/usr/share/axiom-20170501/src/algebra/EXPRODE.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 | )abbrev package EXPRODE ExpressionSpaceODESolver
++ Author: Manuel Bronstein
++ Date Created: 5 Mar 1990
++ Date Last Updated: 30 September 1993
++ References:
++ Dupe99 An Automatic Symbolic-Numeric Taylor Series ODE Solver
++ Description:
++ Taylor series solutions of explicit ODE's;
ExpressionSpaceODESolver(R, F) : SIG == CODE where
R : Join(OrderedSet, IntegralDomain, ConvertibleTo InputForm)
F : FunctionSpace R
K ==> Kernel F
P ==> SparseMultivariatePolynomial(R, K)
OP ==> BasicOperator
SY ==> Symbol
UTS ==> UnivariateTaylorSeries(F, x, center)
MKF ==> MakeUnaryCompiledFunction(F, UTS, UTS)
MKL ==> MakeUnaryCompiledFunction(F, List UTS, UTS)
A1 ==> AnyFunctions1(UTS)
AL1 ==> AnyFunctions1(List UTS)
EQ ==> Equation F
ODE ==> UnivariateTaylorSeriesODESolver(F, UTS)
SIG ==> with
seriesSolve : (EQ, OP, EQ, EQ) -> Any
++ seriesSolve(eq,y,x=a, y a = b) returns a Taylor series solution
++ of eq around x = a with initial condition \spad{y(a) = b}.
++ Note that eq must be of the form
++ \spad{f(x, y x) y'(x) + g(x, y x) = h(x, y x)}.
seriesSolve : (EQ, OP, EQ, List F) -> Any
++ seriesSolve(eq,y,x=a,[b0,...,b(n-1)]) returns a Taylor series
++ solution of eq around \spad{x = a} with initial conditions
++ \spad{y(a) = b0}, \spad{y'(a) = b1},
++ \spad{y''(a) = b2}, ...,\spad{y(n-1)(a) = b(n-1)}
++ eq must be of the form
++ \spad{f(x, y x, y'(x),..., y(n-1)(x)) y(n)(x) +
++ g(x,y x,y'(x),...,y(n-1)(x)) = h(x,y x, y'(x),..., y(n-1)(x))}.
seriesSolve : (List EQ, List OP, EQ, List EQ) -> Any
++ seriesSolve([eq1,...,eqn],[y1,...,yn],x = a,[y1 a = b1,...,yn a = bn])
++ returns a taylor series solution of \spad{[eq1,...,eqn]} around
++ \spad{x = a} with initial conditions \spad{yi(a) = bi}.
++ Note that eqi must be of the form
++ \spad{fi(x, y1 x, y2 x,..., yn x) y1'(x) +
++ gi(x, y1 x, y2 x,..., yn x) = h(x, y1 x, y2 x,..., yn x)}.
seriesSolve : (List EQ, List OP, EQ, List F) -> Any
++ seriesSolve([eq1,...,eqn], [y1,...,yn], x=a, [b1,...,bn])
++ is equivalent to
++ \spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x = a,
++ [y1 a = b1,..., yn a = bn])}.
seriesSolve : (List F, List OP, EQ, List F) -> Any
++ seriesSolve([eq1,...,eqn], [y1,...,yn], x=a, [b1,...,bn])
++ is equivalent to
++ \spad{seriesSolve([eq1=0,...,eqn=0], [y1,...,yn], x=a, [b1,...,bn])}.
seriesSolve : (List F, List OP, EQ, List EQ) -> Any
++ seriesSolve([eq1,...,eqn], [y1,...,yn],
++ x = a,[y1 a = b1,..., yn a = bn])
++ is equivalent to
++ \spad{seriesSolve([eq1=0,...,eqn=0], [y1,...,yn], x = a,
++ [y1 a = b1,..., yn a = bn])}.
seriesSolve : (EQ, OP, EQ, F) -> Any
++ seriesSolve(eq,y, x=a, b) is equivalent to
++ \spad{seriesSolve(eq, y, x=a, y a = b)}.
seriesSolve : (F, OP, EQ, F) -> Any
++ seriesSolve(eq, y, x = a, b) is equivalent to
++ \spad{seriesSolve(eq = 0, y, x = a, y a = b)}.
seriesSolve : (F, OP, EQ, EQ) -> Any
++ seriesSolve(eq, y, x = a, y a = b) is equivalent to
++ \spad{seriesSolve(eq=0, y, x=a, y a = b)}.
seriesSolve : (F, OP, EQ, List F) -> Any
++ seriesSolve(eq, y, x = a, [b0,...,bn]) is equivalent to
++ \spad{seriesSolve(eq = 0, y, x = a, [b0,...,b(n-1)])}.
