Return to dgelsd.f CVS log

Up to [local] / rpl / lapack / lapack

Mise à jour de lapack vers la version 3.3.0.

    1:       SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
    2:      $                   WORK, LWORK, IWORK, INFO )
    3: *
    4: *  -- LAPACK driver routine (version 3.2.2) --
    5: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    6: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    7: *     June 2010
    8: *
    9: *     .. Scalar Arguments ..
   10:       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
   11:       DOUBLE PRECISION   RCOND
   12: *     ..
   13: *     .. Array Arguments ..
   14:       INTEGER            IWORK( * )
   15:       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
   16: *     ..
   17: *
   18: *  Purpose
   19: *  =======
   20: *
   21: *  DGELSD computes the minimum-norm solution to a real linear least
   22: *  squares problem:
   23: *      minimize 2-norm(| b - A*x |)
   24: *  using the singular value decomposition (SVD) of A. A is an M-by-N
   25: *  matrix which may be rank-deficient.
   26: *
   27: *  Several right hand side vectors b and solution vectors x can be
   28: *  handled in a single call; they are stored as the columns of the
   29: *  M-by-NRHS right hand side matrix B and the N-by-NRHS solution
   30: *  matrix X.
   31: *
   32: *  The problem is solved in three steps:
   33: *  (1) Reduce the coefficient matrix A to bidiagonal form with
   34: *      Householder transformations, reducing the original problem
   35: *      into a "bidiagonal least squares problem" (BLS)
   36: *  (2) Solve the BLS using a divide and conquer approach.
   37: *  (3) Apply back all the Householder tranformations to solve
   38: *      the original least squares problem.
   39: *
   40: *  The effective rank of A is determined by treating as zero those
   41: *  singular values which are less than RCOND times the largest singular
   42: *  value.
   43: *
   44: *  The divide and conquer algorithm makes very mild assumptions about
   45: *  floating point arithmetic. It will work on machines with a guard
   46: *  digit in add/subtract, or on those binary machines without guard
   47: *  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
   48: *  Cray-2. It could conceivably fail on hexadecimal or decimal machines
   49: *  without guard digits, but we know of none.
   50: *
   51: *  Arguments
   52: *  =========
   53: *
   54: *  M       (input) INTEGER
   55: *          The number of rows of A. M >= 0.
   56: *
   57: *  N       (input) INTEGER
   58: *          The number of columns of A. N >= 0.
   59: *
   60: *  NRHS    (input) INTEGER
   61: *          The number of right hand sides, i.e., the number of columns
   62: *          of the matrices B and X. NRHS >= 0.
   63: *
   64: *  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
   65: *          On entry, the M-by-N matrix A.
   66: *          On exit, A has been destroyed.
   67: *
   68: *  LDA     (input) INTEGER
   69: *          The leading dimension of the array A.  LDA >= max(1,M).
   70: *
   71: *  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
   72: *          On entry, the M-by-NRHS right hand side matrix B.
   73: *          On exit, B is overwritten by the N-by-NRHS solution
   74: *          matrix X.  If m >= n and RANK = n, the residual
   75: *          sum-of-squares for the solution in the i-th column is given
   76: *          by the sum of squares of elements n+1:m in that column.
   77: *
   78: *  LDB     (input) INTEGER
   79: *          The leading dimension of the array B. LDB >= max(1,max(M,N)).
   80: *
   81: *  S       (output) DOUBLE PRECISION array, dimension (min(M,N))
   82: *          The singular values of A in decreasing order.
   83: *          The condition number of A in the 2-norm = S(1)/S(min(m,n)).
   84: *
   85: *  RCOND   (input) DOUBLE PRECISION
   86: *          RCOND is used to determine the effective rank of A.
   87: *          Singular values S(i) <= RCOND*S(1) are treated as zero.
   88: *          If RCOND < 0, machine precision is used instead.
   89: *
   90: *  RANK    (output) INTEGER
   91: *          The effective rank of A, i.e., the number of singular values
   92: *          which are greater than RCOND*S(1).
   93: *
   94: *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
   95: *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   96: *
   97: *  LWORK   (input) INTEGER
   98: *          The dimension of the array WORK. LWORK must be at least 1.
