|
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 |
SUBROUTINE CLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U,
$ LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, $ GIVCOL, LDGCOL, PERM, GIVNUM, C, S, RWORK, $ IWORK, INFO ) * * -- LAPACK routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS, $ SMLSIZ * .. * .. Array Arguments .. INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), $ K( * ), PERM( LDGCOL, * ) REAL C( * ), DIFL( LDU, * ), DIFR( LDU, * ), $ GIVNUM( LDU, * ), POLES( LDU, * ), RWORK( * ), $ S( * ), U( LDU, * ), VT( LDU, * ), Z( LDU, * ) COMPLEX B( LDB, * ), BX( LDBX, * ) * .. * * Purpose * ======= * * CLALSA is an itermediate step in solving the least squares problem * by computing the SVD of the coefficient matrix in compact form (The * singular vectors are computed as products of simple orthorgonal * matrices.). * * If ICOMPQ = 0, CLALSA applies the inverse of the left singular vector * matrix of an upper bidiagonal matrix to the right hand side; and if * ICOMPQ = 1, CLALSA applies the right singular vector matrix to the * right hand side. The singular vector matrices were generated in * compact form by CLALSA. * * Arguments * ========= * * ICOMPQ (input) INTEGER * Specifies whether the left or the right singular vector * matrix is involved. * = 0: Left singular vector matrix * = 1: Right singular vector matrix * * SMLSIZ (input) INTEGER * The maximum size of the subproblems at the bottom of the * computation tree. * * N (input) INTEGER * The row and column dimensions of the upper bidiagonal matrix. * * NRHS (input) INTEGER * The number of columns of B and BX. NRHS must be at least 1. * * B (input/output) COMPLEX array, dimension ( LDB, NRHS ) * On input, B contains the right hand sides of the least * squares problem in rows 1 through M. * On output, B contains the solution X in rows 1 through N. * * LDB (input) INTEGER * The leading dimension of B in the calling subprogram. * LDB must be at least max(1,MAX( M, N ) ). * * BX (output) COMPLEX array, dimension ( LDBX, NRHS ) * On exit, the result of applying the left or right singular * vector matrix to B. * * LDBX (input) INTEGER * The leading dimension of BX. * * U (input) REAL array, dimension ( LDU, SMLSIZ ). * On entry, U contains the left singular vector matrices of all * subproblems at the bottom level. * * LDU (input) INTEGER, LDU = > N. * The leading dimension of arrays U, VT, DIFL, DIFR, * POLES, GIVNUM, and Z. * * VT (input) REAL array, dimension ( LDU, SMLSIZ+1 ). * On entry, VT**H contains the right singular vector matrices of * all subproblems at the bottom level. * * K (input) INTEGER array, dimension ( N ). * * DIFL (input) REAL array, dimension ( LDU, NLVL ). * where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1. * * DIFR (input) REAL array, dimension ( LDU, 2 * NLVL ). * On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record * distances between singular values on the I-th level and * singular values on the (I -1)-th level, and DIFR(*, 2 * I) * record the normalizing factors of the right singular vectors * matrices of subproblems on I-th level. * * Z (input) REAL array, dimension ( LDU, NLVL ). * On entry, Z(1, I) contains the components of the deflation- * adjusted updating row vector for subproblems on the I-th * level. * * POLES (input) REAL