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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.
--
-- - 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,
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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
-- kovacic.spad lodof.spad odeef.spad
)abbrev domain SETMN SetOfMIntegersInOneToN
++ Author: Manuel Bronstein
++ Date Created: 10 January 1994
++ Date Last Updated: 10 January 1994
++ Description:
++ \spadtype{SetOfMIntegersInOneToN} implements the subsets of M integers
++ in the interval \spad{[1..n]}
SetOfMIntegersInOneToN(m, n): Exports == Implementation where
PI ==> PositiveInteger
N ==> NonNegativeInteger
U ==> Union(%, "failed")
n,m: PI
Exports ==> Finite with
incrementKthElement: (%, PI) -> U
++ incrementKthElement(S,k) increments the k^{th} element of S,
++ and returns "failed" if the result is not a set of M integers
++ in \spad{1..n} any more.
replaceKthElement: (%, PI, PI) -> U
++ replaceKthElement(S,k,p) replaces the k^{th} element of S by p,
++ and returns "failed" if the result is not a set of M integers
++ in \spad{1..n} any more.
elements: % -> List PI
++ elements(S) returns the list of the elements of S in increasing order.
setOfMinN: List PI -> %
++ setOfMinN([a_1,...,a_m]) returns the set {a_1,...,a_m}.
++ Error if {a_1,...,a_m} is not a set of M integers in \spad{1..n}.
enumerate: () -> Vector %
++ enumerate() returns a vector of all the sets of M integers in
++ \spad{1..n}.
member?: (PI, %) -> Boolean
++ member?(p, s) returns true is p is in s, false otherwise.
delta: (%, PI, PI) -> N
++ delta(S,k,p) returns the number of elements of S which are strictly
++ between p and the k^{th} element of S.
Implementation ==> add
Rep := Record(bits:Bits, pos:N)
reallyEnumerate: () -> Vector %
enum: (N, N, PI) -> List Bits
all:Reference Vector % := ref empty()
sz:Reference N := ref 0
s1 = s2 == s1.bits =$Bits s2.bits
coerce(s:%):OutputForm == brace [i::OutputForm for i in elements s]
random() == index((1 + (random()$Integer rem size()))::PI)
reallyEnumerate() == [[b, i] for b in enum(m, n, n) for i in 1..]
member?(p, s) == s.bits.p
enumerate() ==
if empty? deref all then setref(all,reallyEnumerate())
deref all
-- enumerates the sets of p integers in 1..q, returns them as sets in 1..n
-- must have p <= q
enum(p, q, n) ==
zero? p or zero? q => empty()
p = q =>
b := new(n, false)$Bits
for i in 1..p repeat b.i := true
[b]
q1 := (q - 1)::N
l := enum((p - 1)::N, q1, n)
if empty? l then l := [new(n, false)$Bits]
for s in l repeat s.q := true
concat!(enum(p, q1, n), l)
size() ==
if zero? deref sz then
setref(sz,binomial(n, m)$IntegerCombinatoricFunctions(Integer) :: N)
deref sz
lookup s ==
if empty? deref all then setref(all,reallyEnumerate())
if zero?(s.pos) then s.pos := position(s, deref all) :: N
s.pos :: PI
index p ==
p > size() => error "index: argument too large"
if empty? deref all then setref(all,reallyEnumerate())
deref(all).p
setOfMinN l ==
s := new(n, false)$Bits
count:N := 0
for i in l repeat
count := count + 1
count > m or zero? i or i > n or s.i =>
error "setOfMinN: improper set of integers"
s.i := true
count < m => error "setOfMinN: improper set of integers"
[s, 0]
elements s ==
b := s.bits
l:List PI := empty()
found:N := 0
i:PI := 1
while found < m repeat
if b.i then
l := concat(i, l)
found := found + 1
i := i + 1
reverse! l
incrementKthElement(s, k) ==
b := s.bits
found:N := 0
i:N := 1
while found < k repeat
if b.i then found := found + 1
i := i + 1
i > n or b.i => "failed"
newb := copy b
newb.i := true
newb.((i-1)::N) := false
[newb, 0]
delta(s, k, p) ==
b := s.bits
count:N := found:N := 0
i:PI := 1
while found < k repeat
if b.i then
found := found + 1
if i > p and found < k then count := count + 1
i := i + 1
count
replaceKthElement(s, k, p) ==
b := s.bits
found:N := 0
i:PI := 1
while found < k repeat
if b.i then found := found + 1
if found < k then i := i + 1
b.p and i ~= p => "failed"
newb := copy b
newb.p := true
newb.i := false
[newb, (i = p => s.pos; 0)]
)abbrev package PREASSOC PrecomputedAssociatedEquations
++ Author: Manuel Bronstein
++ Date Created: 13 January 1994
++ Date Last Updated: 3 February 1994
++ Description:
++ \spadtype{PrecomputedAssociatedEquations} stores some generic
++ precomputations which speed up the computations of the
++ associated equations needed for factoring operators.
