LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zgelsd.f
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00001 *> \brief <b> ZGELSD computes the minimum-norm solution to a linear least squares problem 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 ZGELSD + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgelsd.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
00022 *                          WORK, LWORK, RWORK, IWORK, 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            IWORK( * )
00030 *       DOUBLE PRECISION   RWORK( * ), S( * )
00031 *       COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * )
00032 *       ..
00033 *  
00034 *
00035 *> \par Purpose:
00036 *  =============
00037 *>
00038 *> \verbatim
00039 *>
00040 *> ZGELSD computes the minimum-norm solution to a real linear least
00041 *> squares problem:
00042 *>     minimize 2-norm(| b - A*x |)
00043 *> using the singular value decomposition (SVD) 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 problem is solved in three steps:
00052 *> (1) Reduce the coefficient matrix A to bidiagonal form with
00053 *>     Householder tranformations, reducing the original problem
00054 *>     into a "bidiagonal least squares problem" (BLS)
00055 *> (2) Solve the BLS using a divide and conquer approach.
00056 *> (3) Apply back all the Householder tranformations to solve
00057 *>     the original least squares problem.
00058 *>
00059 *> The effective rank of A is determined by treating as zero those
00060 *> singular values which are less than RCOND times the largest singular
00061 *> value.
00062 *>
00063 *> The divide and conquer algorithm makes very mild assumptions about
00064 *> floating point arithmetic. It will work on machines with a guard
00065 *> digit in add/subtract, or on those binary machines without guard
00066 *> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
00067 *> Cray-2. It could conceivably fail on hexadecimal or decimal machines
00068 *> without guard digits, but we know of none.
00069 *> \endverbatim
00070 *
00071 *  Arguments:
00072 *  ==========
00073 *
00074 *> \param[in] M
00075 *> \verbatim
00076 *>          M is INTEGER
00077 *>          The number of rows of the matrix A. M >= 0.
00078 *> \endverbatim
00079 *>
00080 *> \param[in] N
00081 *> \verbatim
00082 *>          N is INTEGER
00083 *>          The number of columns of the matrix A. N >= 0.
00084 *> \endverbatim
00085 *>
00086 *> \param[in] NRHS
00087 *> \verbatim
00088 *>          NRHS is INTEGER
00089 *>          The number of right hand sides, i.e., the number of columns
00090 *>          of the matrices B and X. NRHS >= 0.
00091 *> \endverbatim
00092 *>
00093 *> \param[in] A
00094 *> \verbatim
00095 *>          A is COMPLEX*16 array, dimension (LDA,N)
00096 *>          On entry, the M-by-N matrix A.
00097 *>          On exit, A has been destroyed.
00098 *> \endverbatim
00099 *>
00100 *> \param[in] LDA
00101 *> \verbatim
00102 *>          LDA is INTEGER
00103 *>          The leading dimension of the array A. LDA >= max(1,M).
00104 *> \endverbatim
00105 *>
00106 *> \param[in,out] B
00107 *> \verbatim
00108 *>          B is COMPLEX*16 array, dimension (LDB,NRHS)
00109 *>          On entry, the M-by-NRHS right hand side matrix B.
00110 *>          On exit, B is overwritten by the N-by-NRHS solution matrix X.
00111 *>          If m >= n and RANK = n, the residual sum-of-squares for
00112 *>          the solution in the i-th column is given by the sum of
00113 *>          squares of the modulus of elements n+1:m in that column.
00114 *> \endverbatim
00115 *>
00116 *> \param[in] LDB
00117 *> \verbatim
00118 *>          LDB is INTEGER
00119 *>          The leading dimension of the array B.  LDB >= max(1,M,N).
00120 *> \endverbatim
00121 *>
00122 *> \param[out] S
00123 *> \verbatim
00124 *>          S is DOUBLE PRECISION array, dimension (min(M,N))
00125 *>          The singular values of A in decreasing order.
00126 *>          The condition number of A in the 2-norm = S(1)/S(min(m,n)).
00127 *> \endverbatim
00128 *>
00129 *> \param[in] RCOND
00130 *> \verbatim
00131 *>          RCOND is DOUBLE PRECISION
00132 *>          RCOND is used to determine the effective rank of A.
00133 *>          Singular values S(i) <= RCOND*S(1) are treated as zero.
00134 *>          If RCOND < 0, machine precision is used instead.
