/usr/share/axiom-20170501/src/algebra/LODEEF.spad is in axiom-source 20170501-3.
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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 | )abbrev package LODEEF ElementaryFunctionLODESolver
++ Author: Manuel Bronstein
++ Date Created: 3 February 1994
++ Date Last Updated: 9 March 1994
++ References:
++ Bron92 Linear Ordinary Differential Equations: Breaking Through the
++ Order 2 Barrier
++ Description:
++ \spad{ElementaryFunctionLODESolver} provides the top-level
++ functions for finding closed form solutions of linear ordinary
++ differential equations and initial value problems.
ElementaryFunctionLODESolver(R, F, L) : SIG == CODE where
R : Join(OrderedSet, EuclideanDomain, RetractableTo Integer,
LinearlyExplicitRingOver Integer, CharacteristicZero)
F : Join(AlgebraicallyClosedFunctionSpace R, TranscendentalFunctionCategory,
PrimitiveFunctionCategory)
L : LinearOrdinaryDifferentialOperatorCategory F
SY ==> Symbol
N ==> NonNegativeInteger
K ==> Kernel F
V ==> Vector F
M ==> Matrix F
UP ==> SparseUnivariatePolynomial F
RF ==> Fraction UP
UPUP==> SparseUnivariatePolynomial RF
P ==> SparseMultivariatePolynomial(R, K)
P2 ==> SparseMultivariatePolynomial(P, K)
LQ ==> LinearOrdinaryDifferentialOperator1 RF
REC ==> Record(particular: F, basis: List F)
U ==> Union(REC, "failed")
ALGOP ==> "%alg"
SIG ==> with
solve : (L, F, SY) -> U
++ solve(op, g, x) returns either a solution of the ordinary differential
++ equation \spad{op y = g} or "failed" if no non-trivial solution can be
++ found; When found, the solution is returned in the form
++ \spad{[h, [b1,...,bm]]} where \spad{h} is a particular solution and
++ and \spad{[b1,...bm]} are linearly independent solutions of the
++ associated homogenuous equation \spad{op y = 0}.
++ A full basis for the solutions of the homogenuous equation
++ is not always returned, only the solutions which were found;
++ \spad{x} is the dependent variable.
solve : (L, F, SY, F, List F) -> Union(F, "failed")
++ solve(op, g, x, a, [y0,...,ym]) returns either the solution
++ of the initial value problem \spad{op y = g, y(a) = y0, y'(a) = y1,...}
++ or "failed" if the solution cannot be found;
++ \spad{x} is the dependent variable.
CODE ==> add
import Kovacic(F, UP)
import ODETools(F, L)
import RationalLODE(F, UP)
import RationalRicDE(F, UP)
import ODEIntegration(R, F)
import ConstantLODE(R, F, L)
import IntegrationTools(R, F)
import ReductionOfOrder(F, L)
import ReductionOfOrder(RF, LQ)
import PureAlgebraicIntegration(R, F, L)
import FunctionSpacePrimitiveElement(R, F)
