 1..n repeat
res(j) := res(j) + c*w(j)
res
lin_comb(u : Vector F, lw : List(Vector F)) : Vector F ==
n := #first(lw)
lin_comb!(u, new(n, 0)$Vector(F), lw)
lin_comb(v : Vector F, lf : List(RF)) : RF ==
res : RF := 0
for i in 1..#v for f in lf repeat
res := res + v(i)::UP*f
res
lin_comb(v : Vector Q, lg : List RF) : RF ==
res : RF := 0
for i in 1..#v for g in lg repeat
res := res + v(i)::F::UP::RF*g
res
lin_comb(v : Vector F, lf : List(GP)) : GP ==
res : GP := 0
for i in 1..#v for f in lf repeat
res := res + v(i)::GP*f
res
lin_comb(v : Vector F, lf : List(UP)) : UP ==
res : UP := 0
for i in 1..#v for f in lf repeat
res := res + v(i)*f
res
)abbrev package EFACTOR ExpressionFactorPolynomial
++ Description: This package implements \spad{factorPolynomial}
++ for Expression.
ExpressionFactorPolynomial(R, F) : Exports == Implementation where
R : Join(GcdDomain, Comparable)
F : FunctionSpace R
UP ==> SparseUnivariatePolynomial F
K ==> Kernel F
MP ==> SparseMultivariatePolynomial(R, K)
UMP ==> SparseUnivariatePolynomial MP
Exports ==> with
factorPolynomial : UP -> Factored(UP)
++ factorPolynomial(p) factors p into irreducible factors over
++ field generated by its coefficients.
factor : (UP, List F) -> Factored(UP)
++ factor(p, [k1, ..., kn]) factors p into irreducible factors over
++ field generated by its coefficients and k1, ..., kn.
poly_factor : UMP -> Factored(UMP)
++ poly_factor(p) should be local but conditional.
Implementation ==> add
if (R is Integer or R is Complex(Integer)) then
Exp ==> IndexedExponents(K)
MULTFACT ==> MultivariateFactorize(K, Exp, R, MP)
poly_factor(p) == factor(p)$MULTFACT
else if MP has PolynomialFactorizationExplicit then
poly_factor(p) == factorPolynomial(p)$MP
else
poly_factor(p) == error "factorPolynomial unimplemented"
dummy := create()$SingletonAsOrderedSet
xs := new()$Symbol
xk := kernel(xs)$K
UPCF2 ==> UnivariatePolynomialCategoryFunctions2
fpoly_factor(p : UP) : Factored(UP) ==
cn := leadingCoefficient(p)
cn = 0 => nilFactor(p, 1)
p := p/cn
cnp := cn::UP
pf := retract(eval(p, dummy, xk::F))@F
p2 := univariate(numer(pf), xk)
fres1 := poly_factor(p2)
res : Factored(UP) := makeFR(cnp, [])
for frec in factorList(fres1) repeat
fr1 := frec.factor
degree(fr1) < 1 => iterate
frec.flag ~= "prime" and frec.flag ~= "irred" => error "impossible"
fru : UP := map(c1 +-> c1::F, fr1
)$UPCF2(MP, UMP, F, UP)
fr2 := fru/leadingCoefficient(fru)
res := res*primeFactor(fr2, frec.exponent)
res
PCQF ==> PolynomialCategoryQuotientFunctions(IndexedExponents K,
K, R, MP, F)
ifactor(p : UP, lk : List K) : Factored(UP) ==
n := degree(p)
cn := leadingCoefficient(p)
cn = 0 => nilFactor(p, 1)
cnp := cn::UP
n < 1 => makeFR(cnp, [])
n = 1 => makeFR(cnp, [["prime", p/cn, 1]])
empty? lk => fpoly_factor(p)
q := minPoly(k1 := reduce(max, lk))
lk2 := [k for k in lk | k ~= k1]
sae := SimpleAlgebraicExtension(F, UP, q)
ups := SparseUnivariatePolynomial sae
p2 : ups := map(x +-> reduce univariate(x, k1, q)$PCQF, p
)$UPCF2(F, UP, sae, ups)
fres1 := factor(p2, x +-> ifactor(x, lk2)
)$InnerAlgFactor(F, UP, sae, ups)
res : Factored(UP) := makeFR(cnp, [])
kf := k1::F
for frec in factorList(fres1) repeat
fr1 := frec.factor
frec.flag ~= "prime" and frec.flag ~= "irred" => error "impossible"
fru : UP := map(c1 +-> lift(c1)(kf), fr1
)$UPCF2(sae, ups, F, UP)
fr2 := fru/leadingCoefficient(fru)
res := res*primeFactor(fr2, frec.exponent)
res
factorPolynomial(p) == ifactor(p, algtower(coefficients(p)))
factor(p, lf) ==
ifactor(p, algtower(concat(lf, coefficients(p))))
)abbrev package RFSSPLIT RootFSSplit
++ Description: This package implements \spad{alg_split_roots}
++ allowing its reuse.
