![]() |
LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
|
00001 *> \brief <b> ZGELSY solves overdetermined or underdetermined systems for GE matrices</b> 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download ZGELSY + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgelsy.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgelsy.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgelsy.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE ZGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, 00022 * WORK, LWORK, RWORK, INFO ) 00023 * 00024 * .. Scalar Arguments .. 00025 * INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK 00026 * DOUBLE PRECISION RCOND 00027 * .. 00028 * .. Array Arguments .. 00029 * INTEGER JPVT( * ) 00030 * DOUBLE PRECISION RWORK( * ) 00031 * COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ) 00032 * .. 00033 * 00034 * 00035 *> \par Purpose: 00036 * ============= 00037 *> 00038 *> \verbatim 00039 *> 00040 *> ZGELSY computes the minimum-norm solution to a complex linear least 00041 *> squares problem: 00042 *> minimize || A * X - B || 00043 *> using a complete orthogonal factorization of A. A is an M-by-N 00044 *> matrix which may be rank-deficient. 00045 *> 00046 *> Several right hand side vectors b and solution vectors x can be 00047 *> handled in a single call; they are stored as the columns of the 00048 *> M-by-NRHS right hand side matrix B and the N-by-NRHS solution 00049 *> matrix X. 00050 *> 00051 *> The routine first computes a QR factorization with column pivoting: 00052 *> A * P = Q * [ R11 R12 ] 00053 *> [ 0 R22 ] 00054 *> with R11 defined as the largest leading submatrix whose estimated 00055 *> condition number is less than 1/RCOND. The order of R11, RANK, 00056 *> is the effective rank of A. 00057 *> 00058 *> Then, R22 is considered to be negligible, and R12 is annihilated 00059 *> by unitary transformations from the right, arriving at the 00060 *> complete orthogonal factorization: 00061 *> A * P = Q * [ T11 0 ] * Z 00062 *> [ 0 0 ] 00063 *> The minimum-norm solution is then 00064 *> X = P * Z**H [ inv(T11)*Q1**H*B ] 00065 *> [ 0 ] 00066 *> where Q1 consists of the first RANK columns of Q. 00067 *> 00068 *> This routine is basically identical to the original xGELSX except 00069 *> three differences: 00070 *> o The permutation of matrix B (the right hand side) is faster and 00071 *> more simple. 00072 *> o The call to the subroutine xGEQPF has been substituted by the 00073 *> the call to the subroutine xGEQP3. This subroutine is a Blas-3 00074 *> version of the QR factorization with column pivoting. 00075 *> o Matrix B (the right hand side) is updated with Blas-3. 00076 *> \endverbatim 00077 * 00078 * Arguments: 00079 * ========== 00080 * 00081 *> \param[in] M 00082 *> \verbatim 00083 *> M is INTEGER 00084 *> The number of rows of the matrix A. M >= 0. 00085 *> \endverbatim 00086 *> 00087 *> \param[in] N 00088 *> \verbatim 00089 *> N is INTEGER 00090 *> The number of columns of the matrix A. N >= 0. 00091 *> \endverbatim 00092 *> 00093 *> \param[in] NRHS 00094 *> \verbatim 00095 *> NRHS is INTEGER 00096 *> The number of right hand sides, i.e., the number of 00097 *> columns of matrices B and X. NRHS >= 0. 00098 *> \endverbatim 00099 *> 00100 *> \param[in,out] A 00101 *> \verbatim 00102 *> A is COMPLEX*16 array, dimension (LDA,N) 00103 *> On entry, the M-by-N matrix A. 00104 *> On exit, A has been overwritten by details of its 00105 *> complete orthogonal factorization. 00106 *> \endverbatim 00107 *> 00108 *> \param[in] LDA 00109 *> \verbatim 00110 *> LDA is INTEGER 00111 *> The leading dimension of the array A. LDA >= max(1,M). 