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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra matstor.spad}
\author{Clifton J. Williamson}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package MATSTOR StorageEfficientMatrixOperations}
<<package MATSTOR StorageEfficientMatrixOperations>>=
)abbrev package MATSTOR StorageEfficientMatrixOperations
++ Author: Clifton J. Williamson
++ Date Created: 18 July 1990
++ Date Last Updated: 18 July 1990
++ Basic Operations:
++ Related Domains: Matrix(R)
++ Also See:
++ AMS Classifications:
++ Keywords: matrix, linear algebra
++ Examples:
++ References:
++ Description:
++ This package provides standard arithmetic operations on matrices.
++ The functions in this package store the results of computations
++ in existing matrices, rather than creating new matrices. This
++ package works only for matrices of type Matrix and uses the
++ internal representation of this type.
StorageEfficientMatrixOperations(R): Exports == Implementation where
R : Ring
M ==> Matrix R
NNI ==> NonNegativeInteger
ARR ==> PrimitiveArray R
REP ==> PrimitiveArray PrimitiveArray R
Exports ==> with
copy_! : (M,M) -> M
++ \spad{copy!(c,a)} copies the matrix \spad{a} into the matrix c.
++ Error: if \spad{a} and c do not have the same
++ dimensions.
plus_! : (M,M,M) -> M
++ \spad{plus!(c,a,b)} computes the matrix sum \spad{a + b} and stores the
++ result in the matrix c.
++ Error: if \spad{a}, b, and c do not have the same dimensions.
minus_! : (M,M) -> M
++ \spad{minus!(c,a)} computes \spad{-a} and stores the result in the
++ matrix c.
++ Error: if a and c do not have the same dimensions.
minus_! : (M,M,M) -> M
++ \spad{!minus!(c,a,b)} computes the matrix difference \spad{a - b}
++ and stores the result in the matrix c.
++ Error: if \spad{a}, b, and c do not have the same dimensions.
leftScalarTimes_! : (M,R,M) -> M
++ \spad{leftScalarTimes!(c,r,a)} computes the scalar product
++ \spad{r * a} and stores the result in the matrix c.
++ Error: if \spad{a} and c do not have the same dimensions.
rightScalarTimes_! : (M,M,R) -> M
++ \spad{rightScalarTimes!(c,a,r)} computes the scalar product
++ \spad{a * r} and stores the result in the matrix c.
++ Error: if \spad{a} and c do not have the same dimensions.
times_! : (M,M,M) -> M
++ \spad{times!(c,a,b)} computes the matrix product \spad{a * b}
++ and stores the result in the matrix c.
++ Error: if \spad{a}, b, and c do not have
++ compatible dimensions.
power_! : (M,M,M,M,NNI) -> M
++ \spad{power!(a,b,c,m,n)} computes m ** n and stores the result in
++ \spad{a}. The matrices b and c are used to store intermediate results.
++ Error: if \spad{a}, b, c, and m are not square
++ and of the same dimensions.
"**" : (M,NNI) -> M
++ \spad{x ** n} computes the n-th power
++ of a square matrix. The power n is assumed greater than 1.
