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diff --git a/src/algebra/lodo.spad.pamphlet b/src/algebra/lodo.spad.pamphlet new file mode 100644 index 00000000..c25901eb --- /dev/null +++ b/src/algebra/lodo.spad.pamphlet @@ -0,0 +1,293 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/algebra lodo.spad} +\author{Manuel Bronstein, Stephen M. Watt} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\section{category LODOCAT LinearOrdinaryDifferentialOperatorCategory} +<<category LODOCAT LinearOrdinaryDifferentialOperatorCategory>>= +)abbrev category LODOCAT LinearOrdinaryDifferentialOperatorCategory +++ Author: Manuel Bronstein +++ Date Created: 9 December 1993 +++ Date Last Updated: 15 April 1994 +++ Keywords: differential operator +++ Description: +++ \spad{LinearOrdinaryDifferentialOperatorCategory} is the category +++ of differential operators with coefficients in a ring A with a given +++ derivation. +++ Multiplication of operators corresponds to functional composition: +++ \spad{(L1 * L2).(f) = L1 L2 f} +LinearOrdinaryDifferentialOperatorCategory(A:Ring): Category == + Join(UnivariateSkewPolynomialCategory A, Eltable(A, A)) with + D: () -> % + ++ D() provides the operator corresponding to a derivation + ++ in the ring \spad{A}. + adjoint: % -> % + ++ adjoint(a) returns the adjoint operator of a. + if A has Field then + symmetricProduct: (%, %) -> % + ++ symmetricProduct(a,b) computes an operator \spad{c} of + ++ minimal order such that the nullspace of \spad{c} is + ++ generated by all the products of a solution of \spad{a} by + ++ a solution of \spad{b}. + symmetricPower : (%, NonNegativeInteger) -> % + ++ symmetricPower(a,n) computes an operator \spad{c} of + ++ minimal order such that the nullspace of \spad{c} is + ++ generated by all the products of \spad{n} solutions + ++ of \spad{a}. + symmetricSquare : % -> % + ++ symmetricSquare(a) computes \spad{symmetricProduct(a,a)} + ++ using a more efficient method. + directSum: (%, %) -> % + ++ directSum(a,b) computes an operator \spad{c} of + ++ minimal order such that the nullspace of \spad{c} is + ++ generated by all the sums of a solution of \spad{a} by + ++ a solution of \spad{b}. + add + m1monom: NonNegativeInteger -> % + + D() == monomial(1, 1) + + m1monom n == + a:A := (odd? n => -1; 1) + monomial(a, n) + + adjoint a == + ans:% := 0 + while a ^= 0 repeat + ans := ans + m1monom(degree a) * leadingCoefficient(a)::% + a := reductum a + ans + + if A has Field then symmetricSquare l == symmetricPower(l, 2) + +@ +\section{package LODOOPS LinearOrdinaryDifferentialOperatorsOps} +<<package LODOOPS LinearOrdinaryDifferentialOperatorsOps>>= +)abbrev package LODOOPS LinearOrdinaryDifferentialOperatorsOps +++ Author: Manuel Bronstein +++ Date Created: 18 January 1994 +++ Date Last Updated: 15 April 1994 +++ Description: +++ \spad{LinearOrdinaryDifferentialOperatorsOps} provides symmetric +++ products and sums for linear ordinary differential operators. +-- Putting those operations here rather than defaults in LODOCAT allows +-- LODOCAT to be defined independently of the derivative used. +-- MB 1/94 +LinearOrdinaryDifferentialOperatorsOps(A, L): Exports == Implementation where + A: Field + L: LinearOrdinaryDifferentialOperatorCategory A + + N ==> NonNegativeInteger + V ==> OrderlyDifferentialVariable Symbol + P ==> DifferentialSparseMultivariatePolynomial(A, Symbol, V) + + Exports ==> with + symmetricProduct: (L, L, A -> A) -> L + ++ symmetricProduct(a,b,D) computes an operator \spad{c} of + ++ minimal order such that the nullspace of \spad{c} is + ++ generated by all the products of a solution of \spad{a} by + ++ a solution of \spad{b}. + ++ D is the derivation to use. + symmetricPower: (L, N, A -> A) -> L + ++ symmetricPower(a,n,D) computes an operator \spad{c} of + ++ minimal order such that the nullspace of \spad{c} is + ++ generated by all the products of \spad{n} solutions + ++ of \spad{a}. + ++ D is the derivation to use. + directSum: (L, L, A -> A) -> L + ++ directSum(a,b,D) computes an operator \spad{c} of + ++ minimal order such that the nullspace of \spad{c} is + ++ generated by all the sums of a solution of \spad{a} by + ++ a solution of \spad{b}. + ++ D is the derivation to use. + + Implementation ==> add + import IntegerCombinatoricFunctions + + var1 := new()$Symbol + var2 := new()$Symbol + + nonTrivial?: Vector A -> Boolean + applyLODO : (L, V) -> P + killer : (P, N, List V, List P, A -> A) -> L + vec2LODO : Vector A -> L + + nonTrivial? v == any?(#1 ^= 0, v)$Vector(A) + vec2LODO v == +/[monomial(v.i, (i-1)::N) for i in 1..