   99: *          The exact minimum amount of workspace needed depends on M,
  100: *          N and NRHS. As long as LWORK is at least
  101: *              12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
  102: *          if M is greater than or equal to N or
  103: *              12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
  104: *          if M is less than N, the code will execute correctly.
  105: *          SMLSIZ is returned by ILAENV and is equal to the maximum
  106: *          size of the subproblems at the bottom of the computation
  107: *          tree (usually about 25), and
  108: *             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
  109: *          For good performance, LWORK should generally be larger.
  110: *
  111: *          If LWORK = -1, then a workspace query is assumed; the routine
  112: *          only calculates the optimal size of the WORK array, returns
  113: *          this value as the first entry of the WORK array, and no error
  114: *          message related to LWORK is issued by XERBLA.
  115: *
  116: *  IWORK   (workspace) INTEGER array, dimension (MAX(1,LIWORK))
  117: *          LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),
  118: *          where MINMN = MIN( M,N ).
  119: *          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
  120: *
  121: *  INFO    (output) INTEGER
  122: *          = 0:  successful exit
  123: *          < 0:  if INFO = -i, the i-th argument had an illegal value.
  124: *          > 0:  the algorithm for computing the SVD failed to converge;
  125: *                if INFO = i, i off-diagonal elements of an intermediate
  126: *                bidiagonal form did not converge to zero.
  127: *
  128: *  Further Details
  129: *  ===============
  130: *
  131: *  Based on contributions by
  132: *     Ming Gu and Ren-Cang Li, Computer Science Division, University of
  133: *       California at Berkeley, USA
  134: *     Osni Marques, LBNL/NERSC, USA
  135: *
  136: *  =====================================================================
  137: *
  138: *     .. Parameters ..
  139:       DOUBLE PRECISION   ZERO, ONE, TWO
  140:       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
  141: *     ..
  142: *     .. Local Scalars ..
  143:       LOGICAL            LQUERY
  144:       INTEGER            IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
  145:      $                   LDWORK, LIWORK, MAXMN, MAXWRK, MINMN, MINWRK,
  146:      $                   MM, MNTHR, NLVL, NWORK, SMLSIZ, WLALSD
  147:       DOUBLE PRECISION   ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
  148: *     ..
  149: *     .. External Subroutines ..
  150:       EXTERNAL           DGEBRD, DGELQF, DGEQRF, DLABAD, DLACPY, DLALSD,
  151:      $                   DLASCL, DLASET, DORMBR, DORMLQ, DORMQR, XERBLA
  152: *     ..
  153: *     .. External Functions ..
  154:       INTEGER            ILAENV
  155:       DOUBLE PRECISION   DLAMCH, DLANGE
  156:       EXTERNAL           ILAENV, DLAMCH, DLANGE
  157: *     ..
  158: *     .. Intrinsic Functions ..
  159:       INTRINSIC          DBLE, INT, LOG, MAX, MIN
  160: *     ..
  161: *     .. Executable Statements ..
  162: *
  163: *     Test the input arguments.
  164: *
  165:       INFO = 0
  166:       MINMN = MIN( M, N )
  167:       MAXMN = MAX( M, N )
  168:       MNTHR = ILAENV( 6, 'DGELSD', ' ', M, N, NRHS, -1 )
  169:       LQUERY = ( LWORK.EQ.-1 )
  170:       IF( M.LT.0 ) THEN
  171:          INFO = -1
  172:       ELSE IF( N.LT.0 ) THEN
  173:          INFO = -2
  174:       ELSE IF( NRHS.LT.0 ) THEN
  175:          INFO = -3
  176:       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
  177:          INFO = -5
  178:       ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
  179:          INFO = -7
  180:       END IF
  181: *
  182:       SMLSIZ = ILAENV( 9, 'DGELSD', ' ', 0, 0, 0, 0 )
  183: *
  184: *     Compute workspace.
  185: *     (Note: Comments in the code beginning "Workspace:" describe the
  186: *     minimal amount of workspace needed at that point in the code,
  187: *     as well as the preferred amount for good performance.
  188: *     NB refers to the optimal block size for the immediately
  189: *     following subroutine, as returned by ILAENV.)
