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--All rights reserved.
--Copyright (C) 2007-2009, Gabriel Dos Reis.
--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.
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-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
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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
--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
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--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
)abbrev domain IMATRIX IndexedMatrix
++ Author: Grabmeier, Gschnitzer, Williamson
++ Date Created: 1987
++ Date Last Updated: July 1990
++ Basic Operations:
++ Related Domains: Matrix, RectangularMatrix, SquareMatrix,
++ StorageEfficientMatrixOperations
++ Also See:
++ AMS Classifications:
++ Keywords: matrix, linear algebra
++ Examples:
++ References:
++ Description:
++ An \spad{IndexedMatrix} is a matrix where the minimal row and column
++ indices are parameters of the type. The domains Row and Col
++ are both IndexedVectors.
++ The index of the 'first' row may be obtained by calling the
++ function \spadfun{minRowIndex}. The index of the 'first' column may
++ be obtained by calling the function \spadfun{minColIndex}. The index of
++ the first element of a 'Row' is the same as the index of the
++ first column in a matrix and vice versa.
IndexedMatrix(R,mnRow,mnCol): Exports == Implementation where
R : Ring
mnRow, mnCol : Integer
Row ==> IndexedVector(R,mnCol)
Col ==> IndexedVector(R,mnRow)
MATLIN ==> MatrixLinearAlgebraFunctions(R,Row,Col,$)
Exports ==> MatrixCategory(R,Row,Col)
Implementation ==>
InnerIndexedTwoDimensionalArray(R,mnRow,mnCol,Row,Col) add
swapRows!(x,i1,i2) ==
(i1 < minRowIndex(x)) or (i1 > maxRowIndex(x)) or _
(i2 < minRowIndex(x)) or (i2 > maxRowIndex(x)) =>
error "swapRows!: index out of range"
i1 = i2 => x
minRow := minRowIndex x
xx := x pretend PrimitiveArray(PrimitiveArray(R))
n1 := i1 - minRow; n2 := i2 - minRow
row1 := qelt(xx,n1)
qsetelt!(xx,n1,qelt(xx,n2))
qsetelt!(xx,n2,row1)
xx pretend $
if R has commutative("*") then
determinant x == determinant(x)$MATLIN
minordet x == minordet(x)$MATLIN
if R has EuclideanDomain then
rowEchelon x == rowEchelon(x)$MATLIN
if R has IntegralDomain then
rank x == rank(x)$MATLIN
nullity x == nullity(x)$MATLIN
nullSpace x == nullSpace(x)$MATLIN
if R has Field then
inverse x == inverse(x)$MATLIN
)abbrev domain MATRIX Matrix
++ Author: Grabmeier, Gschnitzer, Williamson
++ Date Created: 1987
++ Date Last Updated: July 1990
++ Basic Operations:
++ Related Domains: IndexedMatrix, RectangularMatrix, SquareMatrix
++ Also See:
++ AMS Classifications:
++ Keywords: matrix, linear algebra
++ Examples:
++ References:
++ Description:
++ \spadtype{Matrix} is a matrix domain where 1-based indexing is used
++ for both rows and columns.
Matrix(R): Exports == Implementation where
R : Ring
Row ==> Vector R
Col ==> Vector R
mnRow ==> 1
mnCol ==> 1
MATLIN ==> MatrixLinearAlgebraFunctions(R,Row,Col,$)
MATSTOR ==> StorageEfficientMatrixOperations(R)
Exports ==> MatrixCategory(R,Row,Col) with
diagonalMatrix: Vector R -> $
++ \spad{diagonalMatrix(v)} returns a diagonal matrix where the elements
++ of v appear on the diagonal.
if R has ConvertibleTo InputForm then ConvertibleTo InputForm
if R has Field then
inverse: $ -> Union($,"failed")
++ \spad{inverse(m)} returns the inverse of the matrix m.
++ If the matrix is not invertible, "failed" is returned.
++ Error: if the matrix is not square.
-- matrix: Vector Vector R -> $
-- ++ \spad{matrix(v)} converts the vector of vectors v to a matrix, where
-- ++ the vector of vectors is viewed as a vector of the rows of the
-- ++ matrix
-- diagonalMatrix: Vector $ -> $
-- ++ \spad{diagonalMatrix([m1,...,mk])} creates a block diagonal matrix
-- ++ M with block matrices {\em m1},...,{\em mk} down the diagonal,
-- ++ with 0 block matrices elsewhere.