Implementation ==> add
rep : M -> REP
rep m == m pretend REP
copy_!(c,a) ==
m := nrows a; n := ncols a
not((nrows c) = m and (ncols c) = n) =>
error "copy!: matrices of incompatible dimensions"
aa := rep a; cc := rep c
for i in 0..(m-1) repeat
aRow := qelt(aa,i); cRow := qelt(cc,i)
for j in 0..(n-1) repeat
qsetelt_!(cRow,j,qelt(aRow,j))
c
plus_!(c,a,b) ==
m := nrows a; n := ncols a
not((nrows b) = m and (ncols b) = n) =>
error "plus!: matrices of incompatible dimensions"
not((nrows c) = m and (ncols c) = n) =>
error "plus!: matrices of incompatible dimensions"
aa := rep a; bb := rep b; cc := rep c
for i in 0..(m-1) repeat
aRow := qelt(aa,i); bRow := qelt(bb,i); cRow := qelt(cc,i)
for j in 0..(n-1) repeat
qsetelt_!(cRow,j,qelt(aRow,j) + qelt(bRow,j))
c
minus_!(c,a) ==
m := nrows a; n := ncols a
not((nrows c) = m and (ncols c) = n) =>
error "minus!: matrices of incompatible dimensions"
aa := rep a; cc := rep c
for i in 0..(m-1) repeat
aRow := qelt(aa,i); cRow := qelt(cc,i)
for j in 0..(n-1) repeat
qsetelt_!(cRow,j,-qelt(aRow,j))
c
minus_!(c,a,b) ==
m := nrows a; n := ncols a
not((nrows b) = m and (ncols b) = n) =>
error "minus!: matrices of incompatible dimensions"
not((nrows c) = m and (ncols c) = n) =>
error "minus!: matrices of incompatible dimensions"
aa := rep a; bb := rep b; cc := rep c
for i in 0..(m-1) repeat
aRow := qelt(aa,i); bRow := qelt(bb,i); cRow := qelt(cc,i)
for j in 0..(n-1) repeat
qsetelt_!(cRow,j,qelt(aRow,j) - qelt(bRow,j))
c
leftScalarTimes_!(c,r,a) ==
m := nrows a; n := ncols a
not((nrows c) = m and (ncols c) = n) =>
error "leftScalarTimes!: matrices of incompatible dimensions"
aa := rep a; cc := rep c
for i in 0..(m-1) repeat
aRow := qelt(aa,i); cRow := qelt(cc,i)
for j in 0..(n-1) repeat
qsetelt_!(cRow,j,r * qelt(aRow,j))
c
rightScalarTimes_!(c,a,r) ==
m := nrows a; n := ncols a
not((nrows c) = m and (ncols c) = n) =>
error "rightScalarTimes!: matrices of incompatible dimensions"
aa := rep a; cc := rep c
for i in 0..(m-1) repeat
aRow := qelt(aa,i); cRow := qelt(cc,i)
for j in 0..(n-1) repeat
qsetelt_!(cRow,j,qelt(aRow,j) * r)
c
copyCol_!: (ARR,REP,Integer,Integer) -> ARR
copyCol_!(bCol,bb,j,n1) ==
for i in 0..n1 repeat qsetelt_!(bCol,i,qelt(qelt(bb,i),j))
times_!(c,a,b) ==
m := nrows a; n := ncols a; p := ncols b
not((nrows b) = n and (nrows c) = m and (ncols c) = p) =>
error "times!: matrices of incompatible dimensions"
aa := rep a; bb := rep b; cc := rep c
bCol : ARR := new(n,0)
m1 := (m :: Integer) - 1; n1 := (n :: Integer) - 1
for j in 0..(p-1) repeat
copyCol_!(bCol,bb,j,n1)
for i in 0..m1 repeat
aRow := qelt(aa,i); cRow := qelt(cc,i)
sum : R := 0
for k in 0..n1 repeat
sum := sum + qelt(aRow,k) * qelt(bCol,k)
qsetelt_!(cRow,j,sum)
c
power_!(a,b,c,m,p) ==
mm := nrows a; nn := ncols a
not(mm = nn) =>
error "power!: matrix must be square"
not((nrows b) = mm and (ncols b) = nn) =>
error "power!: matrices of incompatible dimensions"
not((nrows c) = mm and (ncols c) = nn) =>
error "power!: matrices of incompatible dimensions"
not((nrows m) = mm and (ncols m) = nn) =>
error "power!: matrices of incompatible dimensions"
flag := false
copy_!(b,m)
repeat
if odd? p then
flag =>
times_!(c,b,a)
copy_!(a,c)
flag := true
copy_!(a,b)
one? p => return a
p := p quo 2
times_!(c,b,b)
copy_!(b,c)
m ** n ==
not square? m => error "**: matrix must be square"
a := copy m; b := copy m; c := copy m
power_!(a,b,c,m,n)
@
\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 MATSTOR StorageEfficientMatrixOperations>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
