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--All rights reserved.
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--met:
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-- - Redistributions of source code must retain the above copyright
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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
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--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
)abbrev package LSMP LinearSystemMatrixPackage
++ Author: P.Gianni, S.Watt
++ Date Created: Summer 1985
++ Date Last Updated:Summer 1990
++ Basic Functions: solve, particularSolution, hasSolution?, rank
++ Related Constructors: LinearSystemMatrixPackage1
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package solves linear system in the matrix form \spad{AX = B}.
LinearSystemMatrixPackage(F, Row, Col, M): Cat == Capsule where
F: Field
Row: FiniteLinearAggregate F with shallowlyMutable
Col: FiniteLinearAggregate F with shallowlyMutable
M : MatrixCategory(F, Row, Col)
N ==> NonNegativeInteger
PartialV ==> Union(Col, "failed")
Both ==> Record(particular: PartialV, basis: List Col)
Cat ==> with
solve : (M, Col) -> Both
++ solve(A,B) finds a particular solution of the system \spad{AX = B}
++ and a basis of the associated homogeneous system \spad{AX = 0}.
solve : (M, List Col) -> List Both
++ solve(A,LB) finds a particular soln of the systems \spad{AX = B}
++ and a basis of the associated homogeneous systems \spad{AX = 0}
++ where B varies in the list of column vectors LB.
particularSolution: (M, Col) -> PartialV
++ particularSolution(A,B) finds a particular solution of the linear
++ system \spad{AX = B}.
hasSolution?: (M, Col) -> Boolean
++ hasSolution?(A,B) tests if the linear system \spad{AX = B}
++ has a solution.
rank : (M, Col) -> N
++ rank(A,B) computes the rank of the complete matrix \spad{(A|B)}
++ of the linear system \spad{AX = B}.
Capsule ==> add
systemMatrix : (M, Col) -> M
aSolution : M -> PartialV
-- rank theorem
hasSolution?(A, b) == rank A = rank systemMatrix(A, b)
systemMatrix(m, v) == horizConcat(m, -(v::M))
rank(A, b) == rank systemMatrix(A, b)
particularSolution(A, b) == aSolution rowEchelon systemMatrix(A,b)
-- m should be in row-echelon form.
-- last column of m is -(right-hand-side of system)
aSolution m ==
nvar := (ncols m - 1)::N
rk := maxRowIndex m
while (rk >= minRowIndex m) and every?(zero?, row(m, rk))
repeat rk := dec rk
rk < minRowIndex m => new(nvar, 0)
ck := minColIndex m
while (ck < maxColIndex m) and zero? qelt(m, rk, ck) repeat
ck := inc ck
ck = maxColIndex m => "failed"
sol := new(nvar, 0)$Col
-- find leading elements of diagonal
v := new(nvar, minRowIndex m - 1)$PrimitiveArray(Integer)
for i in minRowIndex m .. rk repeat
j : Integer := 0
while zero? qelt(m, i, j+minColIndex m) repeat j := j + 1
v.j := i
for j in 0..nvar-1 repeat
if v.j >= minRowIndex m then
qsetelt!(sol, j+minIndex sol, - qelt(m, v.j, maxColIndex m))
sol
solve(A:M, b:Col) ==
-- Special case for homogeneous systems.
every?(zero?, b) => [new(ncols A, 0), nullSpace A]
-- General case.
m := rowEchelon systemMatrix(A, b)
[aSolution m,
nullSpace subMatrix(m, minRowIndex m, maxRowIndex m,
minColIndex m, maxColIndex m - 1)]
solve(A:M, l:List Col) ==
null l => [[new(ncols A, 0), nullSpace A]]
nl := (sol0 := solve(A, first l)).basis
cons(sol0,
[[aSolution rowEchelon systemMatrix(A, b), nl]
for b in rest l])
)abbrev package LSMP1 LinearSystemMatrixPackage1
++ Author: R. Sutor
++ Date Created: June, 1994
++ Date Last Updated:
++ Basic Functions: solve, particularSolution, hasSolution?, rank
++ Related Constructors: LinearSystemMatrixPackage
++ Also See:
++ AMS Classifications:
++ Keywords: solve
++ References:
++ Description:
++ This package solves linear system in the matrix form \spad{AX = B}.
++ It is essentially a particular instantiation of the package
++ \spadtype{LinearSystemMatrixPackage} for Matrix and Vector. This
++ package's existence makes it easier to use \spadfun{solve} in the
++ AXIOM interpreter.