  190: *
  191:       MINWRK = 1
  192:       LIWORK = 1
  193:       MINMN = MAX( 1, MINMN )
  194:       NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ+1 ) ) /
  195:      $       LOG( TWO ) ) + 1, 0 )
  196: *
  197:       IF( INFO.EQ.0 ) THEN
  198:          MAXWRK = 0
  199:          LIWORK = 3*MINMN*NLVL + 11*MINMN
  200:          MM = M
  201:          IF( M.GE.N .AND. M.GE.MNTHR ) THEN
  202: *
  203: *           Path 1a - overdetermined, with many more rows than columns.
  204: *
  205:             MM = N
  206:             MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'DGEQRF', ' ', M, N,
  207:      $               -1, -1 ) )
  208:             MAXWRK = MAX( MAXWRK, N+NRHS*
  209:      $               ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, -1 ) )
  210:          END IF
  211:          IF( M.GE.N ) THEN
  212: *
  213: *           Path 1 - overdetermined or exactly determined.
  214: *
  215:             MAXWRK = MAX( MAXWRK, 3*N+( MM+N )*
  216:      $               ILAENV( 1, 'DGEBRD', ' ', MM, N, -1, -1 ) )
  217:             MAXWRK = MAX( MAXWRK, 3*N+NRHS*
  218:      $               ILAENV( 1, 'DORMBR', 'QLT', MM, NRHS, N, -1 ) )
  219:             MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*
  220:      $               ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, N, -1 ) )
  221:             WLALSD = 9*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS+(SMLSIZ+1)**2
  222:             MAXWRK = MAX( MAXWRK, 3*N+WLALSD )
  223:             MINWRK = MAX( 3*N+MM, 3*N+NRHS, 3*N+WLALSD )
  224:          END IF
  225:          IF( N.GT.M ) THEN
  226:             WLALSD = 9*M+2*M*SMLSIZ+8*M*NLVL+M*NRHS+(SMLSIZ+1)**2
  227:             IF( N.GE.MNTHR ) THEN
  228: *
  229: *              Path 2a - underdetermined, with many more columns
  230: *              than rows.
  231: *
  232:                MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
  233:                MAXWRK = MAX( MAXWRK, M*M+4*M+2*M*
  234:      $                  ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
  235:                MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS*
  236:      $                  ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M, -1 ) )
  237:                MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )*
  238:      $                  ILAENV( 1, 'DORMBR', 'PLN', M, NRHS, M, -1 ) )
  239:                IF( NRHS.GT.1 ) THEN
  240:                   MAXWRK = MAX( MAXWRK, M*M+M+M*NRHS )
  241:                ELSE
  242:                   MAXWRK = MAX( MAXWRK, M*M+2*M )
  243:                END IF
  244:                MAXWRK = MAX( MAXWRK, M+NRHS*
  245:      $                  ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, -1 ) )
  246:                MAXWRK = MAX( MAXWRK, M*M+4*M+WLALSD )
  247: !     XXX: Ensure the Path 2a case below is triggered.  The workspace
  248: !     calculation should use queries for all routines eventually.
  249:                MAXWRK = MAX( MAXWRK,
  250:      $              4*M+M*M+MAX( M, 2*M-4, NRHS, N-3*M ) )
  251:             ELSE
  252: *
  253: *              Path 2 - remaining underdetermined cases.
  254: *
  255:                MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'DGEBRD', ' ', M, N,
  256:      $                  -1, -1 )
  257:                MAXWRK = MAX( MAXWRK, 3*M+NRHS*
  258:      $                  ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, N, -1 ) )
  259:                MAXWRK = MAX( MAXWRK, 3*M+M*
  260:      $                  ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, M, -1 ) )
  261:                MAXWRK = MAX( MAXWRK, 3*M+WLALSD )
  262:             END IF
  263:             MINWRK = MAX( 3*M+NRHS, 3*M+M, 3*M+WLALSD )
  264:          END IF
  265:          MINWRK = MIN( MINWRK, MAXWRK )
  266:          WORK( 1 ) = MAXWRK
  267:          IWORK( 1 ) = LIWORK
  268: 
  269:          IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
  270:             INFO = -12
  271:          END IF
  272:       END IF
  273: *
  274:       IF( INFO.NE.0 ) THEN
  275:          CALL XERBLA( 'DGELSD', -INFO )
  276:          RETURN
  277:       ELSE IF( LQUERY ) THEN
  278:          GO TO 10
  279:       END IF
  280: *
  281: *     Quick return if possible.
