]>
Commit | Line | Data |
---|---|---|
98975653 | 1 | /* Integer matrix math routines |
66647d44 | 2 | Copyright (C) 2003, 2004, 2005, 2007, 2008 Free Software Foundation, Inc. |
98975653 DB |
3 | Contributed by Daniel Berlin <dberlin@dberlin.org>. |
4 | ||
5 | This file is part of GCC. | |
6 | ||
7 | GCC is free software; you can redistribute it and/or modify it under | |
8 | the terms of the GNU General Public License as published by the Free | |
9dcd6f09 | 9 | Software Foundation; either version 3, or (at your option) any later |
98975653 DB |
10 | version. |
11 | ||
12 | GCC is distributed in the hope that it will be useful, but WITHOUT ANY | |
13 | WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
14 | FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
15 | for more details. | |
16 | ||
17 | You should have received a copy of the GNU General Public License | |
9dcd6f09 NC |
18 | along with GCC; see the file COPYING3. If not see |
19 | <http://www.gnu.org/licenses/>. */ | |
20 | ||
98975653 DB |
21 | #include "config.h" |
22 | #include "system.h" | |
23 | #include "coretypes.h" | |
726a989a | 24 | #include "tree-flow.h" |
98975653 DB |
25 | #include "lambda.h" |
26 | ||
98975653 DB |
27 | /* Allocate a matrix of M rows x N cols. */ |
28 | ||
29 | lambda_matrix | |
f873b205 | 30 | lambda_matrix_new (int m, int n, struct obstack * lambda_obstack) |
98975653 DB |
31 | { |
32 | lambda_matrix mat; | |
33 | int i; | |
34 | ||
f873b205 LB |
35 | mat = (lambda_matrix) obstack_alloc (lambda_obstack, |
36 | sizeof (lambda_vector *) * m); | |
b8698a0f | 37 | |
98975653 DB |
38 | for (i = 0; i < m; i++) |
39 | mat[i] = lambda_vector_new (n); | |
40 | ||
41 | return mat; | |
42 | } | |
43 | ||
44 | /* Copy the elements of M x N matrix MAT1 to MAT2. */ | |
45 | ||
46 | void | |
47 | lambda_matrix_copy (lambda_matrix mat1, lambda_matrix mat2, | |
48 | int m, int n) | |
49 | { | |
50 | int i; | |
51 | ||
52 | for (i = 0; i < m; i++) | |
53 | lambda_vector_copy (mat1[i], mat2[i], n); | |
54 | } | |
55 | ||
56 | /* Store the N x N identity matrix in MAT. */ | |
57 | ||
58 | void | |
59 | lambda_matrix_id (lambda_matrix mat, int size) | |
60 | { | |
61 | int i, j; | |
62 | ||
63 | for (i = 0; i < size; i++) | |
64 | for (j = 0; j < size; j++) | |
65 | mat[i][j] = (i == j) ? 1 : 0; | |
66 | } | |
67 | ||
f67d92e9 DB |
68 | /* Return true if MAT is the identity matrix of SIZE */ |
69 | ||
70 | bool | |
71 | lambda_matrix_id_p (lambda_matrix mat, int size) | |
72 | { | |
73 | int i, j; | |
74 | for (i = 0; i < size; i++) | |
75 | for (j = 0; j < size; j++) | |
76 | { | |
77 | if (i == j) | |
78 | { | |
79 | if (mat[i][j] != 1) | |
80 | return false; | |
81 | } | |
82 | else | |
83 | { | |
84 | if (mat[i][j] != 0) | |
85 | return false; | |
86 | } | |
87 | } | |
88 | return true; | |
89 | } | |
90 | ||
98975653 DB |
91 | /* Negate the elements of the M x N matrix MAT1 and store it in MAT2. */ |
92 | ||
93 | void | |
94 | lambda_matrix_negate (lambda_matrix mat1, lambda_matrix mat2, int m, int n) | |
95 | { | |
96 | int i; | |
97 | ||
98 | for (i = 0; i < m; i++) | |
99 | lambda_vector_negate (mat1[i], mat2[i], n); | |
100 | } | |