array, dimension ( LDU, 2 * NLVL ). * On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old * singular values involved in the secular equations on the I-th * level. * * GIVPTR (input) INTEGER array, dimension ( N ). * On entry, GIVPTR( I ) records the number of Givens * rotations performed on the I-th problem on the computation * tree. * * GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 * NLVL ). * On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the * locations of Givens rotations performed on the I-th level on * the computation tree. * * LDGCOL (input) INTEGER, LDGCOL = > N. * The leading dimension of arrays GIVCOL and PERM. * * PERM (input) INTEGER array, dimension ( LDGCOL, NLVL ). * On entry, PERM(*, I) records permutations done on the I-th * level of the computation tree. * * GIVNUM (input) REAL array, dimension ( LDU, 2 * NLVL ). * On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S- * values of Givens rotations performed on the I-th level on the * computation tree. * * C (input) REAL array, dimension ( N ). * On entry, if the I-th subproblem is not square, * C( I ) contains the C-value of a Givens rotation related to * the right null space of the I-th subproblem. * * S (input) REAL array, dimension ( N ). * On entry, if the I-th subproblem is not square, * S( I ) contains the S-value of a Givens rotation related to * the right null space of the I-th subproblem. * * RWORK (workspace) REAL array, dimension at least * MAX( (SMLSZ+1)*NRHS*3, N*(1+NRHS) + 2*NRHS ). * * IWORK (workspace) INTEGER array. * The dimension must be at least 3 * N * * INFO (output) INTEGER * = 0: successful exit. * < 0: if INFO = -i, the i-th argument had an illegal value. * * Further Details * =============== * * Based on contributions by * Ming Gu and Ren-Cang Li, Computer Science Division, University of * California at Berkeley, USA * Osni Marques, LBNL/NERSC, USA * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) * .. * .. Local Scalars .. INTEGER I, I1, IC, IM1, INODE, J, JCOL, JIMAG, JREAL, $ JROW, LF, LL, LVL, LVL2, ND, NDB1, NDIML, $ NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, SQRE * .. * .. External Subroutines .. EXTERNAL CCOPY, CLALS0, SGEMM, SLASDT, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC AIMAG, CMPLX, REAL * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 * IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN INFO = -1 ELSE IF( SMLSIZ.LT.3 ) THEN INFO = -2 ELSE IF( N.LT.SMLSIZ ) THEN INFO = -3 ELSE IF( NRHS.LT.1 ) THEN INFO = -4 ELSE IF( LDB.LT.N ) THEN INFO = -6 ELSE IF( LDBX.LT.N ) THEN INFO = -8 ELSE IF( LDU.LT.N ) THEN INFO = -10 ELSE IF( LDGCOL.LT.N ) THEN INFO = -19 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'CLALSA', -INFO ) RETURN END IF * * Book-keeping and setting up the computation tree. * INODE = 1 NDIML = INODE + N NDIMR = NDIML + N * CALL SLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ), $ IWORK( NDIMR ), SMLSIZ ) * * The following code applies back the left singular vector factors. * For applying back the right singular vector factors, go to 170. * IF( ICOMPQ.EQ.1 ) THEN GO TO 170 END IF * * The nodes on the bottom level of the tree were solved * by SLASDQ. The corresponding left and right singular vector * matrices are in explicit form. First apply back the left * singular vector matrices. * NDB1 = ( ND+1 ) / 2 DO 130 I = NDB1, ND * * IC : center row of each node * NL : number of rows of left subproblem * NR : number of rows of right subproblem * NLF: starting row of the left subproblem * NRF: starting row of the right subproblem * I1 = I - 1 IC = IWORK( INODE+I1 ) NL = IWORK( NDIML+I1 ) NR = IWORK( NDIMR+I1 ) NLF = IC - NL NRF = IC + 1 * * Since B and BX are complex, the following call to SGEMM * is performed in two steps (real and imaginary parts). * * CALL SGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU, * $ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX ) * J = NL*NRHS*2 DO 20 JCOL = 1, NRHS DO 10 JROW = NLF, NLF + NL - 1 J = J + 1 RWORK( J ) = REAL( B( JROW, JCOL ) ) 10 CONTINUE 20 CONTINUE CALL SGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU, $ RWORK( 1+NL*NRHS*2 ), NL, ZERO, RWORK( 1 ), NL ) J = NL*NRHS*2 DO 40 JCOL = 1, NRHS DO 30 JROW = NLF, NLF + NL - 1 J = J + 1 RWORK( J ) = AIMAG( B( JROW, JCOL ) ) 30 CONTINUE 40 CONTINUE CALL SGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU, $ RWORK( 1+NL*NRHS*2 ), NL, ZERO, RWORK( 1+NL*NRHS ), $ NL ) JREAL = 0 JIMAG = NL*NRHS DO 60 JCOL = 1, NRHS DO 50 JROW = NLF, NLF + NL - 1 JREAL = JREAL + 1 JIMAG = JIMAG + 1 BX( JROW, JCOL ) = CMPLX( RWORK( JREAL ), $ RWORK( JIMAG ) ) 50 CONTINUE 60 CONTINUE * * Since B and BX are complex, the following call to SGEMM * is performed in two steps (real and imaginary parts). * * CALL SGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU, * $ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX ) * J = NR*NRHS*2 DO 80 JCOL = 1, NRHS DO 70 JROW = NRF, NRF + NR - 1 J = J + 1 RWORK( J ) = REAL( B( JROW, JCOL ) ) 70 CONTINUE 80 CONTINUE CALL SGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU, $ RWORK( 1+NR*NRHS*2 ), NR, ZERO, RWORK( 1 ), NR ) J = NR*NRHS*2 DO 100 JCOL = 1, NRHS DO 90 JROW = NRF, NRF + NR - 1 J = J + 1 RWORK( J ) = AIMAG( B( JROW, JCOL ) ) 90 CONTINUE 100 CONTINUE CALL SGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU, $ RWORK( 1+NR*NRHS*2 ), NR, ZERO, RWORK( 1+NR*NRHS ), $ NR ) JREAL = 0 JIMAG = NR*NRHS DO 120 JCOL = 1, NRHS DO 110 JROW = NRF, NRF + NR - 1 JREAL = JREAL + 1 JIMAG = JIMAG + 1 BX( JROW, JCOL ) = CMPLX( RWORK( JREAL ), $ RWORK( JIMAG ) ) 110 CONTINUE 120 CONTINUE * 130 CONTINUE * * Next copy the rows of B that correspond to unchanged rows * in the bidiagonal matrix to BX. * DO 140 I = 1, ND IC = IWORK( INODE+I-1 ) CALL CCOPY( NRHS, B( IC, 1 ), LDB, BX( IC, 1 ), LDBX ) 140 CONTINUE * * Finally go through the left singular vector matrices of all * the other subproblems bottom-up on the tree. * J = 2**NLVL SQRE = 0 * DO 160 LVL = NLVL, 1, -1 LVL2 = 2*LVL - 1 * * find the first node LF and last node LL on * the current level LVL * IF( LVL.EQ.1 ) THEN LF = 1 LL = 1 ELSE LF = 2**( LVL-1 ) LL = 2*LF - 1 END IF DO 150 I = LF, LL IM1 = I - 1 IC = IWORK( INODE+IM1 ) NL = IWORK( NDIML+IM1 ) NR = IWORK( NDIMR+IM1 ) NLF = IC - NL NRF = IC + 1 J = J - 1 CALL CLALS0( ICOMPQ, NL, NR, SQRE, NRHS, BX( NLF, 1 ), LDBX, $ B( NLF, 1 ), LDB, PERM( NLF, LVL ), $ GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL, $ GIVNUM( NLF, LVL2 ), LDU, POLES( NLF, LVL2 ), $ DIFL( NLF, LVL ), DIFR( NLF, LVL2 ), $ Z( NLF, LVL ), K( J ), C( J ), S( J ), RWORK, $ INFO ) 150 CONTINUE 160 CONTINUE GO TO 330 * * ICOMPQ = 1: