aboutsummaryrefslogtreecommitdiff
path: root/src/algebra/array2.spad.pamphlet
blob: 0a89e53f10ea44367c61c4e86b6fd54ec13a00f4 (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
\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra array2.spad}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{category ARR2CAT TwoDimensionalArrayCategory}
<<category ARR2CAT TwoDimensionalArrayCategory>>=
)abbrev category ARR2CAT TwoDimensionalArrayCategory
++ Two dimensional array categories and domains
++ Author:
++ Date Created: 27 October 1989
++ Date Last Updated: 27 June 1990
++ Keywords: array, data structure
++ Examples:
++ References:
TwoDimensionalArrayCategory(R,Row,Col): Category == Definition where
  ++ TwoDimensionalArrayCategory is a general array category which
  ++ allows different representations and indexing schemes.
  ++ Rows and columns may be extracted with rows returned as objects
  ++ of type Row and columns returned as objects of type Col.
  ++ The index of the 'first' row may be obtained by calling the
  ++ function 'minRowIndex'.  The index of the 'first' column may
  ++ be obtained by calling the function 'minColIndex'.  The index of
  ++ the first element of a 'Row' is the same as the index of the
  ++ first column in an array and vice versa.
  R   : Type
  Row : FiniteLinearAggregate R
  Col : FiniteLinearAggregate R

  Definition == HomogeneousAggregate(R) with

    shallowlyMutable
      ++ one may destructively alter arrays

    finiteAggregate
      ++ two-dimensional arrays are finite

--% Array creation

    new: (NonNegativeInteger,NonNegativeInteger,R) -> %
      ++ new(m,n,r) is an m-by-n array all of whose entries are r
    fill!: (%,R) -> %
      ++ fill!(m,r) fills m with r's

--% Size inquiries

    minRowIndex : % -> Integer
      ++ minRowIndex(m) returns the index of the 'first' row of the array m
    maxRowIndex : % -> Integer
      ++ maxRowIndex(m) returns the index of the 'last' row of the array m
    minColIndex : % -> Integer
      ++ minColIndex(m) returns the index of the 'first' column of the array m
    maxColIndex : % -> Integer
      ++ maxColIndex(m) returns the index of the 'last' column of the array m
    nrows : % -> NonNegativeInteger
      ++ nrows(m) returns the number of rows in the array m
    ncols : % -> NonNegativeInteger
      ++ ncols(m) returns the number of columns in the array m

--% Part extractions

    elt: (%,Integer,Integer) -> R
      ++ elt(m,i,j) returns the element in the ith row and jth
      ++ column of the array m
      ++ error check to determine if indices are in proper ranges
    qelt: (%,Integer,Integer) -> R
      ++ qelt(m,i,j) returns the element in the ith row and jth
      ++ column of the array m
      ++ NO error check to determine if indices are in proper ranges
    elt: (%,Integer,Integer,R) -> R
      ++ elt(m,i,j,r) returns the element in the ith row and jth
      ++ column of the array m, if m has an ith row and a jth column,
      ++ and returns r otherwise
    row: (%,Integer) -> Row
      ++ row(m,i) returns the ith row of m
      ++ error check to determine if index is in proper ranges
    column: (%,Integer) -> Col
      ++ column(m,j) returns the jth column of m
      ++ error check to determine if index is in proper ranges
    parts: % -> List R
      ++ parts(m) returns a list of the elements of m in row major order

--% Part assignments

    setelt: (%,Integer,Integer,R) -> R
      -- will become setelt!
      ++ setelt(m,i,j,r) sets the element in the ith row and jth
      ++ column of m to r
      ++ error check to determine if indices are in proper ranges
    qsetelt!: (%,Integer,Integer,R) -> R
      ++ qsetelt!(m,i,j,r) sets the element in the ith row and jth
      ++ column of m to r
      ++ NO error check to determine if indices are in proper ranges
    setRow!: (%,Integer,Row) -> %
      ++ setRow!(m,i,v) sets to ith row of m to v
    setColumn!: (%,Integer,Col) -> %
      ++ setColumn!(m,j,v) sets to jth column of m to v