-- vectorOfVectors: $ -> Vector Vector R
-- ++ \spad{vectorOfVectors(m)} returns the rows of the matrix m as a
-- ++ vector of vectors
Implementation ==>
InnerIndexedTwoDimensionalArray(R,mnRow,mnCol,Row,Col) add
minr ==> minRowIndex
maxr ==> maxRowIndex
minc ==> minColIndex
maxc ==> maxColIndex
mini ==> minIndex
maxi ==> maxIndex
minRowIndex x == mnRow
minColIndex x == mnCol
swapRows!(x,i1,i2) ==
(i1 < minRowIndex(x)) or (i1 > maxRowIndex(x)) or _
(i2 < minRowIndex(x)) or (i2 > maxRowIndex(x)) =>
error "swapRows!: index out of range"
i1 = i2 => x
minRow := minRowIndex x
xx := x pretend PrimitiveArray(PrimitiveArray(R))
n1 := i1 - minRow; n2 := i2 - minRow
row1 := qelt(xx,n1)
qsetelt!(xx,n1,qelt(xx,n2))
qsetelt!(xx,n2,row1)
xx pretend $
positivePower:($,Integer,NonNegativeInteger) -> $
positivePower(x,n,nn) ==
one? n => x
-- no need to allocate space for 3 additional matrices
n = 2 => x * x
n = 3 => x * x * x
n = 4 => (y := x * x; y * y)
a := new(nn,nn,0) pretend Matrix(R)
b := new(nn,nn,0) pretend Matrix(R)
c := new(nn,nn,0) pretend Matrix(R)
xx := x pretend Matrix(R)
power!(a,b,c,xx,n :: NonNegativeInteger)$MATSTOR pretend $
x:$ ** n:NonNegativeInteger ==
not((nn := nrows x) = ncols x) =>
error "**: matrix must be square"
zero? n => scalarMatrix(nn,1)
positivePower(x,n,nn)
if R has commutative("*") then
determinant x == determinant(x)$MATLIN
minordet x == minordet(x)$MATLIN
if R has EuclideanDomain then
rowEchelon x == rowEchelon(x)$MATLIN
if R has IntegralDomain then
rank x == rank(x)$MATLIN
nullity x == nullity(x)$MATLIN
nullSpace x == nullSpace(x)$MATLIN
if R has Field then
inverse x == inverse(x)$MATLIN
x:$ ** n:Integer ==
nn := nrows x
not(nn = ncols x) =>
error "**: matrix must be square"
zero? n => scalarMatrix(nn,1)
positive? n => positivePower(x,n,nn)
(xInv := inverse x) case "failed" =>
error "**: matrix must be invertible"
positivePower(xInv :: $,-n,nn)
-- matrix(v: Vector Vector R) ==
-- (rows := # v) = 0 => new(0,0,0)
-- -- error check: this is a top level function
-- cols := # v.mini(v)
-- for k in (mini(v) + 1)..maxi(v) repeat
-- cols ~= # v.k => error "matrix: rows of different lengths"
-- ans := new(rows,cols,0)
-- for i in minr(ans)..maxr(ans) for k in mini(v)..maxi(v) repeat
-- vv := v.k
-- for j in minc(ans)..maxc(ans) for l in mini(vv)..maxi(vv) repeat
-- ans(i,j) := vv.l
-- ans
diagonalMatrix(v: Vector R) ==
n := #v; ans := zero(n,n)
for i in minr(ans)..maxr(ans) for j in minc(ans)..maxc(ans) _
for k in mini(v)..maxi(v) repeat qsetelt!(ans,i,j,qelt(v,k))
ans
-- diagonalMatrix(vec: Vector $) ==
-- rows : NonNegativeInteger := 0
-- cols : NonNegativeInteger := 0
-- for r in mini(vec)..maxi(vec) repeat
-- mat := vec.r
-- rows := rows + nrows mat; cols := cols + ncols mat
-- ans := zero(rows,cols)
-- loR := minr ans; loC := minc ans
-- for r in mini(vec)..maxi(vec) repeat
-- mat := vec.r
-- hiR := loR + nrows(mat) - 1; hiC := loC + nrows(mat) - 1
-- for i in loR..hiR for k in minr(mat)..maxr(mat) repeat
-- for j in loC..hiC for l in minc(mat)..maxc(mat) repeat
-- ans(i,j) := mat(k,l)
-- loR := hiR + 1; loC := hiC + 1
-- ans
-- vectorOfVectors x ==