00135 *> \endverbatim
00136 *>
00137 *> \param[out] RANK
00138 *> \verbatim
00139 *>          RANK is INTEGER
00140 *>          The effective rank of A, i.e., the number of singular values
00141 *>          which are greater than RCOND*S(1).
00142 *> \endverbatim
00143 *>
00144 *> \param[out] WORK
00145 *> \verbatim
00146 *>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
00147 *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00148 *> \endverbatim
00149 *>
00150 *> \param[in] LWORK
00151 *> \verbatim
00152 *>          LWORK is INTEGER
00153 *>          The dimension of the array WORK. LWORK must be at least 1.
00154 *>          The exact minimum amount of workspace needed depends on M,
00155 *>          N and NRHS. As long as LWORK is at least
00156 *>              2*N + N*NRHS
00157 *>          if M is greater than or equal to N or
00158 *>              2*M + M*NRHS
00159 *>          if M is less than N, the code will execute correctly.
00160 *>          For good performance, LWORK should generally be larger.
00161 *>
00162 *>          If LWORK = -1, then a workspace query is assumed; the routine
00163 *>          only calculates the optimal size of the array WORK and the
00164 *>          minimum sizes of the arrays RWORK and IWORK, and returns
00165 *>          these values as the first entries of the WORK, RWORK and
00166 *>          IWORK arrays, and no error message related to LWORK is issued
00167 *>          by XERBLA.
00168 *> \endverbatim
00169 *>
00170 *> \param[out] RWORK
00171 *> \verbatim
00172 *>          RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
00173 *>          LRWORK >=
00174 *>             10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
00175 *>             MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
00176 *>          if M is greater than or equal to N or
00177 *>             10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
00178 *>             MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
00179 *>          if M is less than N, the code will execute correctly.
00180 *>          SMLSIZ is returned by ILAENV and is equal to the maximum
00181 *>          size of the subproblems at the bottom of the computation
00182 *>          tree (usually about 25), and
00183 *>             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
00184 *>          On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.
00185 *> \endverbatim
00186 *>
00187 *> \param[out] IWORK
00188 *> \verbatim
00189 *>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
00190 *>          LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
00191 *>          where MINMN = MIN( M,N ).
00192 *>          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
00193 *> \endverbatim
00194 *>
00195 *> \param[out] INFO
00196 *> \verbatim
00197 *>          INFO is INTEGER
00198 *>          = 0: successful exit
00199 *>          < 0: if INFO = -i, the i-th argument had an illegal value.
00200 *>          > 0:  the algorithm for computing the SVD failed to converge;
00201 *>                if INFO = i, i off-diagonal elements of an intermediate
00202 *>                bidiagonal form did not converge to zero.
00203 *> \endverbatim
00204 *
00205 *  Authors:
00206 *  ========
00207 *
00208 *> \author Univ. of Tennessee 
00209 *> \author Univ. of California Berkeley 
00210 *> \author Univ. of Colorado Denver 
00211 *> \author NAG Ltd. 
00212 *
00213 *> \date November 2011
00214 *
00215 *> \ingroup complex16GEsolve
00216 *
00217 *> \par Contributors:
00218 *  ==================
00219 *>
00220 *>     Ming Gu and Ren-Cang Li, Computer Science Division, University of
00221 *>       California at Berkeley, USA \n
00222 *>     Osni Marques, LBNL/NERSC, USA \n
00223 *
00224 *  =====================================================================
00225       SUBROUTINE ZGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
00226      $                   WORK, LWORK, RWORK, IWORK, INFO )
00227 *
00228 *  -- LAPACK driver routine (version 3.4.0) --
00229 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00230 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00231 *     November 2011
00232 *
00233 *     .. Scalar Arguments ..
00234       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
00235       DOUBLE PRECISION   RCOND
00236 *     ..
00237 *     .. Array Arguments ..
00238       INTEGER            IWORK( * )
00239       DOUBLE PRECISION   RWORK( * ), S( * )
00240       COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * )
00241 *     ..
00242 *
00243 *  =====================================================================
00244 *
00245 *     .. Parameters ..
00246       DOUBLE PRECISION   ZERO, ONE, TWO
00247       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
00248       COMPLEX*16         CZERO
00249       PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ) )
00250 *     ..
00251 *     .. Local Scalars ..
00252       LOGICAL            LQUERY
00253       INTEGER            IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
00254      $                   LDWORK, LIWORK, LRWORK, MAXMN, MAXWRK, MINMN,
00255      $                   MINWRK, MM, MNTHR, NLVL, NRWORK, NWORK, SMLSIZ
00256       DOUBLE PRECISION   ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
00257 *     ..
