LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dgelsd.f
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00001 *> \brief <b> DGELSD 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 DGELSD + dependencies 
00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelsd.f"> 
00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelsd.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelsd.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
00022 *                          WORK, LWORK, 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   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
00031 *       ..
00032 *  
00033 *
00034 *> \par Purpose:
00035 *  =============
00036 *>
00037 *> \verbatim
00038 *>
00039 *> DGELSD computes the minimum-norm solution to a real linear least
00040 *> squares problem:
00041 *>     minimize 2-norm(| b - A*x |)
00042 *> using the singular value decomposition (SVD) of A. A is an M-by-N
00043 *> matrix which may be rank-deficient.
00044 *>
00045 *> Several right hand side vectors b and solution vectors x can be
00046 *> handled in a single call; they are stored as the columns of the
00047 *> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
00048 *> matrix X.
00049 *>
00050 *> The problem is solved in three steps:
00051 *> (1) Reduce the coefficient matrix A to bidiagonal form with
00052 *>     Householder transformations, reducing the original problem
00053 *>     into a "bidiagonal least squares problem" (BLS)
00054 *> (2) Solve the BLS using a divide and conquer approach.
00055 *> (3) Apply back all the Householder tranformations to solve
00056 *>     the original least squares problem.
00057 *>
00058 *> The effective rank of A is determined by treating as zero those
00059 *> singular values which are less than RCOND times the largest singular
00060 *> value.
00061 *>
00062 *> The divide and conquer algorithm makes very mild assumptions about
00063 *> floating point arithmetic. It will work on machines with a guard
00064 *> digit in add/subtract, or on those binary machines without guard
00065 *> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
00066 *> Cray-2. It could conceivably fail on hexadecimal or decimal machines
00067 *> without guard digits, but we know of none.
00068 *> \endverbatim
00069 *
00070 *  Arguments:
00071 *  ==========
00072 *
00073 *> \param[in] M
00074 *> \verbatim
00075 *>          M is INTEGER
00076 *>          The number of rows of A. M >= 0.
00077 *> \endverbatim
00078 *>
00079 *> \param[in] N
00080 *> \verbatim
00081 *>          N is INTEGER
00082 *>          The number of columns of A. N >= 0.
00083 *> \endverbatim
00084 *>
00085 *> \param[in] NRHS
00086 *> \verbatim
00087 *>          NRHS is INTEGER
00088 *>          The number of right hand sides, i.e., the number of columns
00089 *>          of the matrices B and X. NRHS >= 0.
00090 *> \endverbatim
00091 *>
00092 *> \param[in] A
00093 *> \verbatim
00094 *>          A is DOUBLE PRECISION array, dimension (LDA,N)
00095 *>          On entry, the M-by-N matrix A.
00096 *>          On exit, A has been destroyed.
00097 *> \endverbatim
00098 *>
00099 *> \param[in] LDA
00100 *> \verbatim
00101 *>          LDA is INTEGER
00102 *>          The leading dimension of the array A.  LDA >= max(1,M).
00103 *> \endverbatim
00104 *>
00105 *> \param[in,out] B
00106 *> \verbatim
00107 *>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
00108 *>          On entry, the M-by-NRHS right hand side matrix B.
00109 *>          On exit, B is overwritten by the N-by-NRHS solution
00110 *>          matrix X.  If m >= n and RANK = n, the residual
00111 *>          sum-of-squares for the solution in the i-th column is given
00112 *>          by the sum of squares of elements n+1:m in that column.
00113 *> \endverbatim
00114 *>
00115 *> \param[in] LDB
00116 *> \verbatim
00117 *>          LDB is INTEGER
00118 *>          The leading dimension of the array B. LDB >= max(1,max(M,N)).
00119 *> \endverbatim
00120 *>
00121 *> \param[out] S
00122 *> \verbatim
00123 *>          S is DOUBLE PRECISION array, dimension (min(M,N))
00124 *>          The singular values of A in decreasing order.
00125 *>          The condition number of A in the 2-norm = S(1)/S(min(m,n)).
00126 *> \endverbatim
00127 *>
00128 *> \param[in] RCOND
00129 *> \verbatim
00130 *>          RCOND is DOUBLE PRECISION
00131 *>          RCOND is used to determine the effective rank of A.
00132 *>          Singular values S(i) <= RCOND*S(1) are treated as zero.
00133 *>          If RCOND < 0, machine precision is used instead.