00112 *> \endverbatim 00113 *> 00114 *> \param[in,out] B 00115 *> \verbatim 00116 *> B is COMPLEX*16 array, dimension (LDB,NRHS) 00117 *> On entry, the M-by-NRHS right hand side matrix B. 00118 *> On exit, the N-by-NRHS solution matrix X. 00119 *> \endverbatim 00120 *> 00121 *> \param[in] LDB 00122 *> \verbatim 00123 *> LDB is INTEGER 00124 *> The leading dimension of the array B. LDB >= max(1,M,N). 00125 *> \endverbatim 00126 *> 00127 *> \param[in,out] JPVT 00128 *> \verbatim 00129 *> JPVT is INTEGER array, dimension (N) 00130 *> On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted 00131 *> to the front of AP, otherwise column i is a free column. 00132 *> On exit, if JPVT(i) = k, then the i-th column of A*P 00133 *> was the k-th column of A. 00134 *> \endverbatim 00135 *> 00136 *> \param[in] RCOND 00137 *> \verbatim 00138 *> RCOND is DOUBLE PRECISION 00139 *> RCOND is used to determine the effective rank of A, which 00140 *> is defined as the order of the largest leading triangular 00141 *> submatrix R11 in the QR factorization with pivoting of A, 00142 *> whose estimated condition number < 1/RCOND. 00143 *> \endverbatim 00144 *> 00145 *> \param[out] RANK 00146 *> \verbatim 00147 *> RANK is INTEGER 00148 *> The effective rank of A, i.e., the order of the submatrix 00149 *> R11. This is the same as the order of the submatrix T11 00150 *> in the complete orthogonal factorization of A. 00151 *> \endverbatim 00152 *> 00153 *> \param[out] WORK 00154 *> \verbatim 00155 *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) 00156 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00157 *> \endverbatim 00158 *> 00159 *> \param[in] LWORK 00160 *> \verbatim 00161 *> LWORK is INTEGER 00162 *> The dimension of the array WORK. 00163 *> The unblocked strategy requires that: 00164 *> LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS ) 00165 *> where MN = min(M,N). 00166 *> The block algorithm requires that: 00167 *> LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS ) 00168 *> where NB is an upper bound on the blocksize returned 00169 *> by ILAENV for the routines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR, 00170 *> and ZUNMRZ. 00171 *> 00172 *> If LWORK = -1, then a workspace query is assumed; the routine 00173 *> only calculates the optimal size of the WORK array, returns 00174 *> this value as the first entry of the WORK array, and no error 00175 *> message related to LWORK is issued by XERBLA. 00176 *> \endverbatim 00177 *> 00178 *> \param[out] RWORK 00179 *> \verbatim 00180 *> RWORK is DOUBLE PRECISION array, dimension (2*N) 00181 *> \endverbatim 00182 *> 00183 *> \param[out] INFO 00184 *> \verbatim 00185 *> INFO is INTEGER 00186 *> = 0: successful exit 00187 *> < 0: if INFO = -i, the i-th argument had an illegal value 00188 *> \endverbatim 00189 * 00190 * Authors: 00191 * ======== 00192 * 00193 *> \author Univ. of Tennessee 00194 *> \author Univ. of California Berkeley 00195 *> \author Univ. of Colorado Denver 00196 *> \author NAG Ltd. 00197 * 00198 *> \date November 2011 00199 * 00200 *> \ingroup complex16GEsolve 00201 * 00202 *> \par Contributors: 00203 * ================== 00204 *> 00205 *> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA \n 00206 *> E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n 00207 *> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n 00208 *> 00209 * ===================================================================== 00210 SUBROUTINE ZGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, 00211 $ WORK, LWORK, RWORK, INFO ) 00212 * 00213 * -- LAPACK driver routine (version 3.4.0) -- 00214 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00215 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00216 * November 2011 00217 * 00218 * .. Scalar Arguments .. 00219 INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK 00220 DOUBLE PRECISION RCOND 00221 * .. 00222 * .. Array Arguments .. 