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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra solvedio.spad}
\author{Albrecht Fortenbacher}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package DIOSP DiophantineSolutionPackage}
<<package DIOSP DiophantineSolutionPackage>>=
)abbrev package DIOSP DiophantineSolutionPackage
++ Author: A. Fortenbacher
++ Date Created: 29 March 1991
++ Date Last Updated: 29 March 1991
++ Basic Operations: dioSolve
++ Related Constructors: Equation, Vector
++ Also See:
++ AMS Classifications:
++ Keywords: Diophantine equation, nonnegative solutions,
++ basis, depth-first-search
++ Reference:
++ M. Clausen, A. Fortenbacher: Efficient Solution of
++ Linear Diophantine Equations. in JSC (1989) 8, 201-216
++ Description:
++ any solution of a homogeneous linear Diophantine equation
++ can be represented as a sum of minimal solutions, which
++ form a "basis" (a minimal solution cannot be represented
++ as a nontrivial sum of solutions)
++ in the case of an inhomogeneous linear Diophantine equation,
++ each solution is the sum of a inhomogeneous solution and
++ any number of homogeneous solutions
++ therefore, it suffices to compute two sets:
++ 1. all minimal inhomogeneous solutions
++ 2. all minimal homogeneous solutions
++ the algorithm implemented is a completion procedure, which
++ enumerates all solutions in a recursive depth-first-search
++ it can be seen as finding monotone paths in a graph
++ for more details see Reference
DiophantineSolutionPackage(): Cat == Capsule where
B ==> Boolean
I ==> Integer
NI ==> NonNegativeInteger
LI ==> List(I)
VI ==> Vector(I)
VNI ==> Vector(NI)
POLI ==> Polynomial(I)
EPOLI ==> Equation(POLI)
LPOLI ==> List(POLI)
S ==> Symbol
LS ==> List(S)
ListSol ==> List(VNI)
Solutions ==> Record(varOrder: LS, inhom: Union(ListSol,"failed"),
hom: ListSol)
Node ==> Record(vert: VI, free: B)
Graph ==> Record(vn: Vector(Node), dim : NI, zeroNode: I)
Cat ==> with
dioSolve: EPOLI -> Solutions
++ dioSolve(u) computes a basis of all minimal solutions for
++ linear homogeneous Diophantine equation u,
++ then all minimal solutions of inhomogeneous equation
Capsule ==> add
import I
import POLI
-- local function specifications
initializeGraph: (LPOLI, I) -> Graph
createNode: (I, VI, NI, I) -> Node
findSolutions: (VNI, I, I, I, Graph, B) -> ListSol
verifyMinimality: (VNI, Graph, B) -> B
verifySolution: (VNI, I, I, I, Graph) -> B
-- exported functions
dioSolve(eq) ==
p := lhs(eq) - rhs(eq)
n := totalDegree(p)
n = 0 or n > 1 =>
error "a linear Diophantine equation is expected"
mon := empty()$LPOLI
c : I := 0
for x in monomials(p) repeat
ground?(x) =>
c := ground(x) :: I
mon := cons(x, mon)$LPOLI
graph := initializeGraph(mon, c)
sol := zero(graph.dim)$VNI
hs := findSolutions(sol, graph.zeroNode, 1, 1, graph, true)
ihs : ListSol :=
c = 0 => [sol]
findSolutions(sol, graph.zeroNode + c, 1, 1, graph, false)
vars := [first(variables(x))$LS for x in mon]
[vars, if empty?(ihs)$ListSol then "failed" else ihs, hs]
-- local functions
initializeGraph(mon, c) ==
coeffs := vector([first(coefficients(x))$LI for x in mon])$VI
k := #coeffs
m := min(c, reduce(min, coeffs)$VI)
n := max(c, reduce(max, coeffs)$VI)
[[createNode(i, coeffs, k, 1 - m) for i in m..n], k, 1 - m]
createNode(ind, coeffs, k, zeroNode) ==
-- create vertices from node ind to other nodes
v := zero(k)$VI
for i in 1..k repeat
positive? ind =>
negative? coeffs.i =>
v.i := zeroNode + ind + coeffs.i
positive? coeffs.i =>
v.i := zeroNode + ind + coeffs.i
[v, true]
findSolutions(sol, ind, m, n, graph, flag) ==
-- return all solutions (paths) from node ind to node zeroNode
sols := empty()$ListSol
node := graph.vn.ind
node.free =>
node.free := false
v := node.vert
k := if ind < graph.zeroNode then m else n
for i in k..graph.dim repeat
x := sol.i
positive? v.i => -- vertex exists to other node
sol.i := x + 1
v.i = graph.zeroNode => -- solution found
verifyMinimality(sol, graph, flag) =>
sols := cons(copy(sol)$VNI, sols)$ListSol
sol.i := x
sol.i := x
s :=
ind < graph.zeroNode =>
findSolutions(sol, v.i, i, n, graph, flag)
findSolutions(sol, v.i, m, i, graph, flag)
sols := append(s, sols)$ListSol
sol.i := x
node.free := true
sols
sols
verifyMinimality(sol, graph, flag) ==
-- test whether sol contains a minimal homogeneous solution
flag => -- sol is a homogeneous solution
i: I := 1
while sol.i = 0 repeat
i := i + 1
x := sol.i
sol.i := (x - 1) :: NI
flag := verifySolution(sol, graph.zeroNode, 1, 1, graph)
sol.i := x
flag
verifySolution(sol, graph.zeroNode, 1, 1, graph)
verifySolution(sol, ind, m, n, graph) ==
-- test whether sol contains a path from ind to zeroNode
flag := true
node := graph.vn.ind
v := node.vert
k := if ind < graph.zeroNode then m else n
for i in k..graph.dim while flag repeat
x := sol.i
positive? x and positive? v.i => -- vertex exists to other node
sol.i := (x - 1) :: NI
v.i = graph.zeroNode => -- solution found
flag := false
sol.i := x
flag :=
ind < graph.zeroNode =>
verifySolution(sol, v.i, i, n, graph)
verifySolution(sol, v.i, m, i, graph)
sol.i := x
flag
@
\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>>
<<package DIOSP DiophantineSolutionPackage>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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