--% Map and Zip

    map: (R -> R,%) -> %
      ++ map(f,a) returns \spad{b}, where \spad{b(i,j) = f(a(i,j))} for all \spad{i, j}
    map!: (R -> R,%) -> %
      ++ map!(f,a)  assign \spad{a(i,j)} to \spad{f(a(i,j))} for all \spad{i, j}
    map:((R,R) -> R,%,%) -> %
      ++ map(f,a,b) returns \spad{c}, where \spad{c(i,j) = f(a(i,j),b(i,j))}
      ++ for all \spad{i, j}
    map:((R,R) -> R,%,%,R) -> %
      ++ map(f,a,b,r) returns \spad{c}, where \spad{c(i,j) = f(a(i,j),b(i,j))} when both
      ++ \spad{a(i,j)} and \spad{b(i,j)} exist;
      ++ else \spad{c(i,j) = f(r, b(i,j))} when \spad{a(i,j)} does not exist;
      ++ else \spad{c(i,j) = f(a(i,j),r)} when \spad{b(i,j)} does not exist;
      ++ otherwise \spad{c(i,j) = f(r,r)}.

   add

--% Predicates

    any?(f,m) ==
      for i in minRowIndex(m)..maxRowIndex(m) repeat
        for j in minColIndex(m)..maxColIndex(m) repeat
          f(qelt(m,i,j)) => return true
      false

    every?(f,m) ==
      for i in minRowIndex(m)..maxRowIndex(m) repeat
        for j in minColIndex(m)..maxColIndex(m) repeat
          not f(qelt(m,i,j)) => return false
      true

    size?(m,n) == nrows(m) * ncols(m) = n
    less?(m,n) == nrows(m) * ncols(m) < n
    more?(m,n) == nrows(m) * ncols(m) > n

--% Size inquiries

    # m == nrows(m) * ncols(m)

--% Part extractions

    elt(m,i,j,r) ==
      i < minRowIndex(m) or i > maxRowIndex(m) => r
      j < minColIndex(m) or j > maxColIndex(m) => r
      qelt(m,i,j)

    count(f:R -> Boolean,m:%) ==
      num : NonNegativeInteger := 0
      for i in minRowIndex(m)..maxRowIndex(m) repeat
        for j in minColIndex(m)..maxColIndex(m) repeat
          if f(qelt(m,i,j)) then num := num + 1
      num

    parts m ==
      entryList : List R := nil()
      for i in maxRowIndex(m)..minRowIndex(m) by -1 repeat
        for j in maxColIndex(m)..minColIndex(m) by -1 repeat
          entryList := concat(qelt(m,i,j),entryList)
      entryList

--% Creation
    -- array creation requires an initial element used to
    -- populate the array.  This is a best effort attempt
    -- to supply such element, when semantics permits.
    sampleElement(): R ==
      R has sample: () -> R => sample()$R
      NIL$Lisp    -- better obfuscation welcome.

    copy m ==
      ans := new(nrows m,ncols m,sampleElement())
      for i in minRowIndex(m)..maxRowIndex(m) repeat
        for j in minColIndex(m)..maxColIndex(m) repeat
          qsetelt!(ans,i,j,qelt(m,i,j))
      ans

    fill!(m,r) ==
      for i in minRowIndex(m)..maxRowIndex(m) repeat
        for j in minColIndex(m)..maxColIndex(m) repeat
          qsetelt!(m,i,j,r)
      m

    map(f,m) ==
      ans := new(nrows m,ncols m,sampleElement())
      for i in minRowIndex(m)..maxRowIndex(m) repeat
        for j in minColIndex(m)..maxColIndex(m) repeat
          qsetelt!(ans,i,j,f(qelt(m,i,j)))
      ans

    map!(f,m) ==
      for i in minRowIndex(m)..maxRowIndex(m) repeat
        for j in minColIndex(m)..maxColIndex(m) repeat
          qsetelt!(m,i,j,f(qelt(m,i,j)))
      m

    map(f,m,n) ==
      (nrows(m) ~= nrows(n)) or (ncols(m) ~= ncols(n)) =>
        error "map: arguments must have same dimensions"
      ans := new(nrows m,ncols m,sampleElement())
      for i in minRowIndex(m)..maxRowIndex(m) repeat
        for j in minColIndex(m)..maxColIndex(m) repeat
          qsetelt!(ans,i,j,f(qelt(m,i,j),qelt(n,i,j)))
      ans