00223 INTEGER JPVT( * ) 00224 DOUBLE PRECISION RWORK( * ) 00225 COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ) 00226 * .. 00227 * 00228 * ===================================================================== 00229 * 00230 * .. Parameters .. 00231 INTEGER IMAX, IMIN 00232 PARAMETER ( IMAX = 1, IMIN = 2 ) 00233 DOUBLE PRECISION ZERO, ONE 00234 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00235 COMPLEX*16 CZERO, CONE 00236 PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), 00237 $ CONE = ( 1.0D+0, 0.0D+0 ) ) 00238 * .. 00239 * .. Local Scalars .. 00240 LOGICAL LQUERY 00241 INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKOPT, MN, 00242 $ NB, NB1, NB2, NB3, NB4 00243 DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMAX, SMAXPR, SMIN, SMINPR, 00244 $ SMLNUM, WSIZE 00245 COMPLEX*16 C1, C2, S1, S2 00246 * .. 00247 * .. External Subroutines .. 00248 EXTERNAL DLABAD, XERBLA, ZCOPY, ZGEQP3, ZLAIC1, ZLASCL, 00249 $ ZLASET, ZTRSM, ZTZRZF, ZUNMQR, ZUNMRZ 00250 * .. 00251 * .. External Functions .. 00252 INTEGER ILAENV 00253 DOUBLE PRECISION DLAMCH, ZLANGE 00254 EXTERNAL ILAENV, DLAMCH, ZLANGE 00255 * .. 00256 * .. Intrinsic Functions .. 00257 INTRINSIC ABS, DBLE, DCMPLX, MAX, MIN 00258 * .. 00259 * .. Executable Statements .. 00260 * 00261 MN = MIN( M, N ) 00262 ISMIN = MN + 1 00263 ISMAX = 2*MN + 1 00264 * 00265 * Test the input arguments. 00266 * 00267 INFO = 0 00268 NB1 = ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 ) 00269 NB2 = ILAENV( 1, 'ZGERQF', ' ', M, N, -1, -1 ) 00270 NB3 = ILAENV( 1, 'ZUNMQR', ' ', M, N, NRHS, -1 ) 00271 NB4 = ILAENV( 1, 'ZUNMRQ', ' ', M, N, NRHS, -1 ) 00272 NB = MAX( NB1, NB2, NB3, NB4 ) 00273 LWKOPT = MAX( 1, MN+2*N+NB*( N+1 ), 2*MN+NB*NRHS ) 00274 WORK( 1 ) = DCMPLX( LWKOPT ) 00275 LQUERY = ( LWORK.EQ.-1 ) 00276 IF( M.LT.0 ) THEN 00277 INFO = -1 00278 ELSE IF( N.LT.0 ) THEN 00279 INFO = -2 00280 ELSE IF( NRHS.LT.0 ) THEN 00281 INFO = -3 00282 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 00283 INFO = -5 00284 ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN 00285 INFO = -7 00286 ELSE IF( LWORK.LT.( MN+MAX( 2*MN, N+1, MN+NRHS ) ) .AND. .NOT. 00287 $ LQUERY ) THEN 00288 INFO = -12 00289 END IF 00290 * 00291 IF( INFO.NE.0 ) THEN 00292 CALL XERBLA( 'ZGELSY', -INFO ) 00293 RETURN 00294 ELSE IF( LQUERY ) THEN 00295 RETURN 00296 END IF 00297 * 00298 * Quick return if possible 00299 * 00300 IF( MIN( M, N, NRHS ).EQ.0 ) THEN 00301 RANK = 0 00302 RETURN 00303 END IF 00304 * 00305 * Get machine parameters 00306 * 00307 SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' ) 00308 BIGNUM = ONE / SMLNUM 00309 CALL DLABAD( SMLNUM, BIGNUM ) 00310 * 00311 * Scale A, B if max entries outside range [SMLNUM,BIGNUM] 00312 * 00313 ANRM = ZLANGE( 'M', M, N, A, LDA, RWORK ) 00314 IASCL = 0 00315 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN 00316 * 00317 * Scale matrix norm up to SMLNUM 00318 * 00319 CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO ) 00320 IASCL = 1 00321 ELSE IF( ANRM.GT.BIGNUM ) THEN 00322 * 00323 * Scale matrix norm down to BIGNUM 00324 * 00325 CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO ) 00326 IASCL = 2 00327 ELSE IF( ANRM.EQ.ZERO ) THEN 00328 * 00329 * Matrix all zero. Return zero solution. 00330 * 00331 CALL ZLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB ) 00332 RANK = 0 00333 GO TO 70 00334 END IF 00335 * 00336 BNRM = ZLANGE( 'M', M, NRHS, B, LDB, RWORK ) 00337 IBSCL = 0 00338 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN 00339 * 00340 * Scale matrix norm up to SMLNUM 00341 * 00342 CALL ZLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO ) 00343 IBSCL = 1 00344 ELSE IF( BNRM.GT.BIGNUM ) THEN 00345 * 00346 * Scale matrix norm down to BIGNUM 00347 * 00348 CALL ZLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO ) 00349 IBSCL = 2 00350 END IF 00351 * 00352 * Compute QR factorization with column pivoting of A: 00353 * A * P = Q * R 00354 * 00355 CALL ZGEQP3( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ), 00356 $ LWORK-MN, RWORK, INFO ) 00357 WSIZE = MN + DBLE( WORK( MN+1 ) ) 00358 * 00359 * complex workspace: MN+NB*(N+1). real workspace 2*N. 00360 * Details of Householder rotations stored in WORK(1:MN). 