aboutsummaryrefslogtreecommitdiff
path: root/src/algebra/matrix.spad.pamphlet
blob: 6aded1983a02fca97175cdb0994d9921037220e3 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra matrix.spad}
\author{Johannes Grabmeier, Oswald Gschnitzer, Clifton J. Williamson}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject

\section{domain MATRIX Matrix}
<<domain MATRIX Matrix>>=
)abbrev domain MATRIX Matrix
++ Author: Grabmeier, Gschnitzer, Williamson
++ Date Created: 1987
++ Date Last Updated: July 1990
++ Basic Operations:
++ Related Domains: 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
  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 == InnerTwoDimensionalArray(R,Row,Col) add
    minr ==> minRowIndex
    maxr ==> maxRowIndex
    minc ==> minColIndex
    maxc ==> maxColIndex
    mini ==> minIndex
    maxi ==> maxIndex
 
    minRowIndex x == 1
    minColIndex x == 1
 
    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)

@
\section{domain RMATRIX RectangularMatrix}
<<domain RMATRIX RectangularMatrix>>=
)abbrev domain RMATRIX RectangularMatrix
++ Author: Grabmeier, Gschnitzer, Williamson
++ Date Created: 1987
++ Date Last Updated: July 1990
++ Basic Operations:
++ Related Domains: 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
 
    0 == per new(m,n,0)$Matrix(R)
 
    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)

@
\section{domain SQMATRIX SquareMatrix}
<<domain SQMATRIX SquareMatrix>>=
)abbrev domain SQMATRIX SquareMatrix
++ Author: Grabmeier, Gschnitzer, Williamson
++ Date Created: 1987
++ Date Last Updated: July 1990
++ Basic Operations:
++ Related Domains: 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
 
    0 == scalarMatrix 0
    1 == scalarMatrix 1

    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)


@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--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.
--
--    - 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>>

<<domain MATRIX Matrix>>
<<domain RMATRIX RectangularMatrix>>
<<domain SQMATRIX SquareMatrix>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}