  282: *
  283:       IF( M.EQ.0 .OR. N.EQ.0 ) THEN
  284:          RANK = 0
  285:          RETURN
  286:       END IF
  287: *
  288: *     Get machine parameters.
  289: *
  290:       EPS = DLAMCH( 'P' )
  291:       SFMIN = DLAMCH( 'S' )
  292:       SMLNUM = SFMIN / EPS
  293:       BIGNUM = ONE / SMLNUM
  294:       CALL DLABAD( SMLNUM, BIGNUM )
  295: *
  296: *     Scale A if max entry outside range [SMLNUM,BIGNUM].
  297: *
  298:       ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
  299:       IASCL = 0
  300:       IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
  301: *
  302: *        Scale matrix norm up to SMLNUM.
  303: *
  304:          CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
  305:          IASCL = 1
  306:       ELSE IF( ANRM.GT.BIGNUM ) THEN
  307: *
  308: *        Scale matrix norm down to BIGNUM.
  309: *
  310:          CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
  311:          IASCL = 2
  312:       ELSE IF( ANRM.EQ.ZERO ) THEN
  313: *
  314: *        Matrix all zero. Return zero solution.
  315: *
  316:          CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
  317:          CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
  318:          RANK = 0
  319:          GO TO 10
  320:       END IF
  321: *
  322: *     Scale B if max entry outside range [SMLNUM,BIGNUM].
  323: *
  324:       BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
  325:       IBSCL = 0
  326:       IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
  327: *
  328: *        Scale matrix norm up to SMLNUM.
  329: *
  330:          CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
  331:          IBSCL = 1
  332:       ELSE IF( BNRM.GT.BIGNUM ) THEN
  333: *
  334: *        Scale matrix norm down to BIGNUM.
  335: *
  336:          CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
  337:          IBSCL = 2
  338:       END IF
  339: *
  340: *     If M < N make sure certain entries of B are zero.
  341: *
  342:       IF( M.LT.N )
  343:      $   CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
  344: *
  345: *     Overdetermined case.
  346: *
  347:       IF( M.GE.N ) THEN
  348: *
  349: *        Path 1 - overdetermined or exactly determined.
  350: *
  351:          MM = M
  352:          IF( M.GE.MNTHR ) THEN
  353: *
  354: *           Path 1a - overdetermined, with many more rows than columns.
  355: *
  356:             MM = N
  357:             ITAU = 1
  358:             NWORK = ITAU + N
  359: *
  360: *           Compute A=Q*R.
  361: *           (Workspace: need 2*N, prefer N+N*NB)
  362: *
  363:             CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
  364:      $                   LWORK-NWORK+1, INFO )
  365: *
  366: *           Multiply B by transpose(Q).
  367: *           (Workspace: need N+NRHS, prefer N+NRHS*NB)
  368: *
  369:             CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B,
  370:      $                   LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
  371: *
  372: *           Zero out below R.
  373: *
  374:             IF( N.GT.1 ) THEN
  375:                CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
  376:             END IF
  377:          END IF
  378: *
  379:          IE = 1
  380:          ITAUQ = IE + N
  381:          ITAUP = ITAUQ + N
  382:          NWORK = ITAUP + N
  383: *
  384: *        Bidiagonalize R in A.
  385: *        (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
  386: *
  387:          CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
  388:      $                WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
  389:      $                INFO )
  390: *
  391: *        Multiply B by transpose of left bidiagonalizing vectors of R.
  392: *        (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
  393: *
  394:          CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ),
  395:      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
  396: *
  397: *        Solve the bidiagonal least squares problem.
  398: *
  399:          CALL DLALSD( 'U', SMLSIZ, N, NRHS, S, WORK( IE ), B, LDB,
  400:      $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
  401:          IF( INFO.NE.0 ) THEN
  402:             GO TO 10
  403:          END IF
  404: *
  405: *        Multiply B by right bidiagonalizing vectors of R.
  406: *
  407:          CALL DORMBR( 'P', 'L', 'N', N, NRHS, N, A, LDA, WORK( ITAUP ),
  408:      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
  409: *
  410:       ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
  411:      $         MAX( M, 2*M-4, NRHS, N-3*M, WLALSD ) ) THEN
  412: *
  413: *        Path 2a - underdetermined, with many more columns than rows
  414: *        and sufficient workspace for an efficient algorithm.