LinearSystemMatrixPackage1(F): Cat == Capsule where
F: Field
Row ==> Vector F
Col ==> Vector F
M ==> Matrix(F)
LL ==> List List F
N ==> NonNegativeInteger
PartialV ==> Union(Col, "failed")
Both ==> Record(particular: PartialV, basis: List Col)
LSMP ==> LinearSystemMatrixPackage(F, Row, Col, M)
Cat ==> with
solve : (M, Col) -> Both
++ solve(A,B) finds a particular solution of the system \spad{AX = B}
++ and a basis of the associated homogeneous system \spad{AX = 0}.
solve : (LL, Col) -> Both
++ solve(A,B) finds a particular solution of the system \spad{AX = B}
++ and a basis of the associated homogeneous system \spad{AX = 0}.
solve : (M, List Col) -> List Both
++ solve(A,LB) finds a particular soln of the systems \spad{AX = B}
++ and a basis of the associated homogeneous systems \spad{AX = 0}
++ where B varies in the list of column vectors LB.
solve : (LL, List Col) -> List Both
++ solve(A,LB) finds a particular soln of the systems \spad{AX = B}
++ and a basis of the associated homogeneous systems \spad{AX = 0}
++ where B varies in the list of column vectors LB.
particularSolution: (M, Col) -> PartialV
++ particularSolution(A,B) finds a particular solution of the linear
++ system \spad{AX = B}.
hasSolution?: (M, Col) -> Boolean
++ hasSolution?(A,B) tests if the linear system \spad{AX = B}
++ has a solution.
rank : (M, Col) -> N
++ rank(A,B) computes the rank of the complete matrix \spad{(A|B)}
++ of the linear system \spad{AX = B}.
Capsule ==> add
solve(m : M, c: Col): Both == solve(m,c)$LSMP
solve(ll : LL, c: Col): Both == solve(matrix(ll)$M,c)$LSMP
solve(m : M, l : List Col): List Both == solve(m, l)$LSMP
solve(ll : LL, l : List Col): List Both == solve(matrix(ll)$M, l)$LSMP
particularSolution (m : M, c : Col): PartialV == particularSolution(m, c)$LSMP
hasSolution?(m :M, c : Col): Boolean == hasSolution?(m, c)$LSMP
rank(m : M, c : Col): N == rank(m, c)$LSMP
)abbrev package LSPP LinearSystemPolynomialPackage
++ Author: P.Gianni
++ Date Created: Summer 1985
++ Date Last Updated: Summer 1993
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References: SystemSolvePackage
++ Description:
++ this package finds the solutions of linear systems presented as a
++ list of polynomials.
LinearSystemPolynomialPackage(R, E, OV, P): Cat == Capsule where
R : IntegralDomain
OV : OrderedSet
E : OrderedAbelianMonoidSup
P : PolynomialCategory(R,E,OV)
F ==> Fraction P
NNI ==> NonNegativeInteger
V ==> Vector
M ==> Matrix
Soln ==> Record(particular: Union(V F, "failed"), basis: List V F)
Cat == with
linSolve: (List P, List OV) -> Soln
++ linSolve(lp,lvar) finds the solutions of the linear system
++ of polynomials lp = 0 with respect to the list of symbols lvar.
Capsule == add
---- Local Functions ----
poly2vect: (P, List OV) -> Record(coefvec: V F, reductum: F)
intoMatrix: (List P, List OV) -> Record(mat: M F, vec: V F)
poly2vect(p : P, vs : List OV) : Record(coefvec: V F, reductum: F) ==
coefs := new(#vs, 0)$(V F)
for v in vs for i in 1.. while p ~= 0 repeat
u := univariate(p, v)
degree u = 0 => "next v"
coefs.i := (c := leadingCoefficient u)::F
p := p - monomial(c,v, 1)
[coefs, p :: F]
intoMatrix(ps : List P, vs : List OV ) : Record(mat: M F, vec: V F) ==
m := zero(#ps, #vs)$M(F)
v := new(#ps, 0)$V(F)
for p in ps for i in 1.. repeat
totalDegree(p,vs) > 1 => error "The system is not linear"
r := poly2vect(p,vs)
m:=setRow!(m,i,r.coefvec)
v.i := - r.reductum
[m, v]
linSolve(ps, vs) ==
r := intoMatrix(ps, vs)
solve(r.mat, r.vec)$LinearSystemMatrixPackage(F,V F,V F,M F)
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