PrecomputedAssociatedEquations(R, L): Exports == Implementation where
R: IntegralDomain
L: LinearOrdinaryDifferentialOperatorCategory R
PI ==> PositiveInteger
N ==> NonNegativeInteger
A ==> PrimitiveArray R
U ==> Union(Matrix R, "failed")
Exports ==> with
firstUncouplingMatrix: (L, PI) -> U
++ firstUncouplingMatrix(op, m) returns the matrix A such that
++ \spad{A w = (W',W'',...,W^N)} in the corresponding associated
++ equations for right-factors of order m of op.
++ Returns "failed" if the matrix A has not been precomputed for
++ the particular combination \spad{degree(L), m}.
Implementation ==> add
A32: L -> U
A42: L -> U
A425: (A, A, A) -> List R
A426: (A, A, A) -> List R
makeMonic: L -> Union(A, "failed")
diff:L := D()
firstUncouplingMatrix(op, m) ==
n := degree op
n = 3 and m = 2 => A32 op
n = 4 and m = 2 => A42 op
"failed"
makeMonic op ==
lc := leadingCoefficient op
a:A := new(n := degree op, 0)
for i in 0..(n-1)::N repeat
(u := coefficient(op, i) exquo lc) case "failed" => return "failed"
a.i := - (u::R)
a
A32 op ==
(u := makeMonic op) case "failed" => "failed"
a := u::A
matrix [[0, 1, 0], [a.1, a.2, 1],
[diff(a.1) + a.1 * a.2 - a.0, diff(a.2) + a.2**2 + a.1, 2 * a.2]]
A42 op ==
(u := makeMonic op) case "failed" => "failed"
a := u::A
a':A := new(4, 0)
a'':A := new(4, 0)
for i in 0..3 repeat
a'.i := diff(a.i)
a''.i := diff(a'.i)
matrix [[0, 1, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0], [a.1,a.2,0,a.3,2::R,0],
[a'.1 + a.1 * a.3 - 2 * a.0, a'.2 + a.2 * a.3 + a.1, 3 * a.2,
a'.3 + a.3 ** 2 + a.2, 3 * a.3, 2::R],
A425(a, a', a''), A426(a, a', a'')]
A425(a, a', a'') ==
[a''.1 + 2 * a.1 * a'.3 + a.3 * a'.1 - 2 * a'.0 + a.1 * a.3 ** 2
- 3 * a.0 * a.3 + a.1 * a.2,
a''.2 + 2 * a.2 * a'.3 + a.3 * a'.2 + 2 * a'.1 + a.2 * a.3 ** 2
+ a.1 * a.3 + a.2 ** 2 - 4 * a.0,
4 * a'.2 + 4 * a.2 * a.3 - a.1,
a''.3 + 3 * a.3 * a'.3 + 2 * a'.2 + a.3 ** 3 + 2 * a.2 * a.3 + a.1,
4 * a'.3 + 4 * a.3 ** 2 + 4 * a.2, 5 * a.3]
A426(a, a', a'') ==
[diff(a''.1) + 3 * a.1 * a''.3 + a.3 * a''.1 - 2 * a''.0
+ (3 * a'.1 + 5 * a.1 * a.3 - 7 * a.0) * a'.3 + 3 * a.1 * a'.2
+ (a.3 ** 2 + a.2) * a'.1 - 3 * a.3 * a'.0 + a.1 * a.3 ** 3
- 4 * a.0 * a.3 ** 2 + 2 * a.1 * a.2 * a.3 - 4 * a.0 * a.2 + a.1 ** 2,
diff(a''.2) + 3 * a.2 * a''.3 + a.3 * a''.2 + 3 * a''.1
+ (3*a'.2 + 5*a.2 * a.3 + 3 * a.1) * a'.3 + (a.3**2 + 4*a.2)*a'.2
+ 2 * a.3 * a'.1 - 6 * a'.0 + a.2 * a.3 ** 3 + a.1 * a.3 ** 2
+ (2 * a.2**2 - 8 * a.0) * a.3 + 2 * a.1 * a.2,
5 * a''.2 + 10 * a.2 * a'.3 + 5 * a.3 * a'.2 + a'.1
+ 5 * a.2 * a.3 ** 2 - 4 * a.1 * a.3 + 5 * a.2**2 - 4 * a.0,
diff(a''.3) + 4 * a.3 * a''.3 + 3*a''.2 + 3 * a'.3**2
+ (6 * a.3**2 + 4 * a.2) * a'.3 + 5 * a.3 * a'.2 + 3 * a'.1
+ a.3**4 + 3 * a.2 * a.3**2 + 2 * a.1 * a.3 + a.2**2 - 4*a.0,
5 * a''.3 + 15 * a.3 * a'.3 + 10 * a'.2 + 5 * a.3**3
+ 10 * a.2 * a.3, 9 * a'.3 + 9 * a.3**2 + 4 * a.2]
)abbrev package ASSOCEQ AssociatedEquations
++ Author: Manuel Bronstein
++ Date Created: 10 January 1994
++ Date Last Updated: 3 February 1994
++ Description:
++ \spadtype{AssociatedEquations} provides functions to compute the
++ associated equations needed for factoring operators
AssociatedEquations(R, L):Exports == Implementation where
R: IntegralDomain
L: LinearOrdinaryDifferentialOperatorCategory R
PI ==> PositiveInteger
N ==> NonNegativeInteger
MAT ==> Matrix R
REC ==> Record(minor: List PI, eq: L, minors: List List PI, ops: List L)
Exports ==> with
associatedSystem: (L, PI) -> Record(mat: MAT, vec:Vector List PI)
++ associatedSystem(op, m) returns \spad{[M,w]} such that the
++ m-th associated equation system to L is \spad{w' = M w}.