    map(f,m,n,r) ==
      maxRow := max(maxRowIndex m,maxRowIndex n)
      maxCol := max(maxColIndex m,maxColIndex n)
      ans := new(max(nrows m,nrows n),max(ncols m,ncols n),sampleElement())
      for i in minRowIndex(m)..maxRow repeat
        for j in minColIndex(m)..maxCol repeat
          qsetelt!(ans,i,j,f(elt(m,i,j,r),elt(n,i,j,r)))
      ans

    setRow!(m,i,v) ==
      i < minRowIndex(m) or i > maxRowIndex(m) =>
        error "setRow!: index out of range"
      for j in minColIndex(m)..maxColIndex(m) _
        for k in minIndex(v)..maxIndex(v) repeat
          qsetelt!(m,i,j,v.k)
      m

    setColumn!(m,j,v) ==
      j < minColIndex(m) or j > maxColIndex(m) =>
        error "setColumn!: index out of range"
      for i in minRowIndex(m)..maxRowIndex(m) _
        for k in minIndex(v)..maxIndex(v) repeat
          qsetelt!(m,i,j,v.k)
      m

    if R has _= : (R,R) -> Boolean then

      m = n ==
        eq?(m,n) => true
        (nrows(m) ~= nrows(n)) or (ncols(m) ~= ncols(n)) => false
        for i in minRowIndex(m)..maxRowIndex(m) repeat
          for j in minColIndex(m)..maxColIndex(m) repeat
            not (qelt(m,i,j) = qelt(n,i,j)) => return false
        true

      member?(r,m) ==
        for i in minRowIndex(m)..maxRowIndex(m) repeat
          for j in minColIndex(m)..maxColIndex(m) repeat
            qelt(m,i,j) = r => return true
        false

      count(r:R,m:%) == count(#1 = r,m)

    if Row has shallowlyMutable then

      row(m,i) ==
        i < minRowIndex(m) or i > maxRowIndex(m) =>
          error "row: index out of range"
        v : Row := new(ncols m,sampleElement())
        for j in minColIndex(m)..maxColIndex(m) _
          for k in minIndex(v)..maxIndex(v) repeat
            qsetelt!(v,k,qelt(m,i,j))
        v

    if Col has shallowlyMutable then

      column(m,j) ==
        j < minColIndex(m) or j > maxColIndex(m) =>
          error "column: index out of range"
        v : Col := new(nrows m,sampleElement())
        for i in minRowIndex(m)..maxRowIndex(m) _
          for k in minIndex(v)..maxIndex(v) repeat
            qsetelt!(v,k,qelt(m,i,j))
        v

    if R has CoercibleTo(OutputForm) then

      coerce(m:%) ==
        l : List List OutputForm
        l := [[qelt(m,i,j) :: OutputForm _
                  for j in minColIndex(m)..maxColIndex(m)] _
                  for i in minRowIndex(m)..maxRowIndex(m)]
        matrix l

@
\section{domain IIARRAY2 InnerIndexedTwoDimensionalArray}
<<domain IIARRAY2 InnerIndexedTwoDimensionalArray>>=
)abbrev domain IIARRAY2 InnerIndexedTwoDimensionalArray
InnerIndexedTwoDimensionalArray(R,mnRow,mnCol,Row,Col):_
       Exports == Implementation where
  ++ This is an internal type which provides an implementation of
  ++ 2-dimensional arrays as PrimitiveArray's of PrimitiveArray's.
  R : Type
  mnRow, mnCol : Integer
  Row : FiniteLinearAggregate R
  Col : FiniteLinearAggregate R

  Exports ==> TwoDimensionalArrayCategory(R,Row,Col)

  Implementation ==> add

    Rep := PrimitiveArray PrimitiveArray R

--% Predicates

    empty? m == empty?(m)$Rep

--% Primitive array creation

    empty() == empty()$Rep

    new(rows,cols,a) ==
      rows = 0 =>
        error "new: arrays with zero rows are not supported"
--      cols = 0 =>
--        error "new: arrays with zero columns are not supported"
      arr : PrimitiveArray PrimitiveArray R := new(rows,empty())
      for i in minIndex(arr)..maxIndex(arr) repeat
        qsetelt!(arr,i,new(cols,a))
      arr