RootFSSplit(R, F) : Exports == Implementation where
R : Join(IntegralDomain, RetractableTo Integer, Comparable)
F : FunctionSpace(R)
K ==> Kernel(F)
Alg_Rec ==> Record(funs : List F, nroot : F, npow1 : Integer,
npow2 : Integer)
Exports ==> with
alg_split_roots : (List F, r1 : K, r2 : K) -> List(Alg_Rec)
++ alg_split_roots(lf, r1, r2) splits each f in lf into
++ linear combination of product of powers of r1 and r2
++ which must be independent roots.
alg_split_root0 : (F, K, Integer) -> List F
++ alg_split_root0(f, r, n) returns \spad{[a1, ..., an]}
++ such that \spad{f = a1 + a2*r + ... an*r^(n - 1)}.
++ \spad{r} must be a root and \spad{n} must be its
++ degree.
Implementation == add
P ==> SparseMultivariatePolynomial(R, K)
alg_split_root0(f : F, r : K, n : Integer) : List F ==
n = 2 =>
ef := eval(f, r, -(r::F))
f0 := (f + ef)/(2::F)
f1 := (f - ef)/(2::F)
member?(r, kernels(f0)) =>
print(f0::OutputForm)
error "alg_split_root0: roots did not cancel 1"
f1 := f1/(r::F)
member?(r, kernels(f1)) =>
print(f1::OutputForm)
error "alg_split_root0: roots did not cancel 2"
[f0, f1]
q := univariate(f, r, minPoly r
)$PolynomialCategoryQuotientFunctions(IndexedExponents K,
K, R, P, F)
[coefficient(q, i) for i in 0..(n-1)]
alg_split_root2(lf : List F, r : K, n : Integer) : List List F ==
nlf0 : List List F := [alg_split_root0(f, r, n) for f in lf]
res : List List F := []
for pow1 in (0$Integer)..(n - 1) repeat
nlf1 := nlf0
resi : List(F) := []
while not(empty?(nlf1)) repeat
lf1 := first(nlf1)
resi := cons(first(lf1), resi)
setfirst!(nlf1, rest(lf1))
nlf1 := rest(nlf1)
resi := reverse!(resi)
res := cons(resi, res)
reverse!(res)
alg_split_roots(lf : List F, r1 : K, r2 : K) : List(Alg_Rec) ==
a1 := argument(r1)
a2 := argument(r2)
n1 : Integer := retract(a1(2))@Integer
b1 := a1(1)
n2 : Integer := retract(a2(2))@Integer
b2 := a2(1)
ll1 : List List F := alg_split_root2(lf, r1, n1)
res : List(Alg_Rec) := []
pow1 : Integer
rop := operator(r1)
for l1 in ll1 for pow1 in (0$Integer)..(n1 - 1) repeat
every?((f1 : F) : Boolean +-> f1 = 0, l1) => iterate
ll2 := alg_split_root2(l1, r2, n2)
g1 := gcd(n1, pow1)$Integer
nn1 := (n1 exquo g1)::Integer
np1 := (pow1 exquo g1)::Integer
for l2 in ll2 for pow2 in 0..(n2 - 1) repeat
every?((f1 : F) : Boolean +-> f1 = 0, l2) => iterate
g2 := gcd(n2, pow2)
nn2 := (n2 exquo g2)::Integer
np2 := (pow2 exquo g2)::Integer
nn := lcm(nn1, nn2)
bb1 := b1^(np1*(nn exquo nn1)::Integer)
bb2 := b2^(np2*(nn exquo nn2)::Integer)
nrr :=
pow1 + pow2 > 0 =>
kernel(rop, [bb1*bb2, nn::F])::F
1$F
nl := [f1*nrr for f1 in l2]
res := cons([nl, nrr, pow1, pow2]$Alg_Rec, res)
res
)abbrev package INTPAR1 ParametricTranscendentalIntegration
++ Description: This package implements parametric integration
++ in transcendental case.