import LinearSystemMatrixPackage(F, V, V, M)
import SparseUnivariatePolynomialFunctions2(RF, F)
import FunctionSpaceUnivariatePolynomialFactor(R, F, UP)
import LinearOrdinaryDifferentialOperatorFactorizer(F, UP)
import PolynomialCategoryQuotientFunctions(IndexedExponents K,
K, R, P, F)
upmp : (P, List K) -> P2
downmp : (P2, List K, List P) -> P
xpart : (F, SY) -> F
smpxpart : (P, SY, List K, List P) -> P
multint : (F, List F, SY) -> F
ulodo : (L, K) -> LQ
firstOrder : (F, F, F, SY) -> REC
rfSolve : (L, F, K, SY) -> U
ratlogsol : (LQ, List RF, K, SY) -> List F
expsols : (LQ, K, SY) -> List F
homosolve : (L, LQ, List RF, K, SY) -> List F
homosolve1 : (L, List F, K, SY) -> List F
norf1 : (L, K, SY, N) -> List F
kovode : (LQ, K, SY) -> List F
doVarParams: (L, F, List F, SY) -> U
localmap : (F -> F, L) -> L
algSolve : (L, F, K, List K, SY) -> U
palgSolve : (L, F, K, K, SY) -> U
lastChance : (L, F, SY) -> U
diff := D()$L
smpxpart(p, x, l, lp) == downmp(primitivePart upmp(p, l), l, lp)
downmp(p, l, lp) == ground eval(p, l, lp)
homosolve(lf, op, sols, k, x) == homosolve1(lf, ratlogsol(op,sols,k,x),k,x)
-- left hand side has algebraic (not necessarily pure) coefficients
algSolve(op, g, k, l, x) ==
symbolIfCan(kx := ksec(k, l, x)) case SY => palgSolve(op, g, kx, k, x)
has?(operator kx, ALGOP) =>
rec := primitiveElement(kx::F, k::F)
z := rootOf(rec.prim)
lk:List K := [kx, k]
lv:List F := [(rec.pol1) z, (rec.pol2) z]
(u := solve(localmap((f1:F):F +-> eval(f1, lk, lv), op), _
eval(g, lk, lv), x))
case "failed" => "failed"
rc := u::REC
kz := retract(z)@K
[eval(rc.particular, kz, rec.primelt),
[eval(f, kz, rec.primelt) for f in rc.basis]]
lastChance(op, g, x)
doVarParams(eq, g, bas, x) ==
(u := particularSolution(eq, g, bas, (f1:F):F +-> int(f1, x)))
case "failed" => lastChance(eq, g, x)
[u::F, bas]
lastChance(op, g, x) ==
(degree op)=1 => firstOrder(coefficient(op,0), leadingCoefficient op,g,x)
"failed"
-- solves a0 y + a1 y' = g
-- does not check whether there is a solution in the field generated by
-- a0, a1 and g
firstOrder(a0, a1, g, x) ==
h := xpart(expint(- a0 / a1, x), x)
[h * int((g / h) / a1, x), [h]]
-- xpart(f,x) removes any constant not involving x from f
xpart(f, x) ==
l := reverse_! varselect(tower f, x)
lp := [k::P for k in l]
smpxpart(numer f, x, l, lp) / smpxpart(denom f, x, l, lp)
upmp(p, l) ==
empty? l => p::P2
up := univariate(p, k := first l)
l := rest l
ans:P2 := 0
while up ^= 0 repeat
ans := ans + monomial(upmp(leadingCoefficient up, l), k, degree up)
up := reductum up
ans
-- multint(a, [g1,...,gk], x) returns gk
-- \int(g(k-1) \int(....g1 \int(a))...)