-- vv : Vector Vector R := new(nrows x,0)
-- cols := ncols x
-- for k in mini(vv)..maxi(vv) repeat
-- vv.k := new(cols,0)
-- for i in minr(x)..maxr(x) for k in mini(vv)..maxi(vv) repeat
-- v := vv.k
-- for j in minc(x)..maxc(x) for l in mini(v)..maxi(v) repeat
-- v.l := x(i,j)
-- vv
if R has ConvertibleTo InputForm then
convert(x:$):InputForm ==
convert [convert('matrix)@InputForm,
convert listOfLists x]$List(InputForm)
)abbrev domain RMATRIX RectangularMatrix
++ Author: Grabmeier, Gschnitzer, Williamson
++ Date Created: 1987
++ Date Last Updated: July 1990
++ Basic Operations:
++ Related Domains: IndexedMatrix, Matrix, SquareMatrix
++ Also See:
++ AMS Classifications:
++ Keywords: matrix, linear algebra
++ Examples:
++ References:
++ Description:
++ \spadtype{RectangularMatrix} is a matrix domain where the number of rows
++ and the number of columns are parameters of the domain.
RectangularMatrix(m,n,R): Exports == Implementation where
m,n : NonNegativeInteger
R : Ring
Row ==> DirectProduct(n,R)
Col ==> DirectProduct(m,R)
Exports ==> Join(RectangularMatrixCategory(m,n,R,Row,Col),_
CoercibleTo Matrix R) with
if R has Field then VectorSpace R
if R has ConvertibleTo InputForm then ConvertibleTo InputForm
rectangularMatrix: Matrix R -> $
++ \spad{rectangularMatrix(m)} converts a matrix of type \spadtype{Matrix}
++ to a matrix of type \spad{RectangularMatrix}.
Implementation ==> Matrix R add
minr ==> minRowIndex
maxr ==> maxRowIndex
minc ==> minColIndex
maxc ==> maxColIndex
mini ==> minIndex
maxi ==> maxIndex
ZERO := per new(m,n,0)$Matrix(R)
0 == ZERO
coerce(x:$):OutputForm == rep(x)::OutputForm
matrix(l: List List R) ==
-- error check: this is a top level function
#l ~= m => error "matrix: wrong number of rows"
for ll in l repeat
#ll ~= n => error "matrix: wrong number of columns"
ans : Matrix R := new(m,n,0)
for i in minr(ans)..maxr(ans) for ll in l repeat
for j in minc(ans)..maxc(ans) for r in ll repeat
qsetelt!(ans,i,j,r)
per ans
row(x,i) == directProduct row(rep x,i)
column(x,j) == directProduct column(rep x,j)
coerce(x:$):Matrix(R) == copy rep x
rectangularMatrix x ==
(nrows(x) ~= m) or (ncols(x) ~= n) =>
error "rectangularMatrix: matrix of bad dimensions"
per copy(x)
if R has EuclideanDomain then
rowEchelon x == per rowEchelon(rep x)
if R has IntegralDomain then
rank x == rank rep x
nullity x == nullity rep x
nullSpace x ==
[directProduct c for c in nullSpace rep x]
if R has Field then
dimension() == (m * n) :: CardinalNumber
if R has ConvertibleTo InputForm then
convert(x:$):InputForm ==
convert [convert('rectangularMatrix)@InputForm,
convert(x::Matrix(R))]$List(InputForm)
)abbrev domain SQMATRIX SquareMatrix
++ Author: Grabmeier, Gschnitzer, Williamson
++ Date Created: 1987
++ Date Last Updated: July 1990
++ Basic Operations:
++ Related Domains: IndexedMatrix, Matrix, RectangularMatrix
++ Also See:
++ AMS Classifications:
++ Keywords: matrix, linear algebra
++ Examples:
++ References:
++ Description:
++ \spadtype{SquareMatrix} is a matrix domain of square matrices, where the
++ number of rows (= number of columns) is a parameter of the type.