00258 *     .. External Subroutines ..
00259       EXTERNAL           DLABAD, DLASCL, DLASET, XERBLA, ZGEBRD, ZGELQF,
00260      $                   ZGEQRF, ZLACPY, ZLALSD, ZLASCL, ZLASET, ZUNMBR,
00261      $                   ZUNMLQ, ZUNMQR
00262 *     ..
00263 *     .. External Functions ..
00264       INTEGER            ILAENV
00265       DOUBLE PRECISION   DLAMCH, ZLANGE
00266       EXTERNAL           ILAENV, DLAMCH, ZLANGE
00267 *     ..
00268 *     .. Intrinsic Functions ..
00269       INTRINSIC          INT, LOG, MAX, MIN, DBLE
00270 *     ..
00271 *     .. Executable Statements ..
00272 *
00273 *     Test the input arguments.
00274 *
00275       INFO = 0
00276       MINMN = MIN( M, N )
00277       MAXMN = MAX( M, N )
00278       LQUERY = ( LWORK.EQ.-1 )
00279       IF( M.LT.0 ) THEN
00280          INFO = -1
00281       ELSE IF( N.LT.0 ) THEN
00282          INFO = -2
00283       ELSE IF( NRHS.LT.0 ) THEN
00284          INFO = -3
00285       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
00286          INFO = -5
00287       ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
00288          INFO = -7
00289       END IF
00290 *
00291 *     Compute workspace.
00292 *     (Note: Comments in the code beginning "Workspace:" describe the
00293 *     minimal amount of workspace needed at that point in the code,
00294 *     as well as the preferred amount for good performance.
00295 *     NB refers to the optimal block size for the immediately
00296 *     following subroutine, as returned by ILAENV.)
00297 *
00298       IF( INFO.EQ.0 ) THEN
00299          MINWRK = 1
00300          MAXWRK = 1
00301          LIWORK = 1
00302          LRWORK = 1
00303          IF( MINMN.GT.0 ) THEN
00304             SMLSIZ = ILAENV( 9, 'ZGELSD', ' ', 0, 0, 0, 0 )
00305             MNTHR = ILAENV( 6, 'ZGELSD', ' ', M, N, NRHS, -1 )
00306             NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ + 1 ) ) /
00307      $                  LOG( TWO ) ) + 1, 0 )
00308             LIWORK = 3*MINMN*NLVL + 11*MINMN
00309             MM = M
00310             IF( M.GE.N .AND. M.GE.MNTHR ) THEN
00311 *
00312 *              Path 1a - overdetermined, with many more rows than
00313 *                        columns.
00314 *
00315                MM = N
00316                MAXWRK = MAX( MAXWRK, N*ILAENV( 1, 'ZGEQRF', ' ', M, N,
00317      $                       -1, -1 ) )
00318                MAXWRK = MAX( MAXWRK, NRHS*ILAENV( 1, 'ZUNMQR', 'LC', M,
00319      $                       NRHS, N, -1 ) )
00320             END IF
00321             IF( M.GE.N ) THEN
00322 *
00323 *              Path 1 - overdetermined or exactly determined.
00324 *
00325                LRWORK = 10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
00326      $                  MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
00327                MAXWRK = MAX( MAXWRK, 2*N + ( MM + N )*ILAENV( 1,
00328      $                       'ZGEBRD', ' ', MM, N, -1, -1 ) )
00329                MAXWRK = MAX( MAXWRK, 2*N + NRHS*ILAENV( 1, 'ZUNMBR',
00330      $                       'QLC', MM, NRHS, N, -1 ) )
00331                MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
00332      $                       'ZUNMBR', 'PLN', N, NRHS, N, -1 ) )
00333                MAXWRK = MAX( MAXWRK, 2*N + N*NRHS )
00334                MINWRK = MAX( 2*N + MM, 2*N + N*NRHS )
00335             END IF
00336             IF( N.GT.M ) THEN
00337                LRWORK = 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
00338      $                  MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
00339                IF( N.GE.MNTHR ) THEN
00340 *
00341 *                 Path 2a - underdetermined, with many more columns
00342 *                           than rows.
