LAPACK: SRC/dgelsd.f Source File

Go to the documentation of this file.
00001       SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
00002      $                   WORK, LWORK, IWORK, INFO )
00003 *
00004 *  -- LAPACK driver routine (version 3.2.2) --
00005 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00006 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00007 *     June 2010
00008 *
00009 *     .. Scalar Arguments ..
00010       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
00011       DOUBLE PRECISION   RCOND
00012 *     ..
00013 *     .. Array Arguments ..
00014       INTEGER            IWORK( * )
00015       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
00016 *     ..
00017 *
00018 *  Purpose
00019 *  =======
00020 *
00021 *  DGELSD computes the minimum-norm solution to a real linear least
00022 *  squares problem:
00023 *      minimize 2-norm(| b - A*x |)
00024 *  using the singular value decomposition (SVD) of A. A is an M-by-N
00025 *  matrix which may be rank-deficient.
00026 *
00027 *  Several right hand side vectors b and solution vectors x can be
00028 *  handled in a single call; they are stored as the columns of the
00029 *  M-by-NRHS right hand side matrix B and the N-by-NRHS solution
00030 *  matrix X.
00031 *
00032 *  The problem is solved in three steps:
00033 *  (1) Reduce the coefficient matrix A to bidiagonal form with
00034 *      Householder transformations, reducing the original problem
00035 *      into a "bidiagonal least squares problem" (BLS)
00036 *  (2) Solve the BLS using a divide and conquer approach.
00037 *  (3) Apply back all the Householder tranformations to solve
00038 *      the original least squares problem.
00039 *
00040 *  The effective rank of A is determined by treating as zero those
00041 *  singular values which are less than RCOND times the largest singular
00042 *  value.
00043 *
00044 *  The divide and conquer algorithm makes very mild assumptions about
00045 *  floating point arithmetic. It will work on machines with a guard
00046 *  digit in add/subtract, or on those binary machines without guard
00047 *  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
00048 *  Cray-2. It could conceivably fail on hexadecimal or decimal machines
00049 *  without guard digits, but we know of none.
00050 *
00051 *  Arguments
00052 *  =========
00053 *
00054 *  M       (input) INTEGER
00055 *          The number of rows of A. M >= 0.
00056 *
00057 *  N       (input) INTEGER
00058 *          The number of columns of A. N >= 0.
00059 *
00060 *  NRHS    (input) INTEGER
00061 *          The number of right hand sides, i.e., the number of columns
00062 *          of the matrices B and X. NRHS >= 0.
00063 *
00064 *  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
00065 *          On entry, the M-by-N matrix A.
00066 *          On exit, A has been destroyed.
00067 *
00068 *  LDA     (input) INTEGER
00069 *          The leading dimension of the array A.  LDA >= max(1,M).
00070 *
00071 *  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
00072 *          On entry, the M-by-NRHS right hand side matrix B.
00073 *          On exit, B is overwritten by the N-by-NRHS solution
00074 *          matrix X.  If m >= n and RANK = n, the residual
00075 *          sum-of-squares for the solution in the i-th column is given
00076 *          by the sum of squares of elements n+1:m in that column.
00077 *
00078 *  LDB     (input) INTEGER
00079 *          The leading dimension of the array B. LDB >= max(1,max(M,N)).
00080 *
00081 *  S       (output) DOUBLE PRECISION array, dimension (min(M,N))
00082 *          The singular values of A in decreasing order.
00083 *          The condition number of A in the 2-norm = S(1)/S(min(m,n)).
00084 *
00085 *  RCOND   (input) DOUBLE PRECISION
00086 *          RCOND is used to determine the effective rank of A.
00087 *          Singular values S(i) <= RCOND*S(1) are treated as zero.
00088 *          If RCOND < 0, machine precision is used instead.
00089 *
00090 *  RANK    (output) INTEGER
00091 *          The effective rank of A, i.e., the number of singular values
00092 *          which are greater than RCOND*S(1).
00093 *
00094 *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
00095 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00096 *
00097 *  LWORK   (input) INTEGER
00098 *          The dimension of the array WORK. LWORK must be at least 1.