--% Size inquiries

    minRowIndex m == mnRow
    minColIndex m == mnCol
    maxRowIndex m == nrows m + mnRow - 1
    maxColIndex m == ncols m + mnCol - 1

    nrows m == (# m)$Rep

    ncols m ==
      empty? m => 0
      # m(minIndex(m)$Rep)

--% Part selection/assignment

    qelt(m,i,j) ==
      qelt(qelt(m,i - minRowIndex m)$Rep,j - minColIndex m)

    elt(m:%,i:Integer,j:Integer) ==
      i < minRowIndex(m) or i > maxRowIndex(m) =>
        error "elt: index out of range"
      j < minColIndex(m) or j > maxColIndex(m) =>
        error "elt: index out of range"
      qelt(m,i,j)

    qsetelt!(m,i,j,r) ==
      setelt(qelt(m,i - minRowIndex m)$Rep,j - minColIndex m,r)

    setelt(m:%,i:Integer,j:Integer,r:R) ==
      i < minRowIndex(m) or i > maxRowIndex(m) =>
        error "setelt: index out of range"
      j < minColIndex(m) or j > maxColIndex(m) =>
        error "setelt: index out of range"
      qsetelt!(m,i,j,r)

    if R has SetCategory then
        latex(m : %) : String ==
          s : String := "\left[ \begin{array}{"
          for j in minColIndex(m)..maxColIndex(m) repeat
            s := concat(s,"c")$String
          s := concat(s,"} ")$String
          for i in minRowIndex(m)..maxRowIndex(m) repeat
            for j in minColIndex(m)..maxColIndex(m) repeat
              s := concat(s, latex(qelt(m,i,j))$R)$String
              if j < maxColIndex(m) then s := concat(s, " & ")$String
            if i < maxRowIndex(m) then s := concat(s, " \\ ")$String
          concat(s, "\end{array} \right]")$String

@
\section{domain IARRAY2 IndexedTwoDimensionalArray}
<<domain IARRAY2 IndexedTwoDimensionalArray>>=
)abbrev domain IARRAY2 IndexedTwoDimensionalArray
IndexedTwoDimensionalArray(R,mnRow,mnCol):Exports == Implementation where
  ++ An IndexedTwoDimensionalArray is a 2-dimensional array where
  ++ the minimal row and column indices are parameters of the type.
  ++ Rows and columns are returned as IndexedOneDimensionalArray's with
  ++ minimal indices matching those of the IndexedTwoDimensionalArray.
  ++ The index of the 'first' row may be obtained by calling the
  ++ function 'minRowIndex'.  The index of the 'first' column may
  ++ be obtained by calling the function 'minColIndex'.  The index of
  ++ the first element of a 'Row' is the same as the index of the
  ++ first column in an array and vice versa.
  R : Type
  mnRow, mnCol : Integer
  Row ==> IndexedOneDimensionalArray(R,mnCol)
  Col ==> IndexedOneDimensionalArray(R,mnRow)

  Exports ==> TwoDimensionalArrayCategory(R,Row,Col)

  Implementation ==>
    InnerIndexedTwoDimensionalArray(R,mnRow,mnCol,Row,Col)

@
\section{domain ARRAY2 TwoDimensionalArray}
<<domain ARRAY2 TwoDimensionalArray>>=
)abbrev domain ARRAY2 TwoDimensionalArray
TwoDimensionalArray(R):Exports == Implementation where
  ++ A TwoDimensionalArray is a two dimensional array with
  ++ 1-based indexing for both rows and columns.
  R : Type
  Row ==> OneDimensionalArray R
  Col ==> OneDimensionalArray R

  Exports ==> TwoDimensionalArrayCategory(R,Row,Col) with
    shallowlyMutable
      ++ One may destructively alter TwoDimensionalArray's.

  Implementation ==> InnerIndexedTwoDimensionalArray(R,1,1,Row,Col)

@
\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>>

<<category ARR2CAT TwoDimensionalArrayCategory>>
<<domain IIARRAY2 InnerIndexedTwoDimensionalArray>>
<<domain IARRAY2 IndexedTwoDimensionalArray>>
<<domain ARRAY2 TwoDimensionalArray>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}