00343 *
00344                   MAXWRK = M + M*ILAENV( 1, 'ZGELQF', ' ', M, N, -1,
00345      $                     -1 )
00346                   MAXWRK = MAX( MAXWRK, M*M + 4*M + 2*M*ILAENV( 1,
00347      $                          'ZGEBRD', ' ', M, M, -1, -1 ) )
00348                   MAXWRK = MAX( MAXWRK, M*M + 4*M + NRHS*ILAENV( 1,
00349      $                          'ZUNMBR', 'QLC', M, NRHS, M, -1 ) )
00350                   MAXWRK = MAX( MAXWRK, M*M + 4*M + ( M - 1 )*ILAENV( 1,
00351      $                          'ZUNMLQ', 'LC', N, NRHS, M, -1 ) )
00352                   IF( NRHS.GT.1 ) THEN
00353                      MAXWRK = MAX( MAXWRK, M*M + M + M*NRHS )
00354                   ELSE
00355                      MAXWRK = MAX( MAXWRK, M*M + 2*M )
00356                   END IF
00357                   MAXWRK = MAX( MAXWRK, M*M + 4*M + M*NRHS )
00358 !     XXX: Ensure the Path 2a case below is triggered.  The workspace
00359 !     calculation should use queries for all routines eventually.
00360                   MAXWRK = MAX( MAXWRK,
00361      $                 4*M+M*M+MAX( M, 2*M-4, NRHS, N-3*M ) )
00362                ELSE
00363 *
00364 *                 Path 2 - underdetermined.
00365 *
00366                   MAXWRK = 2*M + ( N + M )*ILAENV( 1, 'ZGEBRD', ' ', M,
00367      $                     N, -1, -1 )
00368                   MAXWRK = MAX( MAXWRK, 2*M + NRHS*ILAENV( 1, 'ZUNMBR',
00369      $                          'QLC', M, NRHS, M, -1 ) )
00370                   MAXWRK = MAX( MAXWRK, 2*M + M*ILAENV( 1, 'ZUNMBR',
00371      $                          'PLN', N, NRHS, M, -1 ) )
00372                   MAXWRK = MAX( MAXWRK, 2*M + M*NRHS )
00373                END IF
00374                MINWRK = MAX( 2*M + N, 2*M + M*NRHS )
00375             END IF
00376          END IF
00377          MINWRK = MIN( MINWRK, MAXWRK )
00378          WORK( 1 ) = MAXWRK
00379          IWORK( 1 ) = LIWORK
00380          RWORK( 1 ) = LRWORK
00381 *
00382          IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
00383             INFO = -12
00384          END IF
00385       END IF
00386 *
00387       IF( INFO.NE.0 ) THEN
00388          CALL XERBLA( 'ZGELSD', -INFO )
00389          RETURN
00390       ELSE IF( LQUERY ) THEN
00391          RETURN
00392       END IF
00393 *
00394 *     Quick return if possible.
00395 *
00396       IF( M.EQ.0 .OR. N.EQ.0 ) THEN
00397          RANK = 0
00398          RETURN
00399       END IF
00400 *
00401 *     Get machine parameters.
00402 *
00403       EPS = DLAMCH( 'P' )
00404       SFMIN = DLAMCH( 'S' )
00405       SMLNUM = SFMIN / EPS
00406       BIGNUM = ONE / SMLNUM
00407       CALL DLABAD( SMLNUM, BIGNUM )
00408 *
00409 *     Scale A if max entry outside range [SMLNUM,BIGNUM].
00410 *
00411       ANRM = ZLANGE( 'M', M, N, A, LDA, RWORK )
00412       IASCL = 0
00413       IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
00414 *
00415 *        Scale matrix norm up to SMLNUM
00416 *
00417          CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
00418          IASCL = 1
00419       ELSE IF( ANRM.GT.BIGNUM ) THEN
00420 *
00421 *        Scale matrix norm down to BIGNUM.
00422 *
00423          CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
00424          IASCL = 2
00425       ELSE IF( ANRM.EQ.ZERO ) THEN
00426 *
00427 *        Matrix all zero. Return zero solution.
00428 *
00429          CALL ZLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
00430          CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
00431          RANK = 0
00432          GO TO 10
00433       END IF
00434 *
00435 *     Scale B if max entry outside range [SMLNUM,BIGNUM].
00436 *
00437       BNRM = ZLANGE( 'M', M, NRHS, B, LDB, RWORK )
00438       IBSCL = 0
00439       IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
00440 *
00441 *        Scale matrix norm up to SMLNUM.
00442 *
00443          CALL ZLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
00444          IBSCL = 1
00445       ELSE IF( BNRM.GT.BIGNUM ) THEN
00446 *
00447 *        Scale matrix norm down to BIGNUM.