00099 *          The exact minimum amount of workspace needed depends on M,
00100 *          N and NRHS. As long as LWORK is at least
00101 *              12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
00102 *          if M is greater than or equal to N or
00103 *              12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
00104 *          if M is less than N, the code will execute correctly.
00105 *          SMLSIZ is returned by ILAENV and is equal to the maximum
00106 *          size of the subproblems at the bottom of the computation
00107 *          tree (usually about 25), and
00108 *             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
00109 *          For good performance, LWORK should generally be larger.
00110 *
00111 *          If LWORK = -1, then a workspace query is assumed; the routine
00112 *          only calculates the optimal size of the WORK array, returns
00113 *          this value as the first entry of the WORK array, and no error
00114 *          message related to LWORK is issued by XERBLA.
00115 *
00116 *  IWORK   (workspace) INTEGER array, dimension (MAX(1,LIWORK))
00117 *          LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),
00118 *          where MINMN = MIN( M,N ).
00119 *          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
00120 *
00121 *  INFO    (output) INTEGER
00122 *          = 0:  successful exit
00123 *          < 0:  if INFO = -i, the i-th argument had an illegal value.
00124 *          > 0:  the algorithm for computing the SVD failed to converge;
00125 *                if INFO = i, i off-diagonal elements of an intermediate
00126 *                bidiagonal form did not converge to zero.
00127 *
00128 *  Further Details
00129 *  ===============
00130 *
00131 *  Based on contributions by
00132 *     Ming Gu and Ren-Cang Li, Computer Science Division, University of
00133 *       California at Berkeley, USA
00134 *     Osni Marques, LBNL/NERSC, USA
00135 *
00136 *  =====================================================================
00137 *
00138 *     .. Parameters ..
00139       DOUBLE PRECISION   ZERO, ONE, TWO
00140       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
00141 *     ..
00142 *     .. Local Scalars ..
00143       LOGICAL            LQUERY
00144       INTEGER            IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
00145      $                   LDWORK, LIWORK, MAXMN, MAXWRK, MINMN, MINWRK,
00146      $                   MM, MNTHR, NLVL, NWORK, SMLSIZ, WLALSD
00147       DOUBLE PRECISION   ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
00148 *     ..
00149 *     .. External Subroutines ..
00150       EXTERNAL           DGEBRD, DGELQF, DGEQRF, DLABAD, DLACPY, DLALSD,
00151      $                   DLASCL, DLASET, DORMBR, DORMLQ, DORMQR, XERBLA
00152 *     ..
00153 *     .. External Functions ..
00154       INTEGER            ILAENV
00155       DOUBLE PRECISION   DLAMCH, DLANGE
00156       EXTERNAL           ILAENV, DLAMCH, DLANGE
00157 *     ..
00158 *     .. Intrinsic Functions ..
00159       INTRINSIC          DBLE, INT, LOG, MAX, MIN
00160 *     ..
00161 *     .. Executable Statements ..
00162 *
00163 *     Test the input arguments.
00164 *
00165       INFO = 0
00166       MINMN = MIN( M, N )
00167       MAXMN = MAX( M, N )
00168       MNTHR = ILAENV( 6, 'DGELSD', ' ', M, N, NRHS, -1 )
00169       LQUERY = ( LWORK.EQ.-1 )
00170       IF( M.LT.0 ) THEN
00171          INFO = -1
00172       ELSE IF( N.LT.0 ) THEN
00173          INFO = -2
00174       ELSE IF( NRHS.LT.0 ) THEN
00175          INFO = -3
00176       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
00177          INFO = -5
00178       ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
00179          INFO = -7
00180       END IF
00181 *
00182       SMLSIZ = ILAENV( 9, 'DGELSD', ' ', 0, 0, 0, 0 )
00183 *
00184 *     Compute workspace.
00185 *     (Note: Comments in the code beginning "Workspace:" describe the
00186 *     minimal amount of workspace needed at that point in the code,
00187 *     as well as the preferred amount for good performance.
00188 *     NB refers to the optimal block size for the immediately
00189 *     following subroutine, as returned by ILAENV.)