101 | ||
102 | /* Take the transpose of matrix MAT1 and store it in MAT2. | |
103 | MAT1 is an M x N matrix, so MAT2 must be N x M. */ | |
104 | ||
105 | void | |
106 | lambda_matrix_transpose (lambda_matrix mat1, lambda_matrix mat2, int m, int n) | |
107 | { | |
108 | int i, j; | |
109 | ||
110 | for (i = 0; i < n; i++) | |
111 | for (j = 0; j < m; j++) | |
112 | mat2[i][j] = mat1[j][i]; | |
113 | } | |
114 | ||
115 | ||
116 | /* Add two M x N matrices together: MAT3 = MAT1+MAT2. */ | |
117 | ||
118 | void | |
119 | lambda_matrix_add (lambda_matrix mat1, lambda_matrix mat2, | |
120 | lambda_matrix mat3, int m, int n) | |
121 | { | |
122 | int i; | |
123 | ||
124 | for (i = 0; i < m; i++) | |
125 | lambda_vector_add (mat1[i], mat2[i], mat3[i], n); | |
126 | } | |
127 | ||
128 | /* MAT3 = CONST1 * MAT1 + CONST2 * MAT2. All matrices are M x N. */ | |
129 | ||
130 | void | |
131 | lambda_matrix_add_mc (lambda_matrix mat1, int const1, | |
132 | lambda_matrix mat2, int const2, | |
133 | lambda_matrix mat3, int m, int n) | |
134 | { | |
135 | int i; | |
136 | ||
137 | for (i = 0; i < m; i++) | |
138 | lambda_vector_add_mc (mat1[i], const1, mat2[i], const2, mat3[i], n); | |
139 | } | |
140 | ||
141 | /* Multiply two matrices: MAT3 = MAT1 * MAT2. | |
142 | MAT1 is an M x R matrix, and MAT2 is R x N. The resulting MAT2 | |
143 | must therefore be M x N. */ | |
144 | ||
145 | void | |
146 | lambda_matrix_mult (lambda_matrix mat1, lambda_matrix mat2, | |
147 | lambda_matrix mat3, int m, int r, int n) | |
148 | { | |
149 | ||
150 | int i, j, k; | |
151 | ||
152 | for (i = 0; i < m; i++) | |
153 | { | |
154 | for (j = 0; j < n; j++) | |
155 | { | |
156 | mat3[i][j] = 0; | |
157 | for (k = 0; k < r; k++) | |
158 | mat3[i][j] += mat1[i][k] * mat2[k][j]; | |
159 | } | |
160 | } | |
161 | } | |
162 | ||
8c27b7d4 | 163 | /* Delete rows r1 to r2 (not including r2). */ |
98975653 DB |
164 | |
165 | void | |
166 | lambda_matrix_delete_rows (lambda_matrix mat, int rows, int from, int to) | |
167 | { | |
168 | int i; | |
169 | int dist; | |
170 | dist = to - from; | |
171 | ||
172 | for (i = to; i < rows; i++) | |
173 | mat[i - dist] = mat[i]; | |
174 | ||
175 | for (i = rows - dist; i < rows; i++) | |
176 | mat[i] = NULL; | |
177 | } | |
178 | ||
179 | /* Swap rows R1 and R2 in matrix MAT. */ | |
180 | ||
181 | void | |
182 | lambda_matrix_row_exchange (lambda_matrix mat, int r1, int r2) | |
183 | { | |
184 | lambda_vector row; | |
185 | ||
186 | row = mat[r1]; | |
187 | mat[r1] = mat[r2]; | |
188 | mat[r2] = row; | |
189 | } | |
190 | ||
191 | /* Add a multiple of row R1 of matrix MAT with N columns to row R2: | |
192 | R2 = R2 + CONST1 * R1. */ | |
193 | ||
194 | void | |
195 | lambda_matrix_row_add (lambda_matrix mat, int n, int r1, int r2, int const1) | |
196 | { | |
197 | int i; | |
198 | ||
199 | if (const1 == 0) | |
200 | return; | |
201 | ||
202 | for (i = 0; i < n; i++) | |
203 | mat[r2][i] += const1 * mat[r1][i]; | |
204 | } | |
205 | ||
206 | /* Negate row R1 of matrix MAT which has N columns. */ | |
207 | ||
208 | void | |
209 | lambda_matrix_row_negate (lambda_matrix mat, int n, int r1) | |
210 | { | |