00134 *> \endverbatim
00135 *>
00136 *> \param[out] RANK
00137 *> \verbatim
00138 *>          RANK is INTEGER
00139 *>          The effective rank of A, i.e., the number of singular values
00140 *>          which are greater than RCOND*S(1).
00141 *> \endverbatim
00142 *>
00143 *> \param[out] WORK
00144 *> \verbatim
00145 *>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
00146 *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00147 *> \endverbatim
00148 *>
00149 *> \param[in] LWORK
00150 *> \verbatim
00151 *>          LWORK is INTEGER
00152 *>          The dimension of the array WORK. LWORK must be at least 1.
00153 *>          The exact minimum amount of workspace needed depends on M,
00154 *>          N and NRHS. As long as LWORK is at least
00155 *>              12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
00156 *>          if M is greater than or equal to N or
00157 *>              12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
00158 *>          if M is less than N, the code will execute correctly.
00159 *>          SMLSIZ is returned by ILAENV and is equal to the maximum
00160 *>          size of the subproblems at the bottom of the computation
00161 *>          tree (usually about 25), and
00162 *>             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
00163 *>          For good performance, LWORK should generally be larger.
00164 *>
00165 *>          If LWORK = -1, then a workspace query is assumed; the routine
00166 *>          only calculates the optimal size of the WORK array, returns
00167 *>          this value as the first entry of the WORK array, and no error
00168 *>          message related to LWORK is issued by XERBLA.
00169 *> \endverbatim
00170 *>
00171 *> \param[out] IWORK
00172 *> \verbatim
00173 *>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
00174 *>          LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),
00175 *>          where MINMN = MIN( M,N ).
00176 *>          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
00177 *> \endverbatim
00178 *>
00179 *> \param[out] INFO
00180 *> \verbatim
00181 *>          INFO is INTEGER
00182 *>          = 0:  successful exit
00183 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
00184 *>          > 0:  the algorithm for computing the SVD failed to converge;
00185 *>                if INFO = i, i off-diagonal elements of an intermediate
00186 *>                bidiagonal form did not converge to zero.
00187 *> \endverbatim
00188 *
00189 *  Authors:
00190 *  ========
00191 *
00192 *> \author Univ. of Tennessee 
00193 *> \author Univ. of California Berkeley 
00194 *> \author Univ. of Colorado Denver 
00195 *> \author NAG Ltd. 
00196 *
00197 *> \date November 2011
00198 *
00199 *> \ingroup doubleGEsolve
00200 *
00201 *> \par Contributors:
00202 *  ==================
00203 *>
00204 *>     Ming Gu and Ren-Cang Li, Computer Science Division, University of
00205 *>       California at Berkeley, USA \n
00206 *>     Osni Marques, LBNL/NERSC, USA \n
00207 *
00208 *  =====================================================================
00209       SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
00210      $                   WORK, LWORK, IWORK, INFO )
00211 *
00212 *  -- LAPACK driver routine (version 3.4.0) --
00213 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00214 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00215 *     November 2011
00216 *
00217 *     .. Scalar Arguments ..
00218       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
00219       DOUBLE PRECISION   RCOND
00220 *     ..
00221 *     .. Array Arguments ..
00222       INTEGER            IWORK( * )
00223       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
00224 *     ..
00225 *
00226 *  =====================================================================
00227 *
00228 *     .. Parameters ..
00229       DOUBLE PRECISION   ZERO, ONE, TWO
00230       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
00231 *     ..
00232 *     .. Local Scalars ..
00233       LOGICAL            LQUERY
00234       INTEGER            IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
00235      $                   LDWORK, LIWORK, MAXMN, MAXWRK, MINMN, MINWRK,
00236      $                   MM, MNTHR, NLVL, NWORK, SMLSIZ, WLALSD
00237       DOUBLE PRECISION   ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
00238 *     ..
00239 *     .. External Subroutines ..
00240       EXTERNAL           DGEBRD, DGELQF, DGEQRF, DLABAD, DLACPY, DLALSD,
00241      $                   DLASCL, DLASET, DORMBR, DORMLQ, DORMQR, XERBLA
00242 *     ..
00243 *     .. External Functions ..
00244       INTEGER            ILAENV
00245       DOUBLE PRECISION   DLAMCH, DLANGE
00246       EXTERNAL           ILAENV, DLAMCH, DLANGE
00247 *     ..
00248 *     .. Intrinsic Functions ..
00249       INTRINSIC          DBLE, INT, LOG, MAX, MIN
00250 *     ..
00251 *     .. Executable Statements ..
00252 *
00253 *     Test the input arguments.
