dgelsd.f

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      SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
     $                   WORK, LWORK, IWORK, INFO )
*
*  -- LAPACK driver routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1999
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DGELSD computes the minimum-norm solution to a real linear least
*  squares problem:
*      minimize 2-norm(| b - A*x |)
*  using the singular value decomposition (SVD) of A. A is an M-by-N
*  matrix which may be rank-deficient.
*
*  Several right hand side vectors b and solution vectors x can be
*  handled in a single call; they are stored as the columns of the
*  M-by-NRHS right hand side matrix B and the N-by-NRHS solution
*  matrix X.
*
*  The problem is solved in three steps:
*  (1) Reduce the coefficient matrix A to bidiagonal form with
*      Householder transformations, reducing the original problem
*      into a "bidiagonal least squares problem" (BLS)
*  (2) Solve the BLS using a divide and conquer approach.
*  (3) Apply back all the Householder tranformations to solve
*      the original least squares problem.
*
*  The effective rank of A is determined by treating as zero those
*  singular values which are less than RCOND times the largest singular
*  value.
*
*  The divide and conquer algorithm makes very mild assumptions about
*  floating point arithmetic. It will work on machines with a guard
*  digit in add/subtract, or on those binary machines without guard
*  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
*  Cray-2. It could conceivably fail on hexadecimal or decimal machines
*  without guard digits, but we know of none.
*
*  Arguments
*  =========
*
*  M       (input) INTEGER
*          The number of rows of A. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of A. N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrices B and X. NRHS >= 0.
*
*  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
*          On entry, the M-by-N matrix A.
*          On exit, A has been destroyed.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
*          On entry, the M-by-NRHS right hand side matrix B.
*          On exit, B is overwritten by the N-by-NRHS solution
*          matrix X.  If m >= n and RANK = n, the residual
*          sum-of-squares for the solution in the i-th column is given
*          by the sum of squares of elements n+1:m in that column.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B. LDB >= max(1,max(M,N)).
*
*  S       (output) DOUBLE PRECISION array, dimension (min(M,N))
*          The singular values of A in decreasing order.
*          The condition number of A in the 2-norm = S(1)/S(min(m,n)).
*
*  RCOND   (input) DOUBLE PRECISION
*          RCOND is used to determine the effective rank of A.
*          Singular values S(i) <= RCOND*S(1) are treated as zero.
*          If RCOND < 0, machine precision is used instead.
*
*  RANK    (output) INTEGER
*          The effective rank of A, i.e., the number of singular values
*          which are greater than RCOND*S(1).
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK. LWORK must be at least 1.
*          The exact minimum amount of workspace needed depends on M,
*          N and NRHS. As long as LWORK is at least
*              12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
*          if M is greater than or equal to N or
*              12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
*          if M is less than N, the code will execute correctly.
*          SMLSIZ is returned by ILAENV and is equal to the maximum
*          size of the subproblems at the bottom of the computation
*          tree (usually about 25), and
*             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
*          For good performance, LWORK should generally be larger.
*
*          If LWORK = -1, then a workspace query is assumed; the routine
*          only calculates the optimal size of the WORK array, returns
*          this value as the first entry of the WORK array, and no error
*          message related to LWORK is issued by XERBLA.
*
*  IWORK   (workspace) INTEGER array, dimension (LIWORK)
*          LIWORK >= 3 * MINMN * NLVL + 11 * MINMN,
*          where MINMN = MIN( M,N ).
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*          > 0:  the algorithm for computing the SVD failed to converge;
*                if INFO = i, i off-diagonal elements of an intermediate
*                bidiagonal form did not converge to zero.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Ren-Cang Li, Computer Science Division, University of
*       California at Berkeley, USA
*     Osni Marques, LBNL/NERSC, USA
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
     $                   LDWORK, MAXMN, MAXWRK, MINMN, MINWRK, MM,
     $                   MNTHR, NLVL, NWORK, SMLSIZ, WLALSD
      DOUBLE PRECISION   ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEBRD, DGELQF, DGEQRF, DLABAD, DLACPY, DLALSD,
     $                   DLASCL, DLASET, DORMBR, DORMLQ, DORMQR, XERBLA
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           ILAENV, DLAMCH, DLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, INT, LOG, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments.