00361 * 00362 * Determine RANK using incremental condition estimation 00363 * 00364 WORK( ISMIN ) = CONE 00365 WORK( ISMAX ) = CONE 00366 SMAX = ABS( A( 1, 1 ) ) 00367 SMIN = SMAX 00368 IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN 00369 RANK = 0 00370 CALL ZLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB ) 00371 GO TO 70 00372 ELSE 00373 RANK = 1 00374 END IF 00375 * 00376 10 CONTINUE 00377 IF( RANK.LT.MN ) THEN 00378 I = RANK + 1 00379 CALL ZLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ), 00380 $ A( I, I ), SMINPR, S1, C1 ) 00381 CALL ZLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ), 00382 $ A( I, I ), SMAXPR, S2, C2 ) 00383 * 00384 IF( SMAXPR*RCOND.LE.SMINPR ) THEN 00385 DO 20 I = 1, RANK 00386 WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 ) 00387 WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 ) 00388 20 CONTINUE 00389 WORK( ISMIN+RANK ) = C1 00390 WORK( ISMAX+RANK ) = C2 00391 SMIN = SMINPR 00392 SMAX = SMAXPR 00393 RANK = RANK + 1 00394 GO TO 10 00395 END IF 00396 END IF 00397 * 00398 * complex workspace: 3*MN. 00399 * 00400 * Logically partition R = [ R11 R12 ] 00401 * [ 0 R22 ] 00402 * where R11 = R(1:RANK,1:RANK) 00403 * 00404 * [R11,R12] = [ T11, 0 ] * Y 00405 * 00406 IF( RANK.LT.N ) 00407 $ CALL ZTZRZF( RANK, N, A, LDA, WORK( MN+1 ), WORK( 2*MN+1 ), 00408 $ LWORK-2*MN, INFO ) 00409 * 00410 * complex workspace: 2*MN. 00411 * Details of Householder rotations stored in WORK(MN+1:2*MN) 00412 * 00413 * B(1:M,1:NRHS) := Q**H * B(1:M,1:NRHS) 00414 * 00415 CALL ZUNMQR( 'Left', 'Conjugate transpose', M, NRHS, MN, A, LDA, 00416 $ WORK( 1 ), B, LDB, WORK( 2*MN+1 ), LWORK-2*MN, INFO ) 00417 WSIZE = MAX( WSIZE, 2*MN+DBLE( WORK( 2*MN+1 ) ) ) 00418 * 00419 * complex workspace: 2*MN+NB*NRHS. 00420 * 00421 * B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS) 00422 * 00423 CALL ZTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK, 00424 $ NRHS, CONE, A, LDA, B, LDB ) 00425 * 00426 DO 40 J = 1, NRHS 00427 DO 30 I = RANK + 1, N 00428 B( I, J ) = CZERO 00429 30 CONTINUE 00430 40 CONTINUE 00431 * 00432 * B(1:N,1:NRHS) := Y**H * B(1:N,1:NRHS) 00433 * 00434 IF( RANK.LT.N ) THEN 00435 CALL ZUNMRZ( 'Left', 'Conjugate transpose', N, NRHS, RANK, 00436 $ N-RANK, A, LDA, WORK( MN+1 ), B, LDB, 00437 $ WORK( 2*MN+1 ), LWORK-2*MN, INFO ) 00438 END IF 00439 * 00440 * complex workspace: 2*MN+NRHS. 00441 * 00442 * B(1:N,1:NRHS) := P * B(1:N,1:NRHS) 00443 * 00444 DO 60 J = 1, NRHS 00445 DO 50 I = 1, N 00446 WORK( JPVT( I ) ) = B( I, J ) 00447 50 CONTINUE 00448 CALL ZCOPY( N, WORK( 1 ), 1, B( 1, J ), 1 ) 00449 60 CONTINUE 00450 * 00451 * complex workspace: N. 00452 * 00453 * Undo scaling 00454 * 00455 IF( IASCL.EQ.1 ) THEN 00456 CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO ) 00457 CALL ZLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA, 00458 $ INFO ) 00459 ELSE IF( IASCL.EQ.2 ) THEN 00460 CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO ) 00461 CALL ZLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA, 00462 $ INFO ) 00463 END IF 00464 IF( IBSCL.EQ.1 ) THEN 00465 CALL ZLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO ) 00466 ELSE IF( IBSCL.EQ.2 ) THEN 00467 CALL ZLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO ) 00468 END IF 00469 * 00470 70 CONTINUE 00471 WORK( 1 ) = DCMPLX( LWKOPT ) 00472 * 00473 RETURN 00474 * 00475 * End of ZGELSY 00476 * 00477 END