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--All rights reserved.
--
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--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.
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-- 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
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--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
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--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
-- Compile order for the differential equation solver:
-- oderf.spad odealg.spad nlode.spad nlinsol.spad riccati.spad odeef.spad
)abbrev package ODESYS SystemODESolver
++ Author: Manuel Bronstein
++ Date Created: 11 June 1991
++ Date Last Updated: 13 April 1994
++ Description: SystemODESolver provides tools for triangulating
++ and solving some systems of linear ordinary differential equations.
++ Keywords: differential equation, ODE, system
SystemODESolver(F, LO): Exports == Implementation where
F : Field
LO: LinearOrdinaryDifferentialOperatorCategory F
N ==> NonNegativeInteger
Z ==> Integer
MF ==> Matrix F
M ==> Matrix LO
V ==> Vector F
UF ==> Union(F, "failed")
UV ==> Union(V, "failed")
REC ==> Record(mat: M, vec: V)
FSL ==> Record(particular: UF, basis: List F)
VSL ==> Record(particular: UV, basis: List V)
SOL ==> Record(particular: F, basis: List F)
USL ==> Union(SOL, "failed")
ER ==> Record(C: MF, g: V, eq: LO, rh: F)
Exports ==> with
triangulate: (MF, V) -> Record(A:MF, eqs: List ER)
++ triangulate(M,v) returns
++ \spad{A,[[C_1,g_1,L_1,h_1],...,[C_k,g_k,L_k,h_k]]}
++ such that under the change of variable \spad{y = A z}, the first
++ order linear system \spad{D y = M y + v} is uncoupled as
++ \spad{D z_i = C_i z_i + g_i} and each \spad{C_i} is a companion
++ matrix corresponding to the scalar equation \spad{L_i z_j = h_i}.
triangulate: (M, V) -> REC
++ triangulate(m, v) returns \spad{[m_0, v_0]} such that \spad{m_0}
++ is upper triangular and the system \spad{m_0 x = v_0} is equivalent
++ to \spad{m x = v}.
solve: (MF,V,(LO,F)->USL) -> Union(Record(particular:V, basis:MF), "failed")
++ solve(m, v, solve) returns \spad{[[v_1,...,v_m], v_p]} such that
++ the solutions in \spad{F} of the system \spad{D x = m x + v} are
++ \spad{v_p + c_1 v_1 + ... + c_m v_m} where the \spad{c_i's} are
++ constants, and the \spad{v_i's} form a basis for the solutions of
++ \spad{D x = m x}.
++ Argument \spad{solve} is a function for solving a single linear
++ ordinary differential equation in \spad{F}.
solveInField: (M, V, (LO, F) -> FSL) -> VSL
++ solveInField(m, v, solve) returns \spad{[[v_1,...,v_m], v_p]} such that
++ the solutions in \spad{F} of the system \spad{m x = v} are
++ \spad{v_p + c_1 v_1 + ... + c_m v_m} where the \spad{c_i's} are
++ constants, and the \spad{v_i's} form a basis for the solutions of
++ \spad{m x = 0}.
++ Argument \spad{solve} is a function for solving a single linear
++ ordinary differential equation in \spad{F}.
Implementation ==> add
import PseudoLinearNormalForm F
applyLodo : (M, Z, V, N) -> F
applyLodo0 : (M, Z, Matrix F, Z, N) -> F
backsolve : (M, V, (LO, F) -> FSL) -> VSL
firstnonzero: (M, Z) -> Z
FSL2USL : FSL -> USL
M2F : M -> Union(MF, "failed")
diff := D()$LO
solve(mm, v, solve) ==
rec := triangulate(mm, v)
sols:List(SOL) := empty()
for e in rec.eqs repeat
(u := solve(e.eq, e.rh)) case "failed" => return "failed"
sols := concat(u::SOL, sols)
n := nrows(rec.A) -- dimension of original vectorspace
k:N := 0 -- sum of sizes of visited companionblocks
i:N := 0 -- number of companionblocks
m:N := 0 -- number of Solutions
part:V := new(n, 0)
-- count first the different solutions
for sol in sols repeat m := m + count(#1 ~= 0, sol.basis)$List(F)
SolMatrix:MF := new(n, m, 0)
m := 0
for sol in reverse! sols repeat
i := i+1
er := rec.eqs.i
nn := #(er.g) -- size of active companionblock
for s in sol.basis repeat
solVec:V := new(n, 0)
-- compute corresponding solution base with recursion (24)
solVec(k+1) := s
for l in 2..nn repeat solVec(k+l) := diff solVec(k+l-1)
m := m+1
setColumn!(SolMatrix, m, solVec)
-- compute with (24) the corresponding components of the part. sol.