00254 *
00255       INFO = 0
00256       MINMN = MIN( M, N )
00257       MAXMN = MAX( M, N )
00258       MNTHR = ILAENV( 6, 'DGELSD', ' ', M, N, NRHS, -1 )
00259       LQUERY = ( LWORK.EQ.-1 )
00260       IF( M.LT.0 ) THEN
00261          INFO = -1
00262       ELSE IF( N.LT.0 ) THEN
00263          INFO = -2
00264       ELSE IF( NRHS.LT.0 ) THEN
00265          INFO = -3
00266       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
00267          INFO = -5
00268       ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
00269          INFO = -7
00270       END IF
00271 *
00272       SMLSIZ = ILAENV( 9, 'DGELSD', ' ', 0, 0, 0, 0 )
00273 *
00274 *     Compute workspace.
00275 *     (Note: Comments in the code beginning "Workspace:" describe the
00276 *     minimal amount of workspace needed at that point in the code,
00277 *     as well as the preferred amount for good performance.
00278 *     NB refers to the optimal block size for the immediately
00279 *     following subroutine, as returned by ILAENV.)
00280 *
00281       MINWRK = 1
00282       LIWORK = 1
00283       MINMN = MAX( 1, MINMN )
00284       NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ+1 ) ) /
00285      $       LOG( TWO ) ) + 1, 0 )
00286 *
00287       IF( INFO.EQ.0 ) THEN
00288          MAXWRK = 0
00289          LIWORK = 3*MINMN*NLVL + 11*MINMN
00290          MM = M
00291          IF( M.GE.N .AND. M.GE.MNTHR ) THEN
00292 *
00293 *           Path 1a - overdetermined, with many more rows than columns.
00294 *
00295             MM = N
00296             MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'DGEQRF', ' ', M, N,
00297      $               -1, -1 ) )
00298             MAXWRK = MAX( MAXWRK, N+NRHS*
00299      $               ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, -1 ) )
00300          END IF
00301          IF( M.GE.N ) THEN
00302 *
00303 *           Path 1 - overdetermined or exactly determined.
00304 *
00305             MAXWRK = MAX( MAXWRK, 3*N+( MM+N )*
00306      $               ILAENV( 1, 'DGEBRD', ' ', MM, N, -1, -1 ) )
00307             MAXWRK = MAX( MAXWRK, 3*N+NRHS*
00308      $               ILAENV( 1, 'DORMBR', 'QLT', MM, NRHS, N, -1 ) )
00309             MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*
00310      $               ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, N, -1 ) )
00311             WLALSD = 9*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS+(SMLSIZ+1)**2
00312             MAXWRK = MAX( MAXWRK, 3*N+WLALSD )
00313             MINWRK = MAX( 3*N+MM, 3*N+NRHS, 3*N+WLALSD )
00314          END IF
00315          IF( N.GT.M ) THEN
00316             WLALSD = 9*M+2*M*SMLSIZ+8*M*NLVL+M*NRHS+(SMLSIZ+1)**2
00317             IF( N.GE.MNTHR ) THEN
00318 *
00319 *              Path 2a - underdetermined, with many more columns
00320 *              than rows.
00321 *
00322                MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
00323                MAXWRK = MAX( MAXWRK, M*M+4*M+2*M*
00324      $                  ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
00325                MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS*
00326      $                  ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M, -1 ) )
00327                MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )*
00328      $                  ILAENV( 1, 'DORMBR', 'PLN', M, NRHS, M, -1 ) )
00329                IF( NRHS.GT.1 ) THEN
00330                   MAXWRK = MAX( MAXWRK, M*M+M+M*NRHS )
00331                ELSE
00332                   MAXWRK = MAX( MAXWRK, M*M+2*M )
00333                END IF
00334                MAXWRK = MAX( MAXWRK, M+NRHS*
00335      $                  ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, -1 ) )
00336                MAXWRK = MAX( MAXWRK, M*M+4*M+WLALSD )
00337 !     XXX: Ensure the Path 2a case below is triggered.  The workspace
00338 !     calculation should use queries for all routines eventually.
00339                MAXWRK = MAX( MAXWRK,
00340      $              4*M+M*M+MAX( M, 2*M-4, NRHS, N-3*M ) )
00341             ELSE
00342 *
00343 *              Path 2 - remaining underdetermined cases.