211 | lambda_vector_negate (mat[r1], mat[r1], n); | |
212 | } | |
213 | ||
214 | /* Multiply row R1 of matrix MAT with N columns by CONST1. */ | |
215 | ||
216 | void | |
217 | lambda_matrix_row_mc (lambda_matrix mat, int n, int r1, int const1) | |
218 | { | |
219 | int i; | |
220 | ||
221 | for (i = 0; i < n; i++) | |
222 | mat[r1][i] *= const1; | |
223 | } | |
224 | ||
225 | /* Exchange COL1 and COL2 in matrix MAT. M is the number of rows. */ | |
226 | ||
227 | void | |
228 | lambda_matrix_col_exchange (lambda_matrix mat, int m, int col1, int col2) | |
229 | { | |
230 | int i; | |
231 | int tmp; | |
232 | for (i = 0; i < m; i++) | |
233 | { | |
234 | tmp = mat[i][col1]; | |
235 | mat[i][col1] = mat[i][col2]; | |
236 | mat[i][col2] = tmp; | |
237 | } | |
238 | } | |
239 | ||
240 | /* Add a multiple of column C1 of matrix MAT with M rows to column C2: | |
241 | C2 = C2 + CONST1 * C1. */ | |
242 | ||
243 | void | |
244 | lambda_matrix_col_add (lambda_matrix mat, int m, int c1, int c2, int const1) | |
245 | { | |
246 | int i; | |
247 | ||
248 | if (const1 == 0) | |
249 | return; | |
250 | ||
251 | for (i = 0; i < m; i++) | |
252 | mat[i][c2] += const1 * mat[i][c1]; | |
253 | } | |
254 | ||
255 | /* Negate column C1 of matrix MAT which has M rows. */ | |
256 | ||
257 | void | |
258 | lambda_matrix_col_negate (lambda_matrix mat, int m, int c1) | |
259 | { | |
260 | int i; | |
261 | ||
262 | for (i = 0; i < m; i++) | |
263 | mat[i][c1] *= -1; | |
264 | } | |
265 | ||
266 | /* Multiply column C1 of matrix MAT with M rows by CONST1. */ | |
267 | ||
268 | void | |
269 | lambda_matrix_col_mc (lambda_matrix mat, int m, int c1, int const1) | |
270 | { | |
271 | int i; | |
272 | ||
273 | for (i = 0; i < m; i++) | |
274 | mat[i][c1] *= const1; | |
275 | } | |
276 | ||
277 | /* Compute the inverse of the N x N matrix MAT and store it in INV. | |
278 | ||
279 | We don't _really_ compute the inverse of MAT. Instead we compute | |
280 | det(MAT)*inv(MAT), and we return det(MAT) to the caller as the function | |
281 | result. This is necessary to preserve accuracy, because we are dealing | |
282 | with integer matrices here. | |
283 | ||
284 | The algorithm used here is a column based Gauss-Jordan elimination on MAT | |
285 | and the identity matrix in parallel. The inverse is the result of applying | |
286 | the same operations on the identity matrix that reduce MAT to the identity | |
287 | matrix. | |
288 | ||
289 | When MAT is a 2 x 2 matrix, we don't go through the whole process, because | |
290 | it is easily inverted by inspection and it is a very common case. */ | |
291 | ||
f873b205 LB |
292 | static int lambda_matrix_inverse_hard (lambda_matrix, lambda_matrix, int, |
293 | struct obstack *); | |
98975653 DB |
294 | |
295 | int | |
f873b205 LB |
296 | lambda_matrix_inverse (lambda_matrix mat, lambda_matrix inv, int n, |
297 | struct obstack * lambda_obstack) | |
98975653 DB |
298 | { |
299 | if (n == 2) | |
300 | { | |
301 | int a, b, c, d, det; | |
302 | a = mat[0][0]; | |
303 | b = mat[1][0]; | |
304 | c = mat[0][1]; | |
b8698a0f | 305 | d = mat[1][1]; |
98975653 DB |
306 | inv[0][0] = d; |
307 | inv[0][1] = -c; | |
308 | inv[1][0] = -b; | |