uncouplingMatrices: MAT -> Vector MAT
++ uncouplingMatrices(M) returns \spad{[A_1,...,A_n]} such that if
++ \spad{y = [y_1,...,y_n]} is a solution of \spad{y' = M y}, then
++ \spad{[$y_j',y_j'',...,y_j^{(n)}$] = $A_j y$} for all j's.
if R has Field then
associatedEquations: (L, PI) -> REC
++ associatedEquations(op, m) returns \spad{[w, eq, lw, lop]}
++ such that \spad{eq(w) = 0} where w is the given minor, and
++ \spad{lw_i = lop_i(w)} for all the other minors.
Implementation ==> add
makeMatrix: (Vector MAT, N) -> MAT
diff:L := D()
makeMatrix(v, n) == matrix [parts row(v.i, n) for i in 1..#v]
associatedSystem(op, m) ==
eq: Vector R
S := SetOfMIntegersInOneToN(m, n := degree(op)::PI)
w := enumerate()$S
s := size()$S
ww:Vector List PI := new(s, empty())
M:MAT := new(s, s, 0)
m1 := (m::Integer - 1)::PI
an := leadingCoefficient op
a:Vector(R) := [- (coefficient(op, j) exquo an)::R for j in 0..n - 1]
for i in 1..s repeat
eq := new(s, 0)
wi := w.i
ww.i := elements wi
for k in 1..m1 repeat
u := incrementKthElement(wi, k::PI)$S
if u case S then eq(lookup(u::S)) := 1
if member?(n, wi) then
for j in 1..n | a.j ~= 0 repeat
u := replaceKthElement(wi, m, j::PI)
if u case S then
eq(lookup(u::S)) := (odd? delta(wi, m, j::PI) => -a.j; a.j)
else
u := incrementKthElement(wi, m)$S
if u case S then eq(lookup(u::S)) := 1
setRow!(M, i, eq)
[M, ww]
uncouplingMatrices m ==
n := nrows m
v:Vector MAT := new(n, zero(1, 0)$MAT)
v.1 := mi := m
for i in 2..n repeat v.i := mi := map(diff #1, mi) + mi * m
[makeMatrix(v, i) for i in 1..n]
if R has Field then
import PrecomputedAssociatedEquations(R, L)
makeop: Vector R -> L
makeeq: (Vector List PI, MAT, N, N) -> REC
computeIt: (L, PI, N) -> REC
makeeq(v, m, i, n) ==
[v.i, makeop row(m, i) - 1, [v.j for j in 1..n | j ~= i],
[makeop row(m, j) for j in 1..n | j ~= i]]
associatedEquations(op, m) ==
(u := firstUncouplingMatrix(op, m)) case "failed" => computeIt(op,m,1)
(v := inverse(u::MAT)) case "failed" => computeIt(op, m, 2)
S := SetOfMIntegersInOneToN(m, degree(op)::PI)
w := enumerate()$S
s := size()$S
ww:Vector List PI := new(s, empty())
for i in 1..s repeat ww.i := elements(w.i)
makeeq(ww, v::MAT, 1, s)
computeIt(op, m, k) ==
rec := associatedSystem(op, m)
a := uncouplingMatrices(rec.mat)
n := #a
for i in k..n repeat
(u := inverse(a.i)) case MAT => return makeeq(rec.vec,u::MAT,i,n)
error "associatedEquations: full degenerate case"
makeop v ==
op:L := 0
for i in 1..#v repeat op := op + monomial(v i, i)
op
)abbrev package LODOF LinearOrdinaryDifferentialOperatorFactorizer
++ Author: Fritz Schwarz, Manuel Bronstein
++ Date Created: 1988
++ Date Last Updated: 3 February 1994
++ Description:
++ \spadtype{LinearOrdinaryDifferentialOperatorFactorizer} provides a
++ factorizer for linear ordinary differential operators whose coefficients
++ are rational functions.