#v] + + symmetricPower(l, m, diff) == + u := var1::V; n := degree l + un := differentiate(u, n) + a := applyLODO(inv(- leadingCoefficient l) * reductum l, u) + killer(u::P ** m, binomial(n + m - 1, n - 1)::N, [un], [a], diff) + +-- returns an operator L such that L(u) = 0, for a given differential +-- polynomial u, given that the differential variables appearing in u +-- satisfy some linear ode's +-- m is a bound on the order of the operator searched. +-- lvar, lval describe the substitution(s) to perform when differentiating +-- the expression u (they encode the fact the the differential variables +-- satisfy some differential equations, which can be seen as the rewrite +-- rules lvar --> lval) +-- diff is the derivation to use + killer(u, m, lvar, lval, diff) == + lu:List P := [u] + for q in 0..m repeat + mat := reducedSystem(matrix([lu])@Matrix(P))@Matrix(A) + (sol := find(nonTrivial?, l := nullSpace mat)) case Vector(A) => + return vec2LODO(sol::Vector(A)) + u := eval(differentiate(u, diff), lvar, lval) + lu := concat_!(lu, [u]) + error "killer: no linear dependence found" + + symmetricProduct(l1, l2, diff) == + u := var1::V; v := var2::V + n1 := degree l1; n2 := degree l2 + un := differentiate(u, n1); vn := differentiate(v, n2) + a := applyLODO(inv(- leadingCoefficient l1) * reductum l1, u) + b := applyLODO(inv(- leadingCoefficient l2) * reductum l2, v) + killer(u::P * v::P, n1 * n2, [un, vn], [a, b], diff) + + directSum(l1, l2, diff) == + u := var1::V; v := var2::V + n1 := degree l1; n2 := degree l2 + un := differentiate(u, n1); vn := differentiate(v, n2) + a := applyLODO(inv(- leadingCoefficient l1) * reductum l1, u) + b := applyLODO(inv(- leadingCoefficient l2) * reductum l2, v) + killer(u::P + v::P, n1 + n2, [un, vn], [a, b], diff) + + applyLODO(l, v) == + p:P := 0 + while l ^= 0 repeat + p := p + monomial(leadingCoefficient(l)::P, + differentiate(v, degree l), 1) + l := reductum l + p + +@ +\section{domain LODO LinearOrdinaryDifferentialOperator} +<<domain LODO LinearOrdinaryDifferentialOperator>>= +)abbrev domain LODO LinearOrdinaryDifferentialOperator +++ Author: Manuel Bronstein +++ Date Created: 9 December 1993 +++ Date Last Updated: 15 April 1994 +++ Keywords: differential operator +++ Description: +++ \spad{LinearOrdinaryDifferentialOperator} defines a ring of +++ differential operators with coefficients in a ring A with a given +++ derivation. +++ Multiplication of operators corresponds to functional composition: +++ \spad{(L1 * L2).(f) = L1 L2 f} +LinearOrdinaryDifferentialOperator(A:Ring, diff: A -> A): + LinearOrdinaryDifferentialOperatorCategory A + == SparseUnivariateSkewPolynomial(A, 1, diff) add + Rep := SparseUnivariateSkewPolynomial(A, 1, diff) + + outputD := "D"@String :: Symbol :: OutputForm + + coerce(l:%):OutputForm == outputForm(l, outputD) + elt(p:%, a:A):A == apply(p, 0, a) + + if A has Field then + import LinearOrdinaryDifferentialOperatorsOps(A, %) + + symmetricProduct(a, b) == symmetricProduct(a, b, diff) + symmetricPower(a, n) == symmetricPower(a, n, diff) + directSum(a, b) == directSum(a, b, diff) + +@ +\section{domain LODO1 LinearOrdinaryDifferentialOperator1} +<<domain LODO1 LinearOrdinaryDifferentialOperator1>>= +)abbrev domain LODO1 LinearOrdinaryDifferentialOperator1 +++ Author: Manuel Bronstein +++ Date Created: 9 December 1993 +++ Date Last Updated: 31 January 1994 +++ Keywords: differential operator +++ Description: +++ \spad{LinearOrdinaryDifferentialOperator1} defines a ring of +++ differential operators with coefficients in a differential ring A. +++ Multiplication of operators corresponds to functional composition: +++ \spad{(L1 * L2).(f) = L1 L2 f} +LinearOrdinaryDifferentialOperator1(A:DifferentialRing) == + LinearOrdinaryDifferentialOperator(A, differentiate$A) + +@ +\section{domain LODO2 LinearOrdinaryDifferentialOperator2} +<<domain LODO2 LinearOrdinaryDifferentialOperator2>>= +)abbrev domain LODO2 LinearOrdinaryDifferentialOperator2 +++ Author: Stephen M. Watt, Manuel Bronstein +++ Date Created: 1986 +++ Date Last Updated: 1 February 1994 +++ Keywords: differential operator +++ Description: +++ \spad{LinearOrdinaryDifferentialOperator2} defines a ring of +++ differential operators with coefficients in a differential ring A +++ and acting on an A-module M. +++ Multiplication of operators corresponds to functional composition: +++ \spad{(L1 * L2).(f) = L1 L2 f} +LinearOrdinaryDifferentialOperator2(A, M): Exports == Implementation where + A: DifferentialRing + M: LeftModule A with + differentiate: $ -> $ + ++ differentiate(x) returns the derivative of x + + Exports ==> Join(LinearOrdinaryDifferentialOperatorCategory A, Eltable(M, M)) + + Implementation ==> LinearOrdinaryDifferentialOperator(A, differentiate$A) add + elt(p:%, m:M):M == + apply(p, differentiate, m)$ApplyUnivariateSkewPolynomial(A, M, %) + +@ +\section{License} +<<license>>= +--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +--All rights reserved. +-- +--Redistribution and use in source and binary forms, with or without +--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, +--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. +@ +<<*>>= +<<license>> + +<<category LODOCAT LinearOrdinaryDifferentialOperatorCategory>> +<<package LODOOPS LinearOrdinaryDifferentialOperatorsOps>> +<<domain LODO LinearOrdinaryDifferentialOperator>> +<<domain LODO1 LinearOrdinaryDifferentialOperator1>> +<<domain LODO2 LinearOrdinaryDifferentialOperator2>> +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} |