309 | inv[1][1] = a; | |
310 | det = (a * d - b * c); | |
311 | if (det < 0) | |
312 | { | |
313 | det *= -1; | |
314 | inv[0][0] *= -1; | |
315 | inv[1][0] *= -1; | |
316 | inv[0][1] *= -1; | |
317 | inv[1][1] *= -1; | |
318 | } | |
319 | return det; | |
320 | } | |
321 | else | |
f873b205 | 322 | return lambda_matrix_inverse_hard (mat, inv, n, lambda_obstack); |
98975653 DB |
323 | } |
324 | ||
325 | /* If MAT is not a special case, invert it the hard way. */ | |
326 | ||
327 | static int | |
f873b205 LB |
328 | lambda_matrix_inverse_hard (lambda_matrix mat, lambda_matrix inv, int n, |
329 | struct obstack * lambda_obstack) | |
98975653 DB |
330 | { |
331 | lambda_vector row; | |
332 | lambda_matrix temp; | |
333 | int i, j; | |
334 | int determinant; | |
335 | ||
f873b205 | 336 | temp = lambda_matrix_new (n, n, lambda_obstack); |
98975653 DB |
337 | lambda_matrix_copy (mat, temp, n, n); |
338 | lambda_matrix_id (inv, n); | |
339 | ||
340 | /* Reduce TEMP to a lower triangular form, applying the same operations on | |
341 | INV which starts as the identity matrix. N is the number of rows and | |
342 | columns. */ | |
343 | for (j = 0; j < n; j++) | |
344 | { | |
345 | row = temp[j]; | |
346 | ||
347 | /* Make every element in the current row positive. */ | |
348 | for (i = j; i < n; i++) | |
349 | if (row[i] < 0) | |
350 | { | |
351 | lambda_matrix_col_negate (temp, n, i); | |
352 | lambda_matrix_col_negate (inv, n, i); | |
353 | } | |
354 | ||
355 | /* Sweep the upper triangle. Stop when only the diagonal element in the | |
356 | current row is nonzero. */ | |
357 | while (lambda_vector_first_nz (row, n, j + 1) < n) | |
358 | { | |
359 | int min_col = lambda_vector_min_nz (row, n, j); | |
360 | lambda_matrix_col_exchange (temp, n, j, min_col); | |
361 | lambda_matrix_col_exchange (inv, n, j, min_col); | |
362 | ||
363 | for (i = j + 1; i < n; i++) | |
364 | { | |
365 | int factor; | |
366 | ||
367 | factor = -1 * row[i]; | |
368 | if (row[j] != 1) | |
369 | factor /= row[j]; | |
370 | ||
371 | lambda_matrix_col_add (temp, n, j, i, factor); | |
372 | lambda_matrix_col_add (inv, n, j, i, factor); | |
373 | } | |
374 | } | |
375 | } | |
376 | ||
377 | /* Reduce TEMP from a lower triangular to the identity matrix. Also compute | |
378 | the determinant, which now is simply the product of the elements on the | |
379 | diagonal of TEMP. If one of these elements is 0, the matrix has 0 as an | |
380 | eigenvalue so it is singular and hence not invertible. */ | |
381 | determinant = 1; | |
382 | for (j = n - 1; j >= 0; j--) | |
383 | { | |
384 | int diagonal; | |
385 | ||
386 | row = temp[j]; | |
387 | diagonal = row[j]; | |
388 | ||
41806d92 NS |
389 | /* The matrix must not be singular. */ |
390 | gcc_assert (diagonal); | |
98975653 DB |
391 | |
392 | determinant = determinant * diagonal; | |
393 | ||
394 | /* If the diagonal is not 1, then multiply the each row by the | |
395 | diagonal so that the middle number is now 1, rather than a | |
396 | rational. */ | |
397 | if (diagonal != 1) | |
398 | { | |
399 | for (i = 0; i < j; i++) | |
400 | lambda_matrix_col_mc (inv, n, i, diagonal); | |
401 | for (i = j + 1; i < n; i++) | |