part(k+1) := sol.particular
for l in 2..nn repeat part(k+l) := diff part(k+l-1) - (er.g)(l-1)
k := k+nn
-- transform these values back to the original system
[rec.A * part, rec.A * SolMatrix]
triangulate(m:MF, v:V) ==
k:N := 0 -- sum of companion-dimensions
rat := normalForm(m, 1, - diff #1)
l := companionBlocks(rat.R, rat.Ainv * v)
ler:List(ER) := empty()
for er in l repeat
n := nrows(er.C) -- dimension of this companion vectorspace
op:LO := 0 -- compute homogeneous equation
for j in 0..n-1 repeat op := op + monomial((er.C)(n, j + 1), j)
op := monomial(1, n) - op
sum:V := new(n::N, 0) -- compute inhomogen Vector (25)
for j in 1..n-1 repeat sum(j+1) := diff(sum j) + (er.g) j
h0:F := 0 -- compute inhomogenity (26)
for j in 1..n repeat h0 := h0 - (er.C)(n, j) * sum j
h0 := h0 + diff(sum n) + (er.g) n
ler := concat([er.C, er.g, op, h0], ler)
k := k + n
[rat.A, ler]
-- like solveInField, but expects a system already triangularized
backsolve(m, v, solve) ==
part:V
r := maxRowIndex m
offset := minIndex v - (mr := minRowIndex m)
while r >= mr and every?(zero?, row(m, r))$Vector(LO) repeat r := r - 1
r < mr => error "backsolve: system has a 0 matrix"
(c := firstnonzero(m, r)) ~= maxColIndex m =>
error "backsolve: undetermined system"
rec := solve(m(r, c), v(r + offset))
dim := (r - mr + 1)::N
if (part? := ((u := rec.particular) case F)) then
part := new(dim, 0) -- particular solution
part(r + offset) := u::F
-- hom is the basis for the homogeneous solutions, each column is a solution
hom:Matrix(F) := new(dim, #(rec.basis), 0)
for i in minColIndex hom .. maxColIndex hom for b in rec.basis repeat
hom(r, i) := b
n:N := 1 -- number of equations already solved
while r > mr repeat
r := r - 1
c := c - 1
firstnonzero(m, r) ~= c => error "backsolve: undetermined system"
positive? degree(eq := m(r, c)) => error "backsolve: pivot of order > 0"
a := leadingCoefficient(eq)::F
if part? then
part(r + offset) := (v(r + offset) - applyLodo(m, r, part, n)) / a
for i in minColIndex hom .. maxColIndex hom repeat
hom(r, i) := - applyLodo0(m, r, hom, i, n)
n := n + 1
bas:List(V) := [column(hom,i) for i in minColIndex hom..maxColIndex hom]
part? => [part, bas]
["failed", bas]
solveInField(m, v, solve) ==
((n := nrows m) = ncols m) and
((u := M2F(diagonalMatrix [diff for i in 1..n] - m)) case MF) =>
(uu := solve(u::MF, v, FSL2USL solve(#1, #2))) case "failed" =>
["failed", empty()]
rc := uu::Record(particular:V, basis:MF)
[rc.particular, [column(rc.basis, i) for i in 1..ncols(rc.basis)]]
rec := triangulate(m, v)
backsolve(rec.mat, rec.vec, solve)
M2F m ==
mf:MF := new(nrows m, ncols m, 0)
for i in minRowIndex m .. maxRowIndex m repeat
for j in minColIndex m .. maxColIndex m repeat
(u := retractIfCan(m(i, j))@Union(F, "failed")) case "failed" =>
return "failed"
mf(i, j) := u::F
mf
FSL2USL rec ==
rec.particular case "failed" => "failed"
[rec.particular::F, rec.basis]
-- returns the index of the first nonzero entry in row r of m
firstnonzero(m, r) ==
for c in minColIndex m .. maxColIndex m repeat
m(r, c) ~= 0 => return c
error "firstnonzero: zero row"
-- computes +/[m(r, i) v(i) for i ranging over the last n columns of m]
applyLodo(m, r, v, n) ==
ans:F := 0
c := maxColIndex m
cv := maxIndex v
for i in 1..n repeat
ans := ans + m(r, c) (v cv)
c := c - 1
cv := cv - 1
ans
-- computes +/[m(r, i) mm(i, c) for i ranging over the last n columns of m]
applyLodo0(m, r, mm, c, n) ==
ans: F := 0
rr := maxRowIndex mm
cc := maxColIndex m
for i in 1..n repeat
ans := ans + m(r, cc) mm(rr, c)
cc := cc - 1
rr := rr - 1
ans
triangulate(m:M, v:V) ==
x := copy m
w := copy v
nrows := maxRowIndex x
ncols := maxColIndex x
minr := i := minRowIndex x
offset := minIndex w - minr
for j in minColIndex x .. ncols repeat
if i > nrows then leave x
rown := minr - 1
for k in i .. nrows repeat
if (x(k, j) ~= 0) and ((rown = minr - 1) or
degree x(k,j) < degree x(rown,j)) then rown := k
rown = minr - 1 => "enuf"
x := swapRows!(x, i, rown)
swap!(w, i + offset, rown + offset)
for k in i+1 .. nrows | x(k, j) ~= 0 repeat
l := rightLcm(x(i,j), x(k,j))
a := rightQuotient(l, x(i, j))
b := rightQuotient(l, x(k, j))
-- l = a x(i,j) = b x(k,j)
for k1 in j+1 .. ncols repeat
x(k, k1) := a * x(i, k1) - b * x(k, k1)
x(k, j) := 0
w(k + offset) := a(w(i + offset)) - b(w(k + offset))
i := i+1
[x, w]
)abbrev package ODERED ReduceLODE
++ Author: Manuel Bronstein
++ Date Created: 19 August 1991
++ Date Last Updated: 11 April 1994
++ Description: Elimination of an algebraic from the coefficentss
++ of a linear ordinary differential equation.