++ Keywords: differential equation, ODE, LODO, factoring
LinearOrdinaryDifferentialOperatorFactorizer(F, UP): Exports == Impl where
F : Join(Field, CharacteristicZero,
RetractableTo Integer, RetractableTo Fraction Integer)
UP: UnivariatePolynomialCategory F
RF ==> Fraction UP
L ==> LinearOrdinaryDifferentialOperator1 RF
Exports ==> with
factor: (L, UP -> List F) -> List L
++ factor(a, zeros) returns the factorisation of a.
++ \spad{zeros} is a zero finder in \spad{UP}.
if F has AlgebraicallyClosedField then
factor: L -> List L
++ factor(a) returns the factorisation of a.
factor1: L -> List L
++ factor1(a) returns the factorisation of a,
++ assuming that a has no first-order right factor.
Impl ==> add
import RationalLODE(F, UP)
import RationalRicDE(F, UP)
-- import AssociatedEquations RF
dd := D()$L
expsol : (L, UP -> List F, UP -> Factored UP) -> Union(RF, "failed")
expsols : (L, UP -> List F, UP -> Factored UP, Boolean) -> List RF
opeval : (L, L) -> L
recurfactor: (L, L, UP -> List F, UP -> Factored UP, Boolean) -> List L
rfactor : (L, L, UP -> List F, UP -> Factored UP, Boolean) -> List L
rightFactor: (L, NonNegativeInteger, UP -> List F, UP -> Factored UP)
-> Union(L, "failed")
innerFactor: (L, UP -> List F, UP -> Factored UP, Boolean) -> List L
factor(l, zeros) == innerFactor(l, zeros, squareFree, true)
expsol(l, zeros, ezfactor) ==
empty?(sol := expsols(l, zeros, ezfactor, false)) => "failed"
first sol
expsols(l, zeros, ezfactor, all?) ==
sol := [differentiate(f)/f for f in ratDsolve(l, 0).basis | f ~= 0]
not(all? or empty? sol) => sol
concat(sol, ricDsolve(l, zeros, ezfactor))
-- opeval(l1, l2) returns l1(l2)
opeval(l1, l2) ==
ans:L := 0
l2n:L := 1
for i in 0..degree l1 repeat
ans := ans + coefficient(l1, i) * l2n
l2n := l2 * l2n
ans
recurfactor(l, r, zeros, ezfactor, adj?) ==
q := rightExactQuotient(l, r)::L
if adj? then q := adjoint q
innerFactor(q, zeros, ezfactor, true)
rfactor(op, r, zeros, ezfactor, adj?) ==
degree r > 1 or not one? leadingCoefficient r =>
recurfactor(op, r, zeros, ezfactor, adj?)
op1 := opeval(op, dd - coefficient(r, 0)::L)
map!(opeval(#1, r), recurfactor(op1, dd, zeros, ezfactor, adj?))
-- r1? is true means look for 1st-order right-factor also
innerFactor(l, zeros, ezfactor, r1?) ==
(n := degree l) <= 1 => [l]
ll := adjoint l
for i in 1..(n quo 2) repeat
(r1? or (i > 1)) and ((u := rightFactor(l,i,zeros,ezfactor)) case L) =>
return concat!(rfactor(l, u::L, zeros, ezfactor, false), u::L)
(2 * i < n) and ((u := rightFactor(ll, i, zeros, ezfactor)) case L) =>
return concat(adjoint(u::L), rfactor(ll, u::L, zeros,ezfactor,true))
[l]
rightFactor(l, n, zeros, ezfactor) ==
one? n =>
(u := expsol(l, zeros, ezfactor)) case "failed" => "failed"
D() - u::RF::L
-- rec := associatedEquations(l, n::PositiveInteger)
-- empty?(sol := expsols(rec.eq, zeros, ezfactor, true)) => "failed"
"failed"
if F has AlgebraicallyClosedField then
zro1: UP -> List F
zro : (UP, UP -> Factored UP) -> List F
zro(p, ezfactor) ==
concat [zro1(r.factor) for r in factors ezfactor p]
zro1 p ==
[zeroOf(map(#1, p)$UnivariatePolynomialCategoryFunctions2(F, UP,
F, SparseUnivariatePolynomial F))]
if F is AlgebraicNumber then
import AlgFactor UP
factor l == innerFactor(l, zro(#1, factor), factor, true)
factor1 l == innerFactor(l, zro(#1, factor), factor, false)
else
factor l == innerFactor(l, zro(#1, squareFree), squareFree, true)
factor1 l == innerFactor(l, zro(#1, squareFree), squareFree, false)
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