402 | lambda_matrix_col_mc (inv, n, i, diagonal); | |
403 | ||
404 | row[j] = diagonal = 1; | |
405 | } | |
406 | ||
407 | /* Sweep the lower triangle column wise. */ | |
408 | for (i = j - 1; i >= 0; i--) | |
409 | { | |
410 | if (row[i]) | |
411 | { | |
412 | int factor = -row[i]; | |
413 | lambda_matrix_col_add (temp, n, j, i, factor); | |
414 | lambda_matrix_col_add (inv, n, j, i, factor); | |
415 | } | |
416 | ||
417 | } | |
418 | } | |
419 | ||
420 | return determinant; | |
421 | } | |
422 | ||
423 | /* Decompose a N x N matrix MAT to a product of a lower triangular H | |
424 | and a unimodular U matrix such that MAT = H.U. N is the size of | |
425 | the rows of MAT. */ | |
426 | ||
427 | void | |
428 | lambda_matrix_hermite (lambda_matrix mat, int n, | |
429 | lambda_matrix H, lambda_matrix U) | |
430 | { | |
431 | lambda_vector row; | |
432 | int i, j, factor, minimum_col; | |
433 | ||
434 | lambda_matrix_copy (mat, H, n, n); | |
435 | lambda_matrix_id (U, n); | |
436 | ||
437 | for (j = 0; j < n; j++) | |
438 | { | |
439 | row = H[j]; | |
440 | ||
441 | /* Make every element of H[j][j..n] positive. */ | |
442 | for (i = j; i < n; i++) | |
443 | { | |
444 | if (row[i] < 0) | |
445 | { | |
446 | lambda_matrix_col_negate (H, n, i); | |
447 | lambda_vector_negate (U[i], U[i], n); | |
448 | } | |
449 | } | |
450 | ||
8e3c61c5 | 451 | /* Stop when only the diagonal element is nonzero. */ |
98975653 DB |
452 | while (lambda_vector_first_nz (row, n, j + 1) < n) |
453 | { | |
454 | minimum_col = lambda_vector_min_nz (row, n, j); | |
455 | lambda_matrix_col_exchange (H, n, j, minimum_col); | |
456 | lambda_matrix_row_exchange (U, j, minimum_col); | |
457 | ||
458 | for (i = j + 1; i < n; i++) | |
459 | { | |
460 | factor = row[i] / row[j]; | |
461 | lambda_matrix_col_add (H, n, j, i, -1 * factor); | |
462 | lambda_matrix_row_add (U, n, i, j, factor); | |
463 | } | |
464 | } | |
465 | } | |
466 | } | |
467 | ||
468 | /* Given an M x N integer matrix A, this function determines an M x | |
469 | M unimodular matrix U, and an M x N echelon matrix S such that | |
470 | "U.A = S". This decomposition is also known as "right Hermite". | |
b8698a0f | 471 | |
98975653 | 472 | Ref: Algorithm 2.1 page 33 in "Loop Transformations for |
8c27b7d4 | 473 | Restructuring Compilers" Utpal Banerjee. */ |
98975653 DB |
474 | |
475 | void | |
476 | lambda_matrix_right_hermite (lambda_matrix A, int m, int n, | |
477 | lambda_matrix S, lambda_matrix U) | |
478 | { | |
479 | int i, j, i0 = 0; | |
480 | ||
481 | lambda_matrix_copy (A, S, m, n); | |
482 | lambda_matrix_id (U, m); | |
483 | ||
484 | for (j = 0; j < n; j++) | |
485 | { | |
486 | if (lambda_vector_first_nz (S[j], m, i0) < m) | |
487 | { | |
488 | ++i0; | |
489 | for (i = m - 1; i >= i0; i--) | |
490 | { | |
491 | while (S[i][j] != 0) | |
492 | { | |
493 | int sigma, factor, a, b; | |
494 | ||
495 | a = S[i-1][j]; | |
496 | b = S[i][j]; | |
497 | sigma = (a * b < 0) ? -1: 1; | |
498 | a = abs (a); | |
499 | b = abs (b); | |
500 | factor = sigma * (a / b); | |
501 | ||
502 | lambda_matrix_row_add (S, n, i, i-1, -factor); | |
503 | lambda_matrix_row_exchange (S, i, i-1); | |