ParametricTranscendentalIntegration(F, UP) : Exports == Implementation where
F : Field
UP : UnivariatePolynomialCategory F
Z ==> Integer
Q ==> Fraction Z
RF ==> Fraction UP
Param_Rec_F ==> Record(ratpart : F, coeffs : Vector F)
L_Param_F ==> List Param_Rec_F
Param_Rec_Q ==> Record(ratpart : RF, coeffs : Vector F)
L_Param_Q ==> List Param_Rec_Q
Param_Rec_QF ==> Record(logands : List F, basis : List Vector Q)
Param_Rec_Q2 ==> Record(logands : List RF, basis : List Vector Q)
Exports ==> with
primextint : (UP -> UP, List F -> L_Param_F,
Matrix F -> List Vector F, List RF) -> L_Param_Q
++ primextint(', ext, csolve, [g1, ..., gn]) returns
++ a basis of solutions of the homogeneous system
++ \spad{h' + c1*g1 + ... + cn*gn = 0}.
++ Argument ext is an extended integration function on F.
++ csolve is solver over constants.
expextint : (UP -> UP, (Z, List F) -> L_Param_F,
Matrix F -> List Vector F, List RF) -> L_Param_Q
++ expextint(', rde, csolve, [g1, ..., gn]) returns
++ a basis of solution of the homogeneous system
++ \spad{h' + c1*g1 + ... + cn*gn = 0}
++ Argument foo is an parametric rde solver on F.
++ csolve is solver over constants.
diffextint : (List UP -> L_Param_F, Matrix F -> List Vector F,
List RF) -> L_Param_F
++ diffextint(ext, csolve, [g1, ..., gn]) is like primextint and
++ expextint but for differentialy transcendental extensions.
unkextint : (List F -> L_Param_F, Matrix F -> List Vector F,
List RF) -> L_Param_F
++ unkextint(ext, csolve, [g1, ..., gn]) is like primextint and
++ expextint but for makes no assumption about generator of
++ the extension.
logextint : (UP -> UP, UP -> Factored(UP), Matrix F -> List Vector Q,
List UP -> Param_Rec_Q2, List(RF)) -> Param_Rec_Q2
++ logextint(der, ufactor, csolve, rec, [g1, ..., gn])
++ returns [[u1, ..., um], bas] giving basis of solution of
++ the homogeneous systym
++ \spad{c1*g1 + ... + cn*gn + c_{n+1}u1'/u1 + ... c_{n+m}um'/um = 0}
monologextint : (List UP, Matrix F -> List Vector Q,
List F -> Param_Rec_QF) -> Param_Rec_Q2
++ monologextint(lup, csolve, rec) is a helper for logextint
Implementation ==> add
N ==> NonNegativeInteger
GP ==> LaurentPolynomial(F, UP)
import from LinearCombinationUtilities(F, UP)
import from LinearCombinationUtilities(Q, SparseUnivariatePolynomial(Q))
import from TranscendentalHermiteIntegration(F, UP)
import from LiftLinearDependence(F, F, UP)
import from LiftLinearDependence(F, Q, UP)
monologextint(lup, csolve, rec1) ==
n0 := #lup
lc0 := [coefficient(p, 0) for p in lup]
lup2 := [p - c0::UP for p in lup for c0 in lc0]
cb := csolve_pol(csolve, lup2)
nlc0 := [lin_comb(bv, lc0) for bv in cb]
(ll, bl) := rec1(nlc0)
empty?(bl) => [[], []]
n1 := #cb
n2 := #ll
n3 := n0 + n2
rbl := [new(n3, 0)$Vector(Q) for bv in bl]
nl := [le::UP::RF for le in ll]
for rbv in rbl for bv in bl repeat
pv := lin_comb(bv(1..n1), cb)
for i in 1..n0 repeat rbv(i) := pv(i)
for i in n0+1..n3 for j in n1+1.. repeat rbv(i) := bv(j)