00344 *
00345                MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'DGEBRD', ' ', M, N,
00346      $                  -1, -1 )
00347                MAXWRK = MAX( MAXWRK, 3*M+NRHS*
00348      $                  ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, N, -1 ) )
00349                MAXWRK = MAX( MAXWRK, 3*M+M*
00350      $                  ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, M, -1 ) )
00351                MAXWRK = MAX( MAXWRK, 3*M+WLALSD )
00352             END IF
00353             MINWRK = MAX( 3*M+NRHS, 3*M+M, 3*M+WLALSD )
00354          END IF
00355          MINWRK = MIN( MINWRK, MAXWRK )
00356          WORK( 1 ) = MAXWRK
00357          IWORK( 1 ) = LIWORK
00358 
00359          IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
00360             INFO = -12
00361          END IF
00362       END IF
00363 *
00364       IF( INFO.NE.0 ) THEN
00365          CALL XERBLA( 'DGELSD', -INFO )
00366          RETURN
00367       ELSE IF( LQUERY ) THEN
00368          GO TO 10
00369       END IF
00370 *
00371 *     Quick return if possible.
00372 *
00373       IF( M.EQ.0 .OR. N.EQ.0 ) THEN
00374          RANK = 0
00375          RETURN
00376       END IF
00377 *
00378 *     Get machine parameters.
00379 *
00380       EPS = DLAMCH( 'P' )
00381       SFMIN = DLAMCH( 'S' )
00382       SMLNUM = SFMIN / EPS
00383       BIGNUM = ONE / SMLNUM
00384       CALL DLABAD( SMLNUM, BIGNUM )
00385 *
00386 *     Scale A if max entry outside range [SMLNUM,BIGNUM].
00387 *
00388       ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
00389       IASCL = 0
00390       IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
00391 *
00392 *        Scale matrix norm up to SMLNUM.
00393 *
00394          CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
00395          IASCL = 1
00396       ELSE IF( ANRM.GT.BIGNUM ) THEN
00397 *
00398 *        Scale matrix norm down to BIGNUM.
00399 *
00400          CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
00401          IASCL = 2
00402       ELSE IF( ANRM.EQ.ZERO ) THEN
00403 *
00404 *        Matrix all zero. Return zero solution.
00405 *
00406          CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
00407          CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
00408          RANK = 0
00409          GO TO 10
00410       END IF
00411 *
00412 *     Scale B if max entry outside range [SMLNUM,BIGNUM].
00413 *
00414       BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
00415       IBSCL = 0
00416       IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
00417 *
00418 *        Scale matrix norm up to SMLNUM.
00419 *
00420          CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
00421          IBSCL = 1
00422       ELSE IF( BNRM.GT.BIGNUM ) THEN
00423 *
00424 *        Scale matrix norm down to BIGNUM.
00425 *
00426          CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
00427          IBSCL = 2
00428       END IF
00429 *
00430 *     If M < N make sure certain entries of B are zero.
00431 *
00432       IF( M.LT.N )
00433      $   CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
00434 *
00435 *     Overdetermined case.
00436 *
00437       IF( M.GE.N ) THEN
00438 *
00439 *        Path 1 - overdetermined or exactly determined.
00440 *
00441          MM = M
00442          IF( M.GE.MNTHR ) THEN
00443 *
00444 *           Path 1a - overdetermined, with many more rows than columns.
00445 *
00446             MM = N
00447             ITAU = 1
00448             NWORK = ITAU + N
00449 *
00450 *           Compute A=Q*R.
00451 *           (Workspace: need 2*N, prefer N+N*NB)
00452 *
00453             CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
00454      $                   LWORK-NWORK+1, INFO )
00455 *
00456 *           Multiply B by transpose(Q).
00457 *           (Workspace: need N+NRHS, prefer N+NRHS*NB)
00458 *
00459             CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B,
00460      $                   LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00461 *
00462 *           Zero out below R.
00463 *
00464             IF( N.GT.1 ) THEN
00465                CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
00466             END IF
00467          END IF
00468 *
00469          IE = 1
00470          ITAUQ = IE + N
00471          ITAUP = ITAUQ + N
00472          NWORK = ITAUP + N
00473 *
00474 *        Bidiagonalize R in A.
00475 *        (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
00476 *
00477          CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
00478      $                WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
00479      $                INFO )
00480 *
00481 *        Multiply B by transpose of left bidiagonalizing vectors of R.
00482 *        (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
00483 *
00484          CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ),
00485      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00486 *
00487 *        Solve the bidiagonal least squares problem.
00488 *
00489          CALL DLALSD( 'U', SMLSIZ, N, NRHS, S, WORK( IE ), B, LDB,
00490      $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
00491          IF( INFO.NE.0 ) THEN
00492             GO TO 10
00493          END IF
00494 *
00495 *        Multiply B by right bidiagonalizing vectors of R.