504 | ||
505 | lambda_matrix_row_add (U, m, i, i-1, -factor); | |
506 | lambda_matrix_row_exchange (U, i, i-1); | |
507 | } | |
508 | } | |
509 | } | |
510 | } | |
511 | } | |
512 | ||
513 | /* Given an M x N integer matrix A, this function determines an M x M | |
514 | unimodular matrix V, and an M x N echelon matrix S such that "A = | |
515 | V.S". This decomposition is also known as "left Hermite". | |
b8698a0f | 516 | |
98975653 | 517 | Ref: Algorithm 2.2 page 36 in "Loop Transformations for |
8c27b7d4 | 518 | Restructuring Compilers" Utpal Banerjee. */ |
98975653 DB |
519 | |
520 | void | |
521 | lambda_matrix_left_hermite (lambda_matrix A, int m, int n, | |
522 | lambda_matrix S, lambda_matrix V) | |
523 | { | |
524 | int i, j, i0 = 0; | |
525 | ||
526 | lambda_matrix_copy (A, S, m, n); | |
527 | lambda_matrix_id (V, m); | |
528 | ||
529 | for (j = 0; j < n; j++) | |
530 | { | |
531 | if (lambda_vector_first_nz (S[j], m, i0) < m) | |
532 | { | |
533 | ++i0; | |
534 | for (i = m - 1; i >= i0; i--) | |
535 | { | |
536 | while (S[i][j] != 0) | |
537 | { | |
538 | int sigma, factor, a, b; | |
539 | ||
540 | a = S[i-1][j]; | |
541 | b = S[i][j]; | |
542 | sigma = (a * b < 0) ? -1: 1; | |
543 | a = abs (a); | |
544 | b = abs (b); | |
545 | factor = sigma * (a / b); | |
546 | ||
547 | lambda_matrix_row_add (S, n, i, i-1, -factor); | |
548 | lambda_matrix_row_exchange (S, i, i-1); | |
549 | ||
550 | lambda_matrix_col_add (V, m, i-1, i, factor); | |
551 | lambda_matrix_col_exchange (V, m, i, i-1); | |
552 | } | |
553 | } | |
554 | } | |
555 | } | |
556 | } | |
557 | ||
8e3c61c5 | 558 | /* When it exists, return the first nonzero row in MAT after row |
98975653 DB |
559 | STARTROW. Otherwise return rowsize. */ |
560 | ||
561 | int | |
562 | lambda_matrix_first_nz_vec (lambda_matrix mat, int rowsize, int colsize, | |
563 | int startrow) | |
564 | { | |
565 | int j; | |
566 | bool found = false; | |
567 | ||
568 | for (j = startrow; (j < rowsize) && !found; j++) | |
569 | { | |
570 | if ((mat[j] != NULL) | |
571 | && (lambda_vector_first_nz (mat[j], colsize, startrow) < colsize)) | |
572 | found = true; | |
573 | } | |
574 | ||
575 | if (found) | |
576 | return j - 1; | |
577 | return rowsize; | |
578 | } | |
579 | ||
98975653 DB |
580 | /* Multiply a vector VEC by a matrix MAT. |
581 | MAT is an M*N matrix, and VEC is a vector with length N. The result | |
582 | is stored in DEST which must be a vector of length M. */ | |
583 | ||
584 | void | |
585 | lambda_matrix_vector_mult (lambda_matrix matrix, int m, int n, | |
586 | lambda_vector vec, lambda_vector dest) | |
587 | { | |
588 | int i, j; | |
589 | ||
590 | lambda_vector_clear (dest, m); | |
591 | for (i = 0; i < m; i++) | |
592 | for (j = 0; j < n; j++) | |
593 | dest[i] += matrix[i][j] * vec[j]; | |
594 | } | |
595 | ||
596 | /* Print out an M x N matrix MAT to OUTFILE. */ | |
597 | ||
598 | void | |
599 | print_lambda_matrix (FILE * outfile, lambda_matrix matrix, int m, int n) | |
600 | { | |
601 | int i; | |
602 | ||
603 | for (i = 0; i < m; i++) | |
604 | print_lambda_vector (outfile, matrix[i], n); | |
605 | fprintf (outfile, "\n"); | |
606 | } | |
607 |