CODE ==> add
checkCompat: (OP, EQ, EQ) -> F
checkOrder1: (F, OP, K, SY, F) -> F
checkOrderN: (F, OP, K, SY, F, NonNegativeInteger) -> F
checkSystem: (F, List K, List F) -> F
div2exquo : F -> F
smp2exquo : P -> F
k2exquo : K -> F
diffRhs : (F, F) -> F
diffRhsK : (K, F) -> F
findCompat : (F, List EQ) -> F
findEq : (K, SY, List F) -> F
localInteger: F -> F
opelt := operator("elt"::Symbol)$OP
opex := operator("fixedPointExquo"::Symbol)$OP
opint := operator("integer"::Symbol)$OP
Rint? := R has IntegerNumberSystem
localInteger n == (Rint? => n; opint n)
diffRhs(f, g) == diffRhsK(retract(f)@K, g)
k2exquo k ==
is?(op := operator k, "%diff"::Symbol) =>
error "Improper differential equation"
kernel(op, [div2exquo f for f in argument k]$List(F))
smp2exquo p ==
map(k2exquo,x+->x::F,p)_
$PolynomialCategoryLifting(IndexedExponents K,K, R, P, F)
div2exquo f ==
((d := denom f) = 1) => f
opex(smp2exquo numer f, smp2exquo d)
-- if g is of the form a * k + b, then return -b/a
diffRhsK(k, g) ==
h := univariate(g, k)
(degree(numer h) <= 1) and ground? denom h =>
- coefficient(numer h, 0) / coefficient(numer h, 1)
error "Improper differential equation"
checkCompat(y, eqx, eqy) ==
lhs(eqy) =$F y(rhs eqx) => rhs eqy
error "Improper initial value"
findCompat(yx, l) ==
for eq in l repeat
yx =$F lhs eq => return rhs eq
error "Improper initial value"
findEq(k, x, sys) ==
k := retract(differentiate(k::F, x))@K
for eq in sys repeat
member?(k, kernels eq) => return eq
error "Improper differential equation"
checkOrder1(diffeq, y, yx, x, sy) ==
div2exquo subst(diffRhs(differentiate(yx::F,x),diffeq),[yx],[sy])
checkOrderN(diffeq, y, yx, x, sy, n) ==
zero? n => error "No initial value(s) given"
m := (minIndex(l := [retract(f := yx::F)@K]$List(K)))::F
lv := [opelt(sy, localInteger m)]$List(F)
for i in 2..n repeat
l := concat(retract(f := differentiate(f, x))@K, l)
lv := concat(opelt(sy, localInteger(m := m + 1)), lv)
div2exquo subst(diffRhs(differentiate(f, x), diffeq), l, lv)
checkSystem(diffeq, yx, lv) ==
for k in kernels diffeq repeat
is?(k, "%diff"::SY) =>
return div2exquo subst(diffRhsK(k, diffeq), yx, lv)
0
seriesSolve(l:List EQ, y:List OP, eqx:EQ, eqy:List EQ) ==
seriesSolve([lhs deq - rhs deq for deq in l]$List(F), y, eqx, eqy)
seriesSolve(l:List EQ, y:List OP, eqx:EQ, y0:List F) ==
seriesSolve([lhs deq - rhs deq for deq in l]$List(F), y, eqx, y0)
seriesSolve(l:List F, ly:List OP, eqx:EQ, eqy:List EQ) ==
seriesSolve(l, ly, eqx,
[findCompat(y rhs eqx, eqy) for y in ly]$List(F))
seriesSolve(diffeq:EQ, y:OP, eqx:EQ, eqy:EQ) ==
seriesSolve(lhs diffeq - rhs diffeq, y, eqx, eqy)
seriesSolve(diffeq:EQ, y:OP, eqx:EQ, y0:F) ==
seriesSolve(lhs diffeq - rhs diffeq, y, eqx, y0)
seriesSolve(diffeq:EQ, y:OP, eqx:EQ, y0:List F) ==
seriesSolve(lhs diffeq - rhs diffeq, y, eqx, y0)
seriesSolve(diffeq:F, y:OP, eqx:EQ, eqy:EQ) ==
seriesSolve(diffeq, y, eqx, checkCompat(y, eqx, eqy))
seriesSolve(diffeq:F, y:OP, eqx:EQ, y0:F) ==
x := symbolIfCan(retract(lhs eqx)@K)::SY
sy := name y
yx := retract(y lhs eqx)@K
f := checkOrder1(diffeq, y, yx, x, sy::F)
center := rhs eqx
coerce(ode1(compiledFunction(f, sy)$MKF, y0)$ODE)$A1
seriesSolve(diffeq:F, y:OP, eqx:EQ, y0:List F) ==
x := symbolIfCan(retract(lhs eqx)@K)::SY
sy := new()$SY
yx := retract(y lhs eqx)@K
f := checkOrderN(diffeq, y, yx, x, sy::F, #y0)
center := rhs eqx
coerce(ode(compiledFunction(f, sy)$MKL, y0)$ODE)$A1
seriesSolve(sys:List F, ly:List OP, eqx:EQ, l0:List F) ==
x := symbolIfCan(kx := retract(lhs eqx)@K)::SY
fsy := (sy := new()$SY)::F
m := (minIndex(l0) - 1)::F
yx := concat(kx, [retract(y lhs eqx)@K for y in ly]$List(K))
lelt := [opelt(fsy, localInteger(m := m+1)) for k in yx]$List(F)
sys := [findEq(k, x, sys) for k in rest yx]
l := [checkSystem(eq, yx, lelt) for eq in sys]$List(F)
center := rhs eqx
coerce(mpsode(l0,[compiledFunction(f,sy)$MKL for f in l])$ODE)$AL1
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