[nl, rbl]
logextint(der, ufactor, csolve, rec, lg) ==
empty?(lg) => [[], []]
n0 := #lg
lghr := [HermiteIntegrate(g, der) for g in lg]
lg1 := [ghr.answer for ghr in lghr]
cb := csolve_rf(csolve, lg1)
lg2 := [ghr.logpart for ghr in lghr]
lg3 := [lin_comb(bv, lg2) for bv in cb]
lden1 := [denom(g) for g in lg3]
(mbas, m2) := gcdDecomposition(vector(lden1))$GcdBasis(UP)
n1 := #lg3
mbasl := entries(mbas)
mbasfl := [[frr.factor for frr in factorList(ufactor(mbasp))]
for mbasp in mbasl]
basl1 := reduce(concat, mbasfl, [])
sl := [#fl1 for fl1 in mbasfl]
n3 := reduce(_+, sl, 0)
m3 := new(n3, n1 + n3, 0)$Matrix(UP)
for i in 1..n1 for g in lg3 repeat
fl : List(UP) := []
jl : List(Z) := []
sl1 := sl
fl1 : List(UP) := []
j0 : Z := 1
j1 : Z := 0
for j in 1..n3 for fj in basl1 repeat
j1 := j1 + 1
if j1 > first(sl1) then
sl1 := rest(sl1)
j1 := 0
j0 := j0 + 1
if m2(j0, i) = 1 then
fl := cons(fj, fl)
jl := cons(j, jl)
nl := decompose(numer(g), fl)$PartialFractionUtilities(F, UP)
for num in nl for j in jl repeat
m3(j, i) := num
lpc : List(RF) := []
for j in 1..n3 for bj in basl1 repeat
dbj := der(bj)
(q, r) := divide(dbj, bj)
m3(j, n1 + j) := r
lpc := cons(q::RF, lpc)
lpc := reverse!(lpc)
rs3 := reducedSystem(m3)
cb2 := csolve(rs3)
cb3 := [bv(1..n1) for bv in cb2]
cb3e := [bv(n1+1..n1+n3) for bv in cb2]
ncb := [lin_comb(bv, cb) for bv in cb3]
nlpc := [lin_comb(bv, lpc) for bv in cb3e]
lg4 := [ghr.polypart::RF + ghr.specpart for ghr in lghr]
lg5 := [lin_comb(bv, lg4) + pc for bv in ncb for pc in nlpc]
lrf : List(RF) := []
lg6 : List(RF) := []
for g in lg5 repeat
den := denom(g)
(q, r) := divide(numer(g), den)
lrf := cons(r/den, lrf)
lg6 := cons(q::RF, lg6)
lrf := reverse!(lrf)
lg6 := reverse!(lg6)
cb4 := csolve_rf(csolve, lrf)
ncb2 := [lin_comb(bv, ncb) for bv in cb4]
cb4e := [lin_comb(bv, cb3e) for bv in cb4]
lg7 := [lin_comb(bv, lg6) for bv in cb4]
n4 := #lg7
(flog, fbas) := rec([retract(g)@UP for g in lg7])
empty?(fbas) => [[], []]
rbasl1 := [up::RF for up in basl1]
nlog := concat(rbasl1, flog)
n5 := #flog
n6 := (n0 + n3 + n5)::N
rbas := [new(n6, 0)$Vector(Q) for bv in fbas]
for rbv in rbas for bv in fbas repeat
bv1 := bv(1..n4)
pv := lin_comb(bv1, ncb2)
for i in 1..n0 repeat rbv(i) := pv(i)
pv := lin_comb(bv1, cb4e)
for i in n0 + 1..n0 + n3 for j in 1..n3 repeat rbv(i) := pv(j)
for i in n0 + n3 + 1..n6 for j in n4 + 1..n4 + n5 repeat
rbv(i) := bv(j)
[nlog, rbas]
RF_to_GP(f : RF) : GP ==
(numer(f)::GP exquo denom(f)::GP)::GP
primextint(der : UP -> UP, ext : List F -> L_Param_F,
csolve : Matrix F -> List Vector F,
lg : List RF) : L_Param_Q ==
empty?(lg) => []
n := #lg
lghr := [HermiteIntegrate(g, der) for g in lg]
lg1 := [ghr.logpart for ghr in lghr]
cb := csolve_rf(csolve, lg1)
a1l := [ghr.answer for ghr in lghr]
lba : List(RF) := [0 for bv in cb]
lg2 := [ghr.polypart for ghr in lghr]
vg2 := vector([lg2])$Vector(UP)
ldg : List(N) := [degree(g2) for g2 in lg2]
d := reduce(max, ldg)