SquareMatrix(ndim,R): Exports == Implementation where
ndim : NonNegativeInteger
R : Ring
Row ==> DirectProduct(ndim,R)
Col ==> DirectProduct(ndim,R)
MATLIN ==> MatrixLinearAlgebraFunctions(R,Row,Col,$)
Exports ==> Join(SquareMatrixCategory(ndim,R,Row,Col),_
CoercibleTo Matrix R) with
new: R -> %
++ \spad{new(c)} constructs a new \spadtype{SquareMatrix}
++ object of dimension \spad{ndim} with initial entries equal
++ to \spad{c}.
transpose: $ -> $
++ \spad{transpose(m)} returns the transpose of the matrix m.
squareMatrix: Matrix R -> $
++ \spad{squareMatrix(m)} converts a matrix of type \spadtype{Matrix}
++ to a matrix of type \spadtype{SquareMatrix}.
-- symdecomp : $ -> Record(sym:$,antisym:$)
-- ++ \spad{symdecomp(m)} decomposes the matrix m as a sum of a symmetric
-- ++ matrix \spad{m1} and an antisymmetric matrix \spad{m2}. The object
-- ++ returned is the Record \spad{[m1,m2]}
-- if R has commutative("*") then
-- minorsVect: -> Vector(Union(R,"uncomputed")) --range: 1..2**n-1
-- ++ \spad{minorsVect(m)} returns a vector of the minors of the matrix m
if R has commutative("*") then central
++ the elements of the Ring R, viewed as diagonal matrices, commute
++ with all matrices and, indeed, are the only matrices which commute
++ with all matrices.
if R has commutative("*") and R has unitsKnown then unitsKnown
++ the invertible matrices are simply the matrices whose determinants
++ are units in the Ring R.
if R has ConvertibleTo InputForm then ConvertibleTo InputForm
Implementation ==> Matrix R add
Rep == Matrix R
minr ==> minRowIndex
maxr ==> maxRowIndex
minc ==> minColIndex
maxc ==> maxColIndex
mini ==> minIndex
maxi ==> maxIndex
ZERO := scalarMatrix 0
0 == ZERO
ONE := scalarMatrix 1
1 == ONE
characteristic == characteristic$R
new c == per new(ndim,ndim,c)$Rep
matrix(l: List List R) ==
-- error check: this is a top level function
#l ~= ndim => error "matrix: wrong number of rows"
for ll in l repeat
#ll ~= ndim => error "matrix: wrong number of columns"
ans := new(ndim,ndim,0)$Rep
for i in minr(ans)..maxr(ans) for ll in l repeat
for j in minc(ans)..maxc(ans) for r in ll repeat
qsetelt!(ans,i,j,r)
per ans
row(x,i) == directProduct row(rep x,i)
column(x,j) == directProduct column(rep x,j)
coerce(x:$):OutputForm == rep(x)::OutputForm
scalarMatrix r == per scalarMatrix(ndim,r)$Matrix(R)
diagonalMatrix l ==
#l ~= ndim =>
error "diagonalMatrix: wrong number of entries in list"
per diagonalMatrix(l)$Matrix(R)
coerce(x: %): Matrix(R) == copy rep x
squareMatrix x ==
(nrows(x) ~= ndim) or (ncols(x) ~= ndim) =>
error "squareMatrix: matrix of bad dimensions"
per copy x
x:% * v:Col ==
directProduct(rep(x) * (v :: Vector(R)))
v:Row * x:$ ==
directProduct((v :: Vector(R)) * rep(x))
x:$ ** n:NonNegativeInteger ==
per(rep(x) ** n)
if R has commutative("*") then
determinant x == determinant rep x
minordet x == minordet rep x
if R has EuclideanDomain then
rowEchelon x == per rowEchelon rep x
if R has IntegralDomain then
rank x == rank rep x
nullity x == nullity rep x
nullSpace x ==
[directProduct c for c in nullSpace rep x]
if R has Field then
dimension == (m * n) :: CardinalNumber
inverse x ==
(u := inverse rep x) case "failed" => "failed"
per(u :: Matrix(R))
x:$ ** n:Integer ==
per(rep(x) ** n)
recip x == inverse x
if R has ConvertibleTo InputForm then
convert(x:$):InputForm ==
convert [convert('squareMatrix)@InputForm,
convert(rep x)@InputForm]$List(InputForm)
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