ReduceLODE(F, L, UP, A, LO): Exports == Implementation where
F : Field
L : LinearOrdinaryDifferentialOperatorCategory F
UP: UnivariatePolynomialCategory F
A : MonogenicAlgebra(F, UP)
LO: LinearOrdinaryDifferentialOperatorCategory A
V ==> Vector F
M ==> Matrix L
Exports ==> with
reduceLODE: (LO, A) -> Record(mat:M, vec:V)
++ reduceLODE(op, g) returns \spad{[m, v]} such that
++ any solution in \spad{A} of \spad{op z = g}
++ is of the form \spad{z = (z_1,...,z_m) . (b_1,...,b_m)} where
++ the \spad{b_i's} are the basis of \spad{A} over \spad{F} returned
++ by \spadfun{basis}() from \spad{A}, and the \spad{z_i's} satisfy the
++ differential system \spad{M.z = v}.
Implementation ==> add
matF2L: Matrix F -> M
diff := D()$L
-- coerces a matrix of elements of F into a matrix of (order 0) L.O.D.O's
matF2L m ==
map(#1::L, m)$MatrixCategoryFunctions2(F, V, V, Matrix F,
L, Vector L, Vector L, M)
-- This follows the algorithm and notation of
-- "The Risch Differential Equation on an Algebraic Curve", M. Bronstein,
-- in 'Proceedings of ISSAC '91', Bonn, BRD, ACM Press, pp.241-246, July 1991.
reduceLODE(l, g) ==
n := rank()$A
-- md is the basic differential matrix (D x I + Dy)
md := matF2L transpose derivationCoordinates(basis(), diff #1)
for i in minRowIndex md .. maxRowIndex md
for j in minColIndex md .. maxColIndex md repeat
md(i, j) := diff + md(i, j)
-- mdi will go through the successive powers of md
mdi := copy md
sys := matF2L(transpose regularRepresentation coefficient(l, 0))
for i in 1..degree l repeat
sys := sys +
matF2L(transpose regularRepresentation coefficient(l, i)) * mdi
mdi := md * mdi
[sys, coordinates g]
)abbrev package ODEPAL PureAlgebraicLODE
++ Author: Manuel Bronstein
++ Date Created: 21 August 1991
++ Date Last Updated: 3 February 1994
++ Description: In-field solution of an linear ordinary differential equation,
++ pure algebraic case.
PureAlgebraicLODE(F, UP, UPUP, R): Exports == Implementation where
F : Join(Field, CharacteristicZero,
RetractableTo Integer, RetractableTo Fraction Integer)
UP : UnivariatePolynomialCategory F
UPUP: UnivariatePolynomialCategory Fraction UP
R : FunctionFieldCategory(F, UP, UPUP)
RF ==> Fraction UP
V ==> Vector RF
U ==> Union(R, "failed")
REC ==> Record(particular: Union(RF, "failed"), basis: List RF)
L ==> LinearOrdinaryDifferentialOperator1 R
LQ ==> LinearOrdinaryDifferentialOperator1 RF
Exports ==> with
algDsolve: (L, R) -> Record(particular: U, basis: List R)
++ algDsolve(op, g) returns \spad{["failed", []]} if the equation
++ \spad{op y = g} has no solution in \spad{R}. Otherwise, it returns
++ \spad{[f, [y1,...,ym]]} where \spad{f} is a particular rational
++ solution and the \spad{y_i's} form a basis for the solutions in
++ \spad{R} of the homogeneous equation.
Implementation ==> add
import RationalLODE(F, UP)
import SystemODESolver(RF, LQ)
import ReduceLODE(RF, LQ, UPUP, R, L)
algDsolve(l, g) ==
rec := reduceLODE(l, g)
sol := solveInField(rec.mat, rec.vec, ratDsolve)
bas:List(R) := [represents v for v in sol.basis]
(u := sol.particular) case V => [represents(u::V), bas]
["failed", bas]
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