dk := retract(der(monomial(1, 1)$UP))@F
cba := [0$F for bv in cb]
nlba : List(RF)
for j in d..0 by -1 repeat
n1 := #cb
lgj : List(F) := []
for i in 1..n repeat
gi := vg2(i)
gij : F := 0
if degree(gi) = j then
gij := leadingCoefficient(gi)
vg2(i) := reductum(gi)
lgj := cons(gij, lgj)
lgj := reverse!(lgj)
lgj1 : List(F) := []
for bv in cb for aa in cba repeat
ff := lin_comb(bv, lgj) + (j + 1)::F*dk*aa
lgj1 := cons(ff, lgj1)
lgj1 := reverse!(lgj1)
b2 := ext(cons(dk, lgj1))
n1p := n1 + 1
cb0 := [(be.coeffs)(2..n1p) for be in b2]
ncb := [lin_comb(bv, cb) for bv in cb0]
cba := [be.ratpart for be in b2]
nlba := [lin_comb(bv, lba) +
monomial(be.ratpart, j)$UP::RF +
monomial((be.coeffs)(1)/(j + 1)::F, j + 1)$UP::RF
for be in b2 for bv in cb0]
cb := ncb
lba := nlba
nlba := [ba - lin_comb(bv, a1l) for bv in cb for ba in lba]
[[ba, bv] for bv in cb for ba in nlba]
expextint(der : UP -> UP, rde : (Z, List F) -> L_Param_F,
csolve : Matrix F -> List Vector F,
lg : List RF) : L_Param_Q ==
empty?(lg) => []
lghr := [HermiteIntegrate(g, der) for g in lg]
lg1 := [ghr.logpart for ghr in lghr]
cb := csolve_rf(csolve, lg1)
a1l := [ghr.answer for ghr in lghr]
lba : List(GP) := [0 for bv in cb]
lg2 := [ghr.polypart::GP + RF_to_GP(ghr.specpart) for ghr in lghr]
vg2 := vector([lg2])$Vector(GP)
ldg : List(Z) := [degree(g2) for g2 in lg2]
d := reduce(max, ldg)
j := d
repeat
last_iter : Boolean := true
lgj : List(F) := []
for i in 1..#vg2 repeat
gi := vg2(i)
gij : F := 0
if gi ~= 0 then
last_iter := false
if degree(gi) = j then
gij := leadingCoefficient(gi)
vg2(i) := reductum(gi)
lgj := cons(gij, lgj)
last_iter => break
lgj := reverse!(lgj)
lgj1 : List(F) := []
for bv in cb repeat
ff := lin_comb(bv, lgj)
lgj1 := cons(ff, lgj1)
lgj1 := reverse!(lgj1)
b2 := rde(j, lgj1)
empty?(b2) => return []
ncb := [lin_comb(be.coeffs, cb) for be in b2]
nlba := [lin_comb(be.coeffs, lba) + monomial(be.ratpart, j)$GP
for be in b2]
cb := ncb
lba := nlba
j := j - 1
lbar := [convert(ba)@RF - lin_comb(bv, a1l)
for bv in cb for ba in lba]
[[bar, bv] for bv in cb for bar in lbar]
diffextint1(trim : RF -> UP, ext : List UP -> L_Param_F,
csolve : Matrix(F) -> List Vector(F),
lg : List RF) : L_Param_F ==
lup := [trim(g) for g in lg]
lg1 := [g - up::RF for g in lg for up in lup]
cb := csolve_rf(csolve, lg1)
lup1 := [lin_comb(bv, lup) for bv in cb]
res1 := ext(lup1)
[[re.ratpart, lin_comb(re.coeffs, cb)] for re in res1]
lin_part(f : RF) : UP ==
p := numer(f) quo denom(f)
monomial(coefficient(p, 1), 1)$UP + coefficient(p, 0)::UP
diffextint(ext, csolve, lg) ==
diffextint1((x : RF) : UP +-> lin_part(x), ext, csolve, lg)
coeff0(f : RF) : UP ==
p := numer(f) quo denom(f)
coefficient(p, 0)::UP
unkextint(ext, csolve, lg) ==
ext1 := (lup : List UP) : L_Param_F +->
lf := [retract(p)@F for p in lup]
ext(lf)
diffextint1((x : RF) : UP +-> coeff0(x), ext1, csolve, lg)
)abbrev package RDEAUX RDEaux
++ Description: This package contains special case for RDE solver.