00190 *
00191       MINWRK = 1
00192       LIWORK = 1
00193       MINMN = MAX( 1, MINMN )
00194       NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ+1 ) ) /
00195      $       LOG( TWO ) ) + 1, 0 )
00196 *
00197       IF( INFO.EQ.0 ) THEN
00198          MAXWRK = 0
00199          LIWORK = 3*MINMN*NLVL + 11*MINMN
00200          MM = M
00201          IF( M.GE.N .AND. M.GE.MNTHR ) THEN
00202 *
00203 *           Path 1a - overdetermined, with many more rows than columns.
00204 *
00205             MM = N
00206             MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'DGEQRF', ' ', M, N,
00207      $               -1, -1 ) )
00208             MAXWRK = MAX( MAXWRK, N+NRHS*
00209      $               ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, -1 ) )
00210          END IF
00211          IF( M.GE.N ) THEN
00212 *
00213 *           Path 1 - overdetermined or exactly determined.
00214 *
00215             MAXWRK = MAX( MAXWRK, 3*N+( MM+N )*
00216      $               ILAENV( 1, 'DGEBRD', ' ', MM, N, -1, -1 ) )
00217             MAXWRK = MAX( MAXWRK, 3*N+NRHS*
00218      $               ILAENV( 1, 'DORMBR', 'QLT', MM, NRHS, N, -1 ) )
00219             MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*
00220      $               ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, N, -1 ) )
00221             WLALSD = 9*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS+(SMLSIZ+1)**2
00222             MAXWRK = MAX( MAXWRK, 3*N+WLALSD )
00223             MINWRK = MAX( 3*N+MM, 3*N+NRHS, 3*N+WLALSD )
00224          END IF
00225          IF( N.GT.M ) THEN
00226             WLALSD = 9*M+2*M*SMLSIZ+8*M*NLVL+M*NRHS+(SMLSIZ+1)**2
00227             IF( N.GE.MNTHR ) THEN
00228 *
00229 *              Path 2a - underdetermined, with many more columns
00230 *              than rows.
00231 *
00232                MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
00233                MAXWRK = MAX( MAXWRK, M*M+4*M+2*M*
00234      $                  ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
00235                MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS*
00236      $                  ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M, -1 ) )
00237                MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )*
00238      $                  ILAENV( 1, 'DORMBR', 'PLN', M, NRHS, M, -1 ) )
00239                IF( NRHS.GT.1 ) THEN
00240                   MAXWRK = MAX( MAXWRK, M*M+M+M*NRHS )
00241                ELSE
00242                   MAXWRK = MAX( MAXWRK, M*M+2*M )
00243                END IF
00244                MAXWRK = MAX( MAXWRK, M+NRHS*
00245      $                  ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, -1 ) )
00246                MAXWRK = MAX( MAXWRK, M*M+4*M+WLALSD )
00247 !     XXX: Ensure the Path 2a case below is triggered.  The workspace
00248 !     calculation should use queries for all routines eventually.
00249                MAXWRK = MAX( MAXWRK,
00250      $              4*M+M*M+MAX( M, 2*M-4, NRHS, N-3*M ) )
00251             ELSE
00252 *
00253 *              Path 2 - remaining underdetermined cases.
00254 *
00255                MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'DGEBRD', ' ', M, N,
00256      $                  -1, -1 )
00257                MAXWRK = MAX( MAXWRK, 3*M+NRHS*
00258      $                  ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, N, -1 ) )
00259                MAXWRK = MAX( MAXWRK, 3*M+M*
00260      $                  ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, M, -1 ) )
00261                MAXWRK = MAX( MAXWRK, 3*M+WLALSD )
00262             END IF
00263             MINWRK = MAX( 3*M+NRHS, 3*M+M, 3*M+WLALSD )
00264          END IF
00265          MINWRK = MIN( MINWRK, MAXWRK )
00266          WORK( 1 ) = MAXWRK
00267          IWORK( 1 ) = LIWORK
00268 
00269          IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
00270             INFO = -12
00271          END IF
00272       END IF
00273 *
00274       IF( INFO.NE.0 ) THEN
00275          CALL XERBLA( 'DGELSD', -INFO )
00276          RETURN
00277       ELSE IF( LQUERY ) THEN
00278          GO TO 10
00279       END IF
00280 *
00281 *     Quick return if possible.