00448 *
00449          CALL ZLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
00450          IBSCL = 2
00451       END IF
00452 *
00453 *     If M < N make sure B(M+1:N,:) = 0
00454 *
00455       IF( M.LT.N )
00456      $   CALL ZLASET( 'F', N-M, NRHS, CZERO, CZERO, B( M+1, 1 ), LDB )
00457 *
00458 *     Overdetermined case.
00459 *
00460       IF( M.GE.N ) THEN
00461 *
00462 *        Path 1 - overdetermined or exactly determined.
00463 *
00464          MM = M
00465          IF( M.GE.MNTHR ) THEN
00466 *
00467 *           Path 1a - overdetermined, with many more rows than columns
00468 *
00469             MM = N
00470             ITAU = 1
00471             NWORK = ITAU + N
00472 *
00473 *           Compute A=Q*R.
00474 *           (RWorkspace: need N)
00475 *           (CWorkspace: need N, prefer N*NB)
00476 *
00477             CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
00478      $                   LWORK-NWORK+1, INFO )
00479 *
00480 *           Multiply B by transpose(Q).
00481 *           (RWorkspace: need N)
00482 *           (CWorkspace: need NRHS, prefer NRHS*NB)
00483 *
00484             CALL ZUNMQR( 'L', 'C', M, NRHS, N, A, LDA, WORK( ITAU ), B,
00485      $                   LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00486 *
00487 *           Zero out below R.
00488 *
00489             IF( N.GT.1 ) THEN
00490                CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO, A( 2, 1 ),
00491      $                      LDA )
00492             END IF
00493          END IF
00494 *
00495          ITAUQ = 1
00496          ITAUP = ITAUQ + N
00497          NWORK = ITAUP + N
00498          IE = 1
00499          NRWORK = IE + N
00500 *
00501 *        Bidiagonalize R in A.
00502 *        (RWorkspace: need N)
00503 *        (CWorkspace: need 2*N+MM, prefer 2*N+(MM+N)*NB)
00504 *
00505          CALL ZGEBRD( MM, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
00506      $                WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
00507      $                INFO )
00508 *
00509 *        Multiply B by transpose of left bidiagonalizing vectors of R.
00510 *        (CWorkspace: need 2*N+NRHS, prefer 2*N+NRHS*NB)
00511 *
00512          CALL ZUNMBR( 'Q', 'L', 'C', MM, NRHS, N, A, LDA, WORK( ITAUQ ),
00513      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00514 *
00515 *        Solve the bidiagonal least squares problem.
00516 *
00517          CALL ZLALSD( 'U', SMLSIZ, N, NRHS, S, RWORK( IE ), B, LDB,
00518      $                RCOND, RANK, WORK( NWORK ), RWORK( NRWORK ),
00519      $                IWORK, INFO )
00520          IF( INFO.NE.0 ) THEN
00521             GO TO 10
00522          END IF
00523 *
00524 *        Multiply B by right bidiagonalizing vectors of R.
00525 *
00526          CALL ZUNMBR( 'P', 'L', 'N', N, NRHS, N, A, LDA, WORK( ITAUP ),
00527      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00528 *
00529       ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
00530      $         MAX( M, 2*M-4, NRHS, N-3*M ) ) THEN
00531 *
00532 *        Path 2a - underdetermined, with many more columns than rows
00533 *        and sufficient workspace for an efficient algorithm.
00534 *
00535          LDWORK = M
00536          IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
00537      $       M*LDA+M+M*NRHS ) )LDWORK = LDA
00538          ITAU = 1
00539          NWORK = M + 1
00540 *
00541 *        Compute A=L*Q.
00542 *        (CWorkspace: need 2*M, prefer M+M*NB)
00543 *
00544          CALL ZGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
00545      $                LWORK-NWORK+1, INFO )
00546          IL = NWORK
00547 *
00548 *        Copy L to WORK(IL), zeroing out above its diagonal.
00549 *
00550          CALL ZLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
00551          CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO, WORK( IL+LDWORK ),
00552      $                LDWORK )
00553          ITAUQ = IL + LDWORK*M
00554          ITAUP = ITAUQ + M
00555          NWORK = ITAUP + M
00556          IE = 1
00557          NRWORK = IE + M
00558 *
00559 *        Bidiagonalize L in WORK(IL).