RDEaux(F) : Exports == Implementation where
F : Field
Z ==> Integer
UP ==> SparseUnivariatePolynomial F
RSOL ==> Record(ans : UP, remainder : UP)
DSOL ==> Record(ans : List(UP), acoeff : UP, eegen : UP, bpar : UP,
lcpar : List UP, dpar : Z)
USOL ==> Union(List(RSOL), DSOL)
Exports ==> with
multi_SPDE : (UP, UP, List(UP), Z, UP -> UP) -> USOL
++ multi_SPDE(a, b, lc, d, der)
SPDE1 : (UP, UP, UP -> UP) -> RSOL
++ SPDE1(b, c, D) solves Q' + b Q = c and returns [Q, r] where
++ r = c - ( Q' + b Q). That is when r is zero then Q is
++ true solution, otherwise r represets unsolved part of c.
++ Moreover def(r) < deg(bQ). Note: SPDE1 assumes that
++ deg(Q') < deg(bQ) for all Q.
Implementation ==> add
EE_Rec ==> Record(coef1 : UP, coef2 : UP)
multi_SPDE(a, b, lc, d, der) ==
d <$Z 0 => [[0, c]$RSOL for c in lc]
every?(zero?, lc) => [[0, 0]$RSOL for c in lc]
ee := extendedEuclidean(a, b)
ee.generator ~= 1 =>
[[0 for c in lc], 1, ee.generator, b, lc, d]$DSOL
lr : List(UP) := []
lnc : List(UP) := []
for c in lc repeat
cc1 := ee.coef1*c
cc2 := ee.coef2*c
qr := divide(cc2, a)
r := qr.remainder
nc := cc1 + b*qr.quotient - der(r)
lnc := cons(nc, lnc)
lr := cons(r, lr)
lr := reverse!(lr)
res1 := multi_SPDE(a, b + der(a), reverse!(lnc), d - degree(a)$UP, der)
res1 case DSOL =>
dres := res1@DSOL
[[a*s2 + r for s2 in dres.ans for r in lr], a*dres.acoeff,
dres.eegen, dres.bpar, dres.lcpar, dres.dpar]$DSOL
[[a*s1.ans + r, s1.remainder]$RSOL for s1 in res1::List(RSOL)
for r in lr]
SPDE1(b, c, der) ==
q : UP := 0
db := (degree b)::Z
lb := leadingCoefficient b
while c ~= 0 repeat
(n := (degree c)::Z - db) < 0 => return [q, c]
qq := monomial((leadingCoefficient c) / lb, n::NonNegativeInteger)
c := c - b*qq - der(qq)
q := q + qq
[q, 0]
)abbrev package RDEPAR ParametricRischDE
++ Description: This package implements parametric RDE solver. Only
++ simplest algebraic cases are implemented, the rest of algebraic
++ case is unimplemented. The code throws errors on inimplmented
++ cases.