00282 *
00283       IF( M.EQ.0 .OR. N.EQ.0 ) THEN
00284          RANK = 0
00285          RETURN
00286       END IF
00287 *
00288 *     Get machine parameters.
00289 *
00290       EPS = DLAMCH( 'P' )
00291       SFMIN = DLAMCH( 'S' )
00292       SMLNUM = SFMIN / EPS
00293       BIGNUM = ONE / SMLNUM
00294       CALL DLABAD( SMLNUM, BIGNUM )
00295 *
00296 *     Scale A if max entry outside range [SMLNUM,BIGNUM].
00297 *
00298       ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
00299       IASCL = 0
00300       IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
00301 *
00302 *        Scale matrix norm up to SMLNUM.
00303 *
00304          CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
00305          IASCL = 1
00306       ELSE IF( ANRM.GT.BIGNUM ) THEN
00307 *
00308 *        Scale matrix norm down to BIGNUM.
00309 *
00310          CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
00311          IASCL = 2
00312       ELSE IF( ANRM.EQ.ZERO ) THEN
00313 *
00314 *        Matrix all zero. Return zero solution.
00315 *
00316          CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
00317          CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
00318          RANK = 0
00319          GO TO 10
00320       END IF
00321 *
00322 *     Scale B if max entry outside range [SMLNUM,BIGNUM].
00323 *
00324       BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
00325       IBSCL = 0
00326       IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
00327 *
00328 *        Scale matrix norm up to SMLNUM.
00329 *
00330          CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
00331          IBSCL = 1
00332       ELSE IF( BNRM.GT.BIGNUM ) THEN
00333 *
00334 *        Scale matrix norm down to BIGNUM.
00335 *
00336          CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
00337          IBSCL = 2
00338       END IF
00339 *
00340 *     If M < N make sure certain entries of B are zero.
00341 *
00342       IF( M.LT.N )
00343      $   CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
00344 *
00345 *     Overdetermined case.
00346 *
00347       IF( M.GE.N ) THEN
00348 *
00349 *        Path 1 - overdetermined or exactly determined.
00350 *
00351          MM = M
00352          IF( M.GE.MNTHR ) THEN
00353 *
00354 *           Path 1a - overdetermined, with many more rows than columns.
00355 *
00356             MM = N
00357             ITAU = 1
00358             NWORK = ITAU + N
00359 *
00360 *           Compute A=Q*R.
00361 *           (Workspace: need 2*N, prefer N+N*NB)
00362 *
00363             CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
00364      $                   LWORK-NWORK+1, INFO )
00365 *
00366 *           Multiply B by transpose(Q).
00367 *           (Workspace: need N+NRHS, prefer N+NRHS*NB)
00368 *
00369             CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B,
00370      $                   LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00371 *
00372 *           Zero out below R.
00373 *
00374             IF( N.GT.1 ) THEN
00375                CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
00376             END IF
00377          END IF
00378 *
00379          IE = 1
00380          ITAUQ = IE + N
00381          ITAUP = ITAUQ + N
00382          NWORK = ITAUP + N
00383 *
00384 *        Bidiagonalize R in A.
00385 *        (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
00386 *
00387          CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
00388      $                WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
00389      $                INFO )
00390 *
00391 *        Multiply B by transpose of left bidiagonalizing vectors of R.
00392 *        (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
00393 *
00394          CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ),
00395      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00396 *
00397 *        Solve the bidiagonal least squares problem.
00398 *
00399          CALL DLALSD( 'U', SMLSIZ, N, NRHS, S, WORK( IE ), B, LDB,
00400      $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
00401          IF( INFO.NE.0 ) THEN
00402             GO TO 10
00403          END IF
00404 *
00405 *        Multiply B by right bidiagonalizing vectors of R.
00406 *
00407          CALL DORMBR( 'P', 'L', 'N', N, NRHS, N, A, LDA, WORK( ITAUP ),
00408      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00409 *
00410       ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
00411      $         MAX( M, 2*M-4, NRHS, N-3*M, WLALSD ) ) THEN
00412 *
00413 *        Path 2a - underdetermined, with many more columns than rows
00414 *        and sufficient workspace for an efficient algorithm.
