zgelsd.f

来自「famous linear algebra library (LAPACK) p」· F 代码 · 共 567 行 · 第 1/2 页

      SUBROUTINE ZGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
     $                   WORK, LWORK, RWORK, IWORK, INFO )
*
*  -- LAPACK driver routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   RWORK( * ), S( * )
      COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  ZGELSD 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 tranformations, 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 the matrix A. M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix 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) COMPLEX*16 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) COMPLEX*16 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 the modulus of elements n+1:m in that column.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,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) COMPLEX*16 array, dimension (MAX(1,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
*              2*N + N*NRHS
*          if M is greater than or equal to N or
*              2*M + M*NRHS
*          if M is less than N, the code will execute correctly.
*          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 array WORK and the
*          minimum sizes of the arrays RWORK and IWORK, and returns
*          these values as the first entries of the WORK, RWORK and
*          IWORK arrays, and no error message related to LWORK is issued
*          by XERBLA.
*
*  RWORK   (workspace) DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
*          LRWORK >=
*              10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
*             (SMLSIZ+1)**2
*          if M is greater than or equal to N or
*             10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*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 )
*          On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.
*
*  IWORK   (workspace) INTEGER array, dimension (MAX(1,LIWORK))
*          LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
*          where MINMN = MIN( M,N ).
*          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
*
*  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 = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
      COMPLEX*16         CZERO
      PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
     $                   LDWORK, LIWORK, LRWORK, MAXMN, MAXWRK, MINMN,
     $                   MINWRK, MM, MNTHR, NLVL, NRWORK, NWORK, SMLSIZ
      DOUBLE PRECISION   ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLABAD, DLASCL, DLASET, XERBLA, ZGEBRD, ZGELQF,
     $                   ZGEQRF, ZLACPY, ZLALSD, ZLASCL, ZLASET, ZUNMBR,
     $                   ZUNMLQ, ZUNMQR
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, ZLANGE
      EXTERNAL           ILAENV, DLAMCH, ZLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          INT, LOG, MAX, MIN, DBLE
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments.
*
      INFO = 0
      MINMN = MIN( M, N )
      MAXMN = MAX( M, N )
      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
*
*     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.)
*
      IF( INFO.EQ.0 ) THEN
         MINWRK = 1
         MAXWRK = 1
         LIWORK = 1
         LRWORK = 1
         IF( MINMN.GT.0 ) THEN
            SMLSIZ = ILAENV( 9, 'ZGELSD', ' ', 0, 0, 0, 0 )
            MNTHR = ILAENV( 6, 'ZGELSD', ' ', M, N, NRHS, -1 )
            NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ + 1 ) ) /
     $                  LOG( TWO ) ) + 1, 0 )
            LIWORK = 3*MINMN*NLVL + 11*MINMN
            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*ILAENV( 1, 'ZGEQRF', ' ', M, N,
     $                       -1, -1 ) )
               MAXWRK = MAX( MAXWRK, NRHS*ILAENV( 1, 'ZUNMQR', 'LC', M,
     $                       NRHS, N, -1 ) )
            END IF
            IF( M.GE.N ) THEN
*
*              Path 1 - overdetermined or exactly determined.
*
               LRWORK = 10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
     $                  ( SMLSIZ + 1 )**2
               MAXWRK = MAX( MAXWRK, 2*N + ( MM + N )*ILAENV( 1,
     $                       'ZGEBRD', ' ', MM, N, -1, -1 ) )
               MAXWRK = MAX( MAXWRK, 2*N + NRHS*ILAENV( 1, 'ZUNMBR',
     $                       'QLC', MM, NRHS, N, -1 ) )
               MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
     $                       'ZUNMBR', 'PLN', N, NRHS, N, -1 ) )
               MAXWRK = MAX( MAXWRK, 2*N + N*NRHS )
               MINWRK = MAX( 2*N + MM, 2*N + N*NRHS )
            END IF
            IF( N.GT.M ) THEN
               LRWORK = 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
     $                  ( SMLSIZ + 1 )**2
               IF( N.GE.MNTHR ) THEN
*
*                 Path 2a - underdetermined, with many more columns
*                           than rows.
*
                  MAXWRK = M + M*ILAENV( 1, 'ZGELQF', ' ', M, N, -1,
     $                     -1 )
                  MAXWRK = MAX( MAXWRK, M*M + 4*M + 2*M*ILAENV( 1,
     $                          'ZGEBRD', ' ', M, M, -1, -1 ) )
                  MAXWRK = MAX( MAXWRK, M*M + 4*M + NRHS*ILAENV( 1,
     $                          'ZUNMBR', 'QLC', M, NRHS, M, -1 ) )
                  MAXWRK = MAX( MAXWRK, M*M + 4*M + ( M - 1 )*ILAENV( 1,
     $                          'ZUNMLQ', 'LC', N, 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*M + 4*M + M*NRHS )
               ELSE
*
*                 Path 2 - underdetermined.
*
                  MAXWRK = 2*M + ( N + M )*ILAENV( 1, 'ZGEBRD', ' ', M,
     $                     N, -1, -1 )
                  MAXWRK = MAX( MAXWRK, 2*M + NRHS*ILAENV( 1, 'ZUNMBR',
     $                          'QLC', M, NRHS, M, -1 ) )
                  MAXWRK = MAX( MAXWRK, 2*M + M*ILAENV( 1, 'ZUNMBR',
     $                          'PLN', N, NRHS, M, -1 ) )
                  MAXWRK = MAX( MAXWRK, 2*M + M*NRHS )
               END IF
               MINWRK = MAX( 2*M + N, 2*M + M*NRHS )
            END IF
         END IF
         MINWRK = MIN( MINWRK, MAXWRK )
         WORK( 1 ) = MAXWRK
         IWORK( 1 ) = LIWORK
         RWORK( 1 ) = LRWORK
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
            INFO = -12