  415: *
  416:          LDWORK = M
  417:          IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
  418:      $       M*LDA+M+M*NRHS, 4*M+M*LDA+WLALSD ) )LDWORK = LDA
  419:          ITAU = 1
  420:          NWORK = M + 1
  421: *
  422: *        Compute A=L*Q.
  423: *        (Workspace: need 2*M, prefer M+M*NB)
  424: *
  425:          CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
  426:      $                LWORK-NWORK+1, INFO )
  427:          IL = NWORK
  428: *
  429: *        Copy L to WORK(IL), zeroing out above its diagonal.
  430: *
  431:          CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
  432:          CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ),
  433:      $                LDWORK )
  434:          IE = IL + LDWORK*M
  435:          ITAUQ = IE + M
  436:          ITAUP = ITAUQ + M
  437:          NWORK = ITAUP + M
  438: *
  439: *        Bidiagonalize L in WORK(IL).
  440: *        (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
  441: *
  442:          CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
  443:      $                WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
  444:      $                LWORK-NWORK+1, INFO )
  445: *
  446: *        Multiply B by transpose of left bidiagonalizing vectors of L.
  447: *        (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
  448: *
  449:          CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
  450:      $                WORK( ITAUQ ), B, LDB, WORK( NWORK ),
  451:      $                LWORK-NWORK+1, INFO )
  452: *
  453: *        Solve the bidiagonal least squares problem.
  454: *
  455:          CALL DLALSD( 'U', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
  456:      $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
  457:          IF( INFO.NE.0 ) THEN
  458:             GO TO 10
  459:          END IF
  460: *
  461: *        Multiply B by right bidiagonalizing vectors of L.
  462: *
  463:          CALL DORMBR( 'P', 'L', 'N', M, NRHS, M, WORK( IL ), LDWORK,
  464:      $                WORK( ITAUP ), B, LDB, WORK( NWORK ),
  465:      $                LWORK-NWORK+1, INFO )
  466: *
  467: *        Zero out below first M rows of B.
  468: *
  469:          CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
  470:          NWORK = ITAU + M
  471: *
  472: *        Multiply transpose(Q) by B.
  473: *        (Workspace: need M+NRHS, prefer M+NRHS*NB)
  474: *
  475:          CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
  476:      $                LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
  477: *
  478:       ELSE
  479: *
  480: *        Path 2 - remaining underdetermined cases.
  481: *
  482:          IE = 1
  483:          ITAUQ = IE + M
  484:          ITAUP = ITAUQ + M
  485:          NWORK = ITAUP + M
  486: *
  487: *        Bidiagonalize A.
  488: *        (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
  489: *
  490:          CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
  491:      $                WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
  492:      $                INFO )
  493: *
  494: *        Multiply B by transpose of left bidiagonalizing vectors.
  495: *        (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
  496: *
  497:          CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ),
  498:      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
  499: *
  500: *        Solve the bidiagonal least squares problem.
  501: *
  502:          CALL DLALSD( 'L', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
  503:      $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
  504:          IF( INFO.NE.0 ) THEN
  505:             GO TO 10
  506:          END IF
  507: *
  508: *        Multiply B by right bidiagonalizing vectors of A.
  509: *
  510:          CALL DORMBR( 'P', 'L', 'N', N, NRHS, M, A, LDA, WORK( ITAUP ),
  511:      $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
  512: *
  513:       END IF
  514: *
  515: *     Undo scaling.
  516: *
  517:       IF( IASCL.EQ.1 ) THEN
  518:          CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
  519:          CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
  520:      $                INFO )
  521:       ELSE IF( IASCL.EQ.2 ) THEN
  522:          CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
  523:          CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
  524:      $                INFO )
  525:       END IF
  526:       IF( IBSCL.EQ.1 ) THEN
  527:          CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
  528:       ELSE IF( IBSCL.EQ.2 ) THEN
  529:          CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
  530:       END IF
  531: *
  532:    10 CONTINUE
  533:       WORK( 1 ) = MAXWRK
  534:       IWORK( 1 ) = LIWORK
  535:       RETURN
  536: *
  537: *     End of DGELSD
  538: *
  539:       END