00496 *
00497          CALL DORMBR( 'P', 'L', 'N', N, NRHS, N, A, LDA, WORK( ITAUP ),
00498      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00499 *
00500       ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
00501      $         MAX( M, 2*M-4, NRHS, N-3*M, WLALSD ) ) THEN
00502 *
00503 *        Path 2a - underdetermined, with many more columns than rows
00504 *        and sufficient workspace for an efficient algorithm.
00505 *
00506          LDWORK = M
00507          IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
00508      $       M*LDA+M+M*NRHS, 4*M+M*LDA+WLALSD ) )LDWORK = LDA
00509          ITAU = 1
00510          NWORK = M + 1
00511 *
00512 *        Compute A=L*Q.
00513 *        (Workspace: need 2*M, prefer M+M*NB)
00514 *
00515          CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
00516      $                LWORK-NWORK+1, INFO )
00517          IL = NWORK
00518 *
00519 *        Copy L to WORK(IL), zeroing out above its diagonal.
00520 *
00521          CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
00522          CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ),
00523      $                LDWORK )
00524          IE = IL + LDWORK*M
00525          ITAUQ = IE + M
00526          ITAUP = ITAUQ + M
00527          NWORK = ITAUP + M
00528 *
00529 *        Bidiagonalize L in WORK(IL).
00530 *        (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
00531 *
00532          CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
00533      $                WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
00534      $                LWORK-NWORK+1, INFO )
00535 *
00536 *        Multiply B by transpose of left bidiagonalizing vectors of L.
00537 *        (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
00538 *
00539          CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
00540      $                WORK( ITAUQ ), B, LDB, WORK( NWORK ),
00541      $                LWORK-NWORK+1, INFO )
00542 *
00543 *        Solve the bidiagonal least squares problem.
00544 *
00545          CALL DLALSD( 'U', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
00546      $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
00547          IF( INFO.NE.0 ) THEN
00548             GO TO 10
00549          END IF
00550 *
00551 *        Multiply B by right bidiagonalizing vectors of L.
00552 *
00553          CALL DORMBR( 'P', 'L', 'N', M, NRHS, M, WORK( IL ), LDWORK,
00554      $                WORK( ITAUP ), B, LDB, WORK( NWORK ),
00555      $                LWORK-NWORK+1, INFO )
00556 *
00557 *        Zero out below first M rows of B.
00558 *
00559          CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
00560          NWORK = ITAU + M
00561 *
00562 *        Multiply transpose(Q) by B.
00563 *        (Workspace: need M+NRHS, prefer M+NRHS*NB)
00564 *
00565          CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
00566      $                LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00567 *
00568       ELSE
00569 *
00570 *        Path 2 - remaining underdetermined cases.
00571 *
00572          IE = 1
00573          ITAUQ = IE + M
00574          ITAUP = ITAUQ + M
00575          NWORK = ITAUP + M
00576 *
00577 *        Bidiagonalize A.
00578 *        (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
00579 *
00580          CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
00581      $                WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
00582      $                INFO )
00583 *
00584 *        Multiply B by transpose of left bidiagonalizing vectors.
00585 *        (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
00586 *
00587          CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ),
00588      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00589 *
00590 *        Solve the bidiagonal least squares problem.
00591 *
00592          CALL DLALSD( 'L', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
00593      $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
00594          IF( INFO.NE.0 ) THEN
00595             GO TO 10
00596          END IF
00597 *
00598 *        Multiply B by right bidiagonalizing vectors of A.
00599 *
00600          CALL DORMBR( 'P', 'L', 'N', N, NRHS, M, A, LDA, WORK( ITAUP ),
00601      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00602 *
00603       END IF
00604 *
00605 *     Undo scaling.
00606 *
00607       IF( IASCL.EQ.1 ) THEN
00608          CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
00609          CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
00610      $                INFO )
00611       ELSE IF( IASCL.EQ.2 ) THEN
00612          CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
00613          CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
00614      $                INFO )
00615       END IF
00616       IF( IBSCL.EQ.1 ) THEN
00617          CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
00618       ELSE IF( IBSCL.EQ.2 ) THEN
00619          CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
00620       END IF
00621 *
00622    10 CONTINUE
00623       WORK( 1 ) = MAXWRK
00624       IWORK( 1 ) = LIWORK
00625       RETURN
00626 *
00627 *     End of DGELSD
00628 *
00629       END
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