ParametricRischDE(R, F) : Exports == Implementation where
R : Join(GcdDomain, Comparable, CharacteristicZero,
RetractableTo Integer, LinearlyExplicitOver Integer)
F : Join(TranscendentalFunctionCategory, AlgebraicallyClosedField,
FunctionSpace R)
N ==> NonNegativeInteger
Z ==> Integer
Q ==> Fraction(Integer)
SY ==> Symbol
LF ==> List F
K ==> Kernel F
LK ==> List K
Param_Rec_F ==> Record(ratpart : F, coeffs : Vector F)
L_Param_F ==> List Param_Rec_F
Partial_F ==> Union(Param_Rec_F, "failed")
Both_F ==> Record(particular : Partial_F, basis : L_Param_F)
UP ==> SparseUnivariatePolynomial F
GP ==> LaurentPolynomial(F, UP)
Param_Rec_UP ==> Record(ratpart : UP, coeffs : Vector F)
L_Param_UP ==> List Param_Rec_UP
Param_Rec_QF ==> Record(logands : List F, basis : List Vector Q)
Exports ==> with
param_rde : (Z, F, LF, SY, LK, (LK, LF) -> L_Param_F,
(LK, LF) -> Param_Rec_QF) -> L_Param_F
++ param_rde(n, f, lg, x, lk, ext, logi) finds basis of
++ solution to the equation
++ dy/dx + n df/dx y + c1 g1 + ... cn gn = 0 where
++ y is in field generated by lk and ci are
++ constants.
param_rde : (Z, F, F, LF, SY, LK, (LK, LF) -> L_Param_F,
(LK, LF) -> Param_Rec_QF) -> Both_F
++ param_rde(n, f, h, lg, x, lk, ext, logi) finds a particular
++ solution and basis of solutions to homogeneous
++ equation for equation
++ dy/dx + n df/dx y + c1 g1 + ... cn gn = h where
++ y is in field generated by lk and ci are
++ constants.
param_rde2 : (F, LF, SY, LK, (LK, LF) -> L_Param_F,
(LK, LF) -> Param_Rec_QF) -> L_Param_F
++ param_rde2(fp, lg, x, lk, ext, logi) finds basis of
++ solution to the equation
++ dy/dx + fp y + c1 g1 + ... cn gn = 0 where
++ y is in field generated by lk and ci are
++ constants.
exp_lower_bound : (UP, GP, Z, Z, LK, F, (LK, LF) -> Param_Rec_QF) -> Z
++ exp_lower_bound(a, b, ob, nc0, lk, eta, logi) computes
++ lower degree bound for solution of \spad{a*D(y) + b*y = c}
++ in exponential case. ob is order of b, nc0 is lower
++ bound on order of c, eta is derivative of the argument
++ of exponential.
Implementation ==> add
MET ==> MonomialExtensionTools(F, UP)
RF ==> Fraction UP
Param_Rec_Q ==> Record(ratpart : RF, coeffs : Vector F)
L_Param_Q ==> List Param_Rec_Q
RSOL ==> Record(ans : UP, remainder : UP)
DSOL ==> Record(ans : List(UP), acoeff : UP, eegen : UP, bpar : UP,
lcpar : List UP, dpar : Z)
P ==> SparseMultivariatePolynomial(R, K)
import from Integer
import from List(Integer)
import from IntegrationTools(R, F)
import from PolynomialCategoryQuotientFunctions(IndexedExponents K,
K, R, P, F)
import from LinearCombinationUtilities(F, UP)
ALGOP ==> '%alg
PRIM ==> 'prim
OPDIFF ==> '%diff
do_spde1 : (UP, List(UP), Z, UP -> UP, Matrix(F) -> Matrix(F))
-> L_Param_UP
do_param_rde : (F, Boolean, LF, SY, K, LK, (LK, LF) -> L_Param_F,
(LK, LF) -> Param_Rec_QF) -> L_Param_F
do_alg_rde : (F, List F, SY, K, LK, (LK, LF) -> L_Param_F,
(LK, LF) -> Param_Rec_QF) -> L_Param_F