00560 *        (RWorkspace: need M)
00561 *        (CWorkspace: need M*M+4*M, prefer M*M+4*M+2*M*NB)
00562 *
00563          CALL ZGEBRD( M, M, WORK( IL ), LDWORK, S, RWORK( IE ),
00564      $                WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
00565      $                LWORK-NWORK+1, INFO )
00566 *
00567 *        Multiply B by transpose of left bidiagonalizing vectors of L.
00568 *        (CWorkspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
00569 *
00570          CALL ZUNMBR( 'Q', 'L', 'C', M, NRHS, M, WORK( IL ), LDWORK,
00571      $                WORK( ITAUQ ), B, LDB, WORK( NWORK ),
00572      $                LWORK-NWORK+1, INFO )
00573 *
00574 *        Solve the bidiagonal least squares problem.
00575 *
00576          CALL ZLALSD( 'U', SMLSIZ, M, NRHS, S, RWORK( IE ), B, LDB,
00577      $                RCOND, RANK, WORK( NWORK ), RWORK( NRWORK ),
00578      $                IWORK, INFO )
00579          IF( INFO.NE.0 ) THEN
00580             GO TO 10
00581          END IF
00582 *
00583 *        Multiply B by right bidiagonalizing vectors of L.
00584 *
00585          CALL ZUNMBR( 'P', 'L', 'N', M, NRHS, M, WORK( IL ), LDWORK,
00586      $                WORK( ITAUP ), B, LDB, WORK( NWORK ),
00587      $                LWORK-NWORK+1, INFO )
00588 *
00589 *        Zero out below first M rows of B.
00590 *
00591          CALL ZLASET( 'F', N-M, NRHS, CZERO, CZERO, B( M+1, 1 ), LDB )
00592          NWORK = ITAU + M
00593 *
00594 *        Multiply transpose(Q) by B.
00595 *        (CWorkspace: need NRHS, prefer NRHS*NB)
00596 *
00597          CALL ZUNMLQ( 'L', 'C', N, NRHS, M, A, LDA, WORK( ITAU ), B,
00598      $                LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00599 *
00600       ELSE
00601 *
00602 *        Path 2 - remaining underdetermined cases.
00603 *
00604          ITAUQ = 1
00605          ITAUP = ITAUQ + M
00606          NWORK = ITAUP + M
00607          IE = 1
00608          NRWORK = IE + M
00609 *
00610 *        Bidiagonalize A.
00611 *        (RWorkspace: need M)
00612 *        (CWorkspace: need 2*M+N, prefer 2*M+(M+N)*NB)
00613 *
00614          CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
00615      $                WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
00616      $                INFO )
00617 *
00618 *        Multiply B by transpose of left bidiagonalizing vectors.
00619 *        (CWorkspace: need 2*M+NRHS, prefer 2*M+NRHS*NB)
00620 *
00621          CALL ZUNMBR( 'Q', 'L', 'C', M, NRHS, N, A, LDA, WORK( ITAUQ ),
00622      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00623 *
00624 *        Solve the bidiagonal least squares problem.
00625 *
00626          CALL ZLALSD( 'L', SMLSIZ, M, NRHS, S, RWORK( IE ), B, LDB,
00627      $                RCOND, RANK, WORK( NWORK ), RWORK( NRWORK ),
00628      $                IWORK, INFO )
00629          IF( INFO.NE.0 ) THEN
00630             GO TO 10
00631          END IF
00632 *
00633 *        Multiply B by right bidiagonalizing vectors of A.
00634 *
00635          CALL ZUNMBR( 'P', 'L', 'N', N, NRHS, M, A, LDA, WORK( ITAUP ),
00636      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00637 *
00638       END IF
00639 *
00640 *     Undo scaling.
00641 *
00642       IF( IASCL.EQ.1 ) THEN
00643          CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
00644          CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
00645      $                INFO )
00646       ELSE IF( IASCL.EQ.2 ) THEN
00647          CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
00648          CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
00649      $                INFO )
00650       END IF
00651       IF( IBSCL.EQ.1 ) THEN
00652          CALL ZLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
00653       ELSE IF( IBSCL.EQ.2 ) THEN
00654          CALL ZLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
00655       END IF
00656 *
00657    10 CONTINUE
00658       WORK( 1 ) = MAXWRK
00659       IWORK( 1 ) = LIWORK
00660       RWORK( 1 ) = LRWORK
00661       RETURN
00662 *
00663 *     End of ZGELSD
00664 *
00665       END
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