multint(a, l, x) ==
for g in l repeat a := g * xpart(int(a, x), x)
a
expsols(op, k, x) ==
(degree op) = 1 =>
firstOrder(multivariate(coefficient(op, 0), k),
multivariate(leadingCoefficient op, k), 0, x).basis
[xpart(expint(multivariate(h, k), x), x) for h in ricDsolve(op, ffactor)]
-- Finds solutions with rational logarithmic derivative
ratlogsol(oper, sols, k, x) ==
bas := [xpart(multivariate(h, k), x) for h in sols]
degree(oper) = #bas => bas -- all solutions are found already
rec := ReduceOrder(oper, sols)
le := expsols(rec.eq, k, x)
int:List(F) := [xpart(multivariate(h, k), x) for h in rec.op]
concat_!([xpart(multivariate(h, k), x) for h in sols],
[multint(e, int, x) for e in le])
homosolve1(oper, sols, k, x) ==
zero?(n := (degree(oper) - #sols)::N) => sols -- all solutions found
rec := ReduceOrder(oper, sols)
int:List(F) := [xpart(h, x) for h in rec.op]
concat_!(sols, [multint(e, int, x) for e in norf1(rec.eq, k, x, n::N)])
-- if the coefficients are rational functions, then the equation does not
-- not have a proper 1st-order right factor over the rational functions
norf1(op, k, x, n) ==
(n = 1) => firstOrder(coefficient(op, 0),leadingCoefficient op,0,x).basis
-- for order > 2, we check that the coeffs are still rational functions
symbolIfCan(kmax vark(coefficients op, x)) case SY =>
eq := ulodo(op, k)
n = 2 => kovode(eq, k, x)
eq := last factor1 eq -- eq cannot have order 1
degree(eq) = 2 =>
empty?(bas := kovode(eq, k, x)) => empty()
homosolve1(op, bas, k, x)
empty()
empty()
kovode(op, k, x) ==
b := coefficient(op, 1)
a := coefficient(op, 2)
(u:= kovacic(coefficient(op, 0), b, a, ffactor)) case "failed" => empty()
p := map(z1+->multivariate(z1, k), u::UPUP)
ba := multivariate(- b / a, k)
-- if p has degree 2 (case 2), then it must be squarefree since the
-- ode is irreducible over the rational functions, so the 2 roots of p
-- are distinct and must yield 2 independent solutions.
degree(p) = 2 => [xpart(expint(ba/(2::F) + e, x), x) for e in zerosOf p]
-- otherwise take 1 root of p and find the 2nd solution by reduction of order
y1 := xpart(expint(ba / (2::F) + zeroOf p, x), x)
[y1, y1 * xpart(int(expint(ba, x) / y1**2, x), x)]
solve(op:L, g:F, x:SY) ==
empty?(l := vark(coefficients op, x)) => constDsolve(op, g, x)
symbolIfCan(k := kmax l) case SY => rfSolve(op, g, k, x)
has?(operator k, ALGOP) => algSolve(op, g, k, l, x)
lastChance(op, g, x)
ulodo(eq, k) ==
op:LQ := 0
while eq ^= 0 repeat
op:= op + monomial(univariate(leadingCoefficient eq, k), degree eq)
eq := reductum eq
op
-- left hand side has rational coefficients
rfSolve(eq, g, k, x) ==
op := ulodo(eq, k)
empty? remove_!(k, varselect(kernels g, x)) => -- rhs is rational
rc := ratDsolve(op, univariate(g, k))
rc.particular case "failed" => -- this implies g ^= 0
doVarParams(eq, g, homosolve(eq, op, rc.basis, k, x), x)
[multivariate(rc.particular::RF, k), homosolve(eq, op, rc.basis, k, x)]
doVarParams(eq, g, homosolve(eq, op, ratDsolve(op, 0).basis, k, x), x)
solve(op, g, x, a, y0) ==
(u := solve(op, g, x)) case "failed" => "failed"
hp := h := (u::REC).particular
b := (u::REC).basis
v:V := new(n := #y0, 0)
kx:K := kernel x
for i in minIndex v .. maxIndex v for yy in y0 repeat
v.i := yy - eval(h, kx, a)
h := diff h
(sol := particularSolution(
map_!((f1:F):F+->eval(f1,kx,a),wronskianMatrix(b,n)), v))
case "failed" => "failed"
for f in b for i in minIndex(s := sol::V) .. repeat
hp := hp + s.i * f
hp
localmap(f, op) ==
ans:L := 0
while op ^= 0 repeat
ans := ans + monomial(f leadingCoefficient op, degree op)
op := reductum op
ans
-- left hand side has pure algebraic coefficients
palgSolve(op, g, kx, k, x) ==
rec := palgLODE(op, g, kx, k, x) -- finds solutions in the coef. field
rec.particular case "failed" =>
doVarParams(op, g, homosolve1(op, rec.basis, k, x), x)
[(rec.particular)::F, homosolve1(op, rec.basis, k, x)]
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