applying back the right singular vector factors. * 170 CONTINUE * * First now go through the right singular vector matrices of all * the tree nodes top-down. * J = 0 DO 190 LVL = 1, NLVL LVL2 = 2*LVL - 1 * * Find the first node LF and last node LL on * the current level LVL. * IF( LVL.EQ.1 ) THEN LF = 1 LL = 1 ELSE LF = 2**( LVL-1 ) LL = 2*LF - 1 END IF DO 180 I = LL, LF, -1 IM1 = I - 1 IC = IWORK( INODE+IM1 ) NL = IWORK( NDIML+IM1 ) NR = IWORK( NDIMR+IM1 ) NLF = IC - NL NRF = IC + 1 IF( I.EQ.LL ) THEN SQRE = 0 ELSE SQRE = 1 END IF J = J + 1 CALL CLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B( NLF, 1 ), LDB, $ BX( NLF, 1 ), LDBX, PERM( NLF, LVL ), $ GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL, $ GIVNUM( NLF, LVL2 ), LDU, POLES( NLF, LVL2 ), $ DIFL( NLF, LVL ), DIFR( NLF, LVL2 ), $ Z( NLF, LVL ), K( J ), C( J ), S( J ), RWORK, $ INFO ) 180 CONTINUE 190 CONTINUE * * The nodes on the bottom level of the tree were solved * by SLASDQ. The corresponding right singular vector * matrices are in explicit form. Apply them back. * NDB1 = ( ND+1 ) / 2 DO 320 I = NDB1, ND I1 = I - 1 IC = IWORK( INODE+I1 ) NL = IWORK( NDIML+I1 ) NR = IWORK( NDIMR+I1 ) NLP1 = NL + 1 IF( I.EQ.ND ) THEN NRP1 = NR ELSE NRP1 = NR + 1 END IF NLF = IC - NL NRF = IC + 1 * * Since B and BX are complex, the following call to SGEMM is * performed in two steps (real and imaginary parts). * * CALL SGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU, * $ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX ) * J = NLP1*NRHS*2 DO 210 JCOL = 1, NRHS DO 200 JROW = NLF, NLF + NLP1 - 1 J = J + 1 RWORK( J ) = REAL( B( JROW, JCOL ) ) 200 CONTINUE 210 CONTINUE CALL SGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU, $ RWORK( 1+NLP1*NRHS*2 ), NLP1, ZERO, RWORK( 1 ), $ NLP1 ) J = NLP1*NRHS*2 DO 230 JCOL = 1, NRHS DO 220 JROW = NLF, NLF + NLP1 - 1 J = J + 1 RWORK( J ) = AIMAG( B( JROW, JCOL ) ) 220 CONTINUE 230 CONTINUE CALL SGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU, $ RWORK( 1+NLP1*NRHS*2 ), NLP1, ZERO, $ RWORK( 1+NLP1*NRHS ), NLP1 ) JREAL = 0 JIMAG = NLP1*NRHS DO 250 JCOL = 1, NRHS DO 240 JROW = NLF, NLF + NLP1 - 1 JREAL = JREAL + 1 JIMAG = JIMAG + 1 BX( JROW, JCOL ) = CMPLX( RWORK( JREAL ), $ RWORK( JIMAG ) ) 240 CONTINUE 250 CONTINUE * * Since B and BX are complex, the following call to SGEMM is * performed in two steps (real and imaginary parts). * * CALL SGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU, * $ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX ) * J = NRP1*NRHS*2 DO 270 JCOL = 1, NRHS DO 260 JROW = NRF, NRF + NRP1 - 1 J = J + 1 RWORK( J ) = REAL( B( JROW, JCOL ) ) 260 CONTINUE 270 CONTINUE CALL SGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU, $ RWORK( 1+NRP1*NRHS*2 ), NRP1, ZERO, RWORK( 1 ), $ NRP1 ) J = NRP1*NRHS*2 DO 290 JCOL = 1, NRHS DO 280 JROW = NRF, NRF + NRP1 - 1 J = J + 1 RWORK( J ) = AIMAG( B( JROW, JCOL ) ) 280 CONTINUE 290 CONTINUE CALL SGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU, $ RWORK( 1+NRP1*NRHS*2 ), NRP1, ZERO, $ RWORK( 1+NRP1*NRHS ), NRP1 ) JREAL = 0 JIMAG = NRP1*NRHS DO 310 JCOL = 1, NRHS DO 300 JROW = NRF, NRF + NRP1 - 1 JREAL = JREAL + 1 JIMAG = JIMAG + 1 BX( JROW, JCOL ) = CMPLX( RWORK( JREAL ), $ RWORK( JIMAG ) ) 300 CONTINUE 310 CONTINUE * 320 CONTINUE * 330 CONTINUE * RETURN * * End of CLALSA * END |