do_diff_rde : (F, List F, SY, K, LK, (LK, LF) -> L_Param_F,
(LK, LF) -> Param_Rec_QF) -> L_Param_F
get_denom(f : RF, lg : List RF, der : UP -> UP) : List(UP) ==
d := normalDenom(f, der)$MET
e0 := lcm [denom(g) for g in lg]$List(UP)
(e, s) := split(e0, der)$MET
gg := gcd(d, e)
h := (gcd(e, differentiate e) exquo gcd(gg, differentiate gg))::UP
[d, h]
Frec ==> Record(fctr : UP, xpnt : Z)
normalize(f : RF, der : UP -> UP) : List(Frec) ==
d := normalDenom(f, der)$MET
g := gcd(d, differentiate(d))
d0 := (d exquo g)::UP
dd := gcd(d0, g)
d1 := (d0 exquo dd)::UP
d2 := (denom(f) exquo d1)::UP
eeu := extendedEuclidean(d2, d1, numer(f))
(a, b) := eeu::Record(coef1 : UP, coef2 : UP)
zk := kernel(new()$Symbol)$K
dd1 := der(d1)
r := resultant(a - zk::F*dd1, d1)
rql := get_rational_roots(r, zk
)$FunctionSpaceRationalRoots(R, F)
rl : List Frec := []
for rq in rql repeat
if (mu := retractIfCan(rq)@Union(Z, "failed")) case Z then
m := mu@Z
if m > 0 then
pi := gcd(a - m::F*dd1, d1)
rl := cons([pi, m], rl)
rl
RF_to_GP(f : RF) : GP == (numer(f)::GP exquo denom(f)::GP)::GP
do_spde1(b : UP, lc : List(UP), der : UP -> UP,
get_rs : Matrix(F) -> Matrix(F)) : L_Param_UP ==
lar := [SPDE1(b, c, der)$RDEaux(F) for c in lc]
la := [ar.ans for ar in lar]
lrem := [ar.remainder for ar in lar]
rs1 : Matrix(F) := reducedSystem(matrix([lrem]))
rs2 := get_rs(rs1)
lkv := nullSpace(rs2)
[[lin_comb(kv, la), kv] for kv in lkv]
param_SPDE(a : UP, b : UP, lc : List(UP), d : Z, der : UP -> UP,
get_rs : Matrix(F) -> Matrix(F),
do_degrad : (RF, List(RF)) -> L_Param_Q,
x : SY) : L_Param_UP ==
dt := der monomial(1, 1)
degree(a) = 0 =>
a ~= 1 => error "param_SPDE: degree(a) = 0 but a ~= 1"
degt := degree(dt) - 1
base_case := dt = 1
(b ~= 0) and (base_case or degree(b) > max(0, degt)) =>
do_spde1(b, lc, der, get_rs)
lcr := [c::RF for c in lc]
res1 := do_degrad(b::RF, lcr)
[[retract(re.ratpart)@UP, re.coeffs] for re in res1]
s1 := multi_SPDE(a, b, lc, d, der)$RDEaux(F)
s1 case List(RSOL) =>
lrs := s1@List(RSOL)
m1 := matrix([[rsol.remainder for rsol in lrs]])$Matrix(UP)
rs1 : Matrix(F) := reducedSystem(m1)
rs2 := get_rs(rs1)
lkv := nullSpace(rs2)
a1l := [rsol.ans for rsol in lrs]
[[lin_comb(kv, a1l), kv] for kv in lkv]
dres := s1::DSOL
g := dres.eegen
a := (a exquo g)::UP
b := (dres.bpar exquo g)::UP
aa := dres.acoeff
oans := dres.ans
lq := [divide(c, g) for c in dres.lcpar]
lr := [q.remainder for q in lq]
rs1 : Matrix(F) := reducedSystem(matrix([lr])$Matrix(UP))
rs2 := get_rs(rs1)
lkv := nullSpace(rs2)
empty?(lkv) => []
lc := [q.quotient for q in lq]
nlc : List(UP) := []
for kv in lkv repeat
nlc := cons(lin_comb(kv, lc), nlc)
nlc := reverse!(nlc)
s2 := param_SPDE(a, b, nlc, dres.dpar, der, get_rs, do_degrad, x)
nres : L_Param_UP := []
for be in s2 repeat
bv := lin_comb(be.coeffs, lkv)
ans1 := lin_comb(bv, oans) + aa*be.ratpart
nres := cons([ans1, bv], nres)
reverse!(nres)