00415 *
00416          LDWORK = M
00417          IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
00418      $       M*LDA+M+M*NRHS, 4*M+M*LDA+WLALSD ) )LDWORK = LDA
00419          ITAU = 1
00420          NWORK = M + 1
00421 *
00422 *        Compute A=L*Q.
00423 *        (Workspace: need 2*M, prefer M+M*NB)
00424 *
00425          CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
00426      $                LWORK-NWORK+1, INFO )
00427          IL = NWORK
00428 *
00429 *        Copy L to WORK(IL), zeroing out above its diagonal.
00430 *
00431          CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
00432          CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ),
00433      $                LDWORK )
00434          IE = IL + LDWORK*M
00435          ITAUQ = IE + M
00436          ITAUP = ITAUQ + M
00437          NWORK = ITAUP + M
00438 *
00439 *        Bidiagonalize L in WORK(IL).
00440 *        (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
00441 *
00442          CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
00443      $                WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
00444      $                LWORK-NWORK+1, INFO )
00445 *
00446 *        Multiply B by transpose of left bidiagonalizing vectors of L.
00447 *        (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
00448 *
00449          CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
00450      $                WORK( ITAUQ ), B, LDB, WORK( NWORK ),
00451      $                LWORK-NWORK+1, INFO )
00452 *
00453 *        Solve the bidiagonal least squares problem.
00454 *
00455          CALL DLALSD( 'U', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
00456      $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
00457          IF( INFO.NE.0 ) THEN
00458             GO TO 10
00459          END IF
00460 *
00461 *        Multiply B by right bidiagonalizing vectors of L.
00462 *
00463          CALL DORMBR( 'P', 'L', 'N', M, NRHS, M, WORK( IL ), LDWORK,
00464      $                WORK( ITAUP ), B, LDB, WORK( NWORK ),
00465      $                LWORK-NWORK+1, INFO )
00466 *
00467 *        Zero out below first M rows of B.
00468 *
00469          CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
00470          NWORK = ITAU + M
00471 *
00472 *        Multiply transpose(Q) by B.
00473 *        (Workspace: need M+NRHS, prefer M+NRHS*NB)
00474 *
00475          CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
00476      $                LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00477 *
00478       ELSE
00479 *
00480 *        Path 2 - remaining underdetermined cases.
00481 *
00482          IE = 1
00483          ITAUQ = IE + M
00484          ITAUP = ITAUQ + M
00485          NWORK = ITAUP + M
00486 *
00487 *        Bidiagonalize A.
00488 *        (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
00489 *
00490          CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
00491      $                WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
00492      $                INFO )
00493 *
00494 *        Multiply B by transpose of left bidiagonalizing vectors.
00495 *        (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
00496 *
00497          CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ),
00498      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00499 *
00500 *        Solve the bidiagonal least squares problem.
00501 *
00502          CALL DLALSD( 'L', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
00503      $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
00504          IF( INFO.NE.0 ) THEN
00505             GO TO 10
00506          END IF
00507 *
00508 *        Multiply B by right bidiagonalizing vectors of A.
00509 *
00510          CALL DORMBR( 'P', 'L', 'N', N, NRHS, M, A, LDA, WORK( ITAUP ),
00511      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
00512 *
00513       END IF
00514 *
00515 *     Undo scaling.
00516 *
00517       IF( IASCL.EQ.1 ) THEN
00518          CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
00519          CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
00520      $                INFO )
00521       ELSE IF( IASCL.EQ.2 ) THEN
00522          CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
00523          CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
00524      $                INFO )
00525       END IF
00526       IF( IBSCL.EQ.1 ) THEN
00527          CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
00528       ELSE IF( IBSCL.EQ.2 ) THEN
00529          CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
00530       END IF
00531 *
00532    10 CONTINUE
00533       WORK( 1 ) = MAXWRK
00534       IWORK( 1 ) = LIWORK
00535       RETURN
00536 *
00537 *     End of DGELSD
00538 *
00539       END