*
      INFO = 0
      MINMN = MIN( M, N )
      MAXMN = MAX( M, N )
      MNTHR = ILAENV( 6, 'DGELSD', ' ', M, N, NRHS, -1 )
      LQUERY = ( LWORK.EQ.-1 )
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
         INFO = -7
      END IF
*
      SMLSIZ = ILAENV( 9, 'DGELSD', ' ', 0, 0, 0, 0 )
*
*     Compute workspace.
*     (Note: Comments in the code beginning "Workspace:" describe the
*     minimal amount of workspace needed at that point in the code,
*     as well as the preferred amount for good performance.
*     NB refers to the optimal block size for the immediately
*     following subroutine, as returned by ILAENV.)
*
      MINWRK = 1
      MINMN = MAX( 1, MINMN )
      NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ+1 ) ) /
     $       LOG( TWO ) )+ 1, 0 )
*
      IF( INFO.EQ.0 ) THEN
         MAXWRK = 0
         MM = M
         IF( M.GE.N .AND. M.GE.MNTHR ) THEN
*
*           Path 1a - overdetermined, with many more rows than columns.
*
            MM = N
            MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'DGEQRF', ' ', M, N,
     $               -1, -1 ) )
            MAXWRK = MAX( MAXWRK, N+NRHS*
     $               ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, -1 ) )
         END IF
         IF( M.GE.N ) THEN
*
*           Path 1 - overdetermined or exactly determined.
*
            MAXWRK = MAX( MAXWRK, 3*N+( MM+N )*
     $               ILAENV( 1, 'DGEBRD', ' ', MM, N, -1, -1 ) )
            MAXWRK = MAX( MAXWRK, 3*N+NRHS*
     $               ILAENV( 1, 'DORMBR', 'QLT', MM, NRHS, N, -1 ) )
            MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*
     $               ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, N, -1 ) )
            WLALSD = 9*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS+(SMLSIZ+1)**2
            MAXWRK = MAX( MAXWRK, 3*N+WLALSD )
            MINWRK = MAX( 3*N+MM, 3*N+NRHS, 3*N+WLALSD )
         END IF
         IF( N.GT.M ) THEN
            WLALSD = 9*M+2*M*SMLSIZ+8*M*NLVL+M*NRHS+(SMLSIZ+1)**2
            IF( N.GE.MNTHR ) THEN
*
*              Path 2a - underdetermined, with many more columns
*              than rows.
*
               MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
               MAXWRK = MAX( MAXWRK, M*M+4*M+2*M*
     $                  ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
               MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS*
     $                  ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M, -1 ) )
               MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )*
     $                  ILAENV( 1, 'DORMBR', 'PLN', M, NRHS, M, -1 ) )
               IF( NRHS.GT.1 ) THEN
                  MAXWRK = MAX( MAXWRK, M*M+M+M*NRHS )
               ELSE
                  MAXWRK = MAX( MAXWRK, M*M+2*M )
               END IF
               MAXWRK = MAX( MAXWRK, M+NRHS*
     $                  ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, -1 ) )
               MAXWRK = MAX( MAXWRK, M*M+4*M+WLALSD )
            ELSE
*
*              Path 2 - remaining underdetermined cases.
*
               MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'DGEBRD', ' ', M, N,
     $                  -1, -1 )
               MAXWRK = MAX( MAXWRK, 3*M+NRHS*
     $                  ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, N, -1 ) )
               MAXWRK = MAX( MAXWRK, 3*M+M*
     $                  ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, M, -1 ) )
               MAXWRK = MAX( MAXWRK, 3*M+WLALSD )
            END IF
            MINWRK = MAX( 3*M+NRHS, 3*M+M, 3*M+WLALSD )
         END IF
         MINWRK = MIN( MINWRK, MAXWRK )
         WORK( 1 ) = MAXWRK
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
            INFO = -12
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN