sgelss.f

来自「famous linear algebra library (LAPACK) p」· F 代码 · 共 618 行 · 第 1/2 页

      SUBROUTINE SGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
     $                   WORK, LWORK, 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
      REAL               RCOND
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  SGELSS 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 effective rank of A is determined by treating as zero those
*  singular values which are less than RCOND times the largest singular
*  value.
*
*  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/output) REAL array, dimension (LDA,N)
*          On entry, the M-by-N matrix A.
*          On exit, the first min(m,n) rows of A are overwritten with
*          its right singular vectors, stored rowwise.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  B       (input/output) REAL 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) REAL 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) REAL
*          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) REAL 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 >= 1, and also:
*          LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
*          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.
*
*  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.
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            BDSPAC, BL, CHUNK, I, IASCL, IBSCL, IE, IL,
     $                   ITAU, ITAUP, ITAUQ, IWORK, LDWORK, MAXMN,
     $                   MAXWRK, MINMN, MINWRK, MM, MNTHR
      REAL               ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, THR
*     ..
*     .. Local Arrays ..
      REAL               VDUM( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           SBDSQR, SCOPY, SGEBRD, SGELQF, SGEMM, SGEMV,
     $                   SGEQRF, SLABAD, SLACPY, SLASCL, SLASET, SORGBR,
     $                   SORMBR, SORMLQ, SORMQR, SRSCL, XERBLA
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      REAL               SLAMCH, SLANGE
      EXTERNAL           ILAENV, SLAMCH, SLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. 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
         IF( MINMN.GT.0 ) THEN
            MM = M
            MNTHR = ILAENV( 6, 'SGELSS', ' ', M, N, NRHS, -1 )
            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, 'SGEQRF', ' ', M,
     $                       N, -1, -1 ) )
               MAXWRK = MAX( MAXWRK, N + NRHS*ILAENV( 1, 'SORMQR', 'LT',
     $                       M, NRHS, N, -1 ) )
            END IF
            IF( M.GE.N ) THEN
*
*              Path 1 - overdetermined or exactly determined
*
*              Compute workspace needed for SBDSQR
*
               BDSPAC = MAX( 1, 5*N )
               MAXWRK = MAX( MAXWRK, 3*N + ( MM + N )*ILAENV( 1,
     $                       'SGEBRD', ' ', MM, N, -1, -1 ) )
               MAXWRK = MAX( MAXWRK, 3*N + NRHS*ILAENV( 1, 'SORMBR',
     $                       'QLT', MM, NRHS, N, -1 ) )
               MAXWRK = MAX( MAXWRK, 3*N + ( N - 1 )*ILAENV( 1,
     $                       'SORGBR', 'P', N, N, N, -1 ) )
               MAXWRK = MAX( MAXWRK, BDSPAC )
               MAXWRK = MAX( MAXWRK, N*NRHS )
               MINWRK = MAX( 3*N + MM, 3*N + NRHS, BDSPAC )
               MAXWRK = MAX( MINWRK, MAXWRK )
            END IF
            IF( N.GT.M ) THEN
*
*              Compute workspace needed for SBDSQR
*
               BDSPAC = MAX( 1, 5*M )
               MINWRK = MAX( 3*M+NRHS, 3*M+N, BDSPAC )
               IF( N.GE.MNTHR ) THEN
*
*                 Path 2a - underdetermined, with many more columns
*                 than rows
*
                  MAXWRK = M + M*ILAENV( 1, 'SGELQF', ' ', M, N, -1,
     $                                  -1 )
                  MAXWRK = MAX( MAXWRK, M*M + 4*M + 2*M*ILAENV( 1,
     $                          'SGEBRD', ' ', M, M, -1, -1 ) )
                  MAXWRK = MAX( MAXWRK, M*M + 4*M + NRHS*ILAENV( 1,
     $                          'SORMBR', 'QLT', M, NRHS, M, -1 ) )
                  MAXWRK = MAX( MAXWRK, M*M + 4*M +
     $                          ( M - 1 )*ILAENV( 1, 'SORGBR', 'P', M,
     $                          M, M, -1 ) )
                  MAXWRK = MAX( MAXWRK, M*M + M + BDSPAC )
                  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, 'SORMLQ',
     $                          'LT', N, NRHS, M, -1 ) )
               ELSE
*
*                 Path 2 - underdetermined
*
                  MAXWRK = 3*M + ( N + M )*ILAENV( 1, 'SGEBRD', ' ', M,
     $                     N, -1, -1 )
                  MAXWRK = MAX( MAXWRK, 3*M + NRHS*ILAENV( 1, 'SORMBR',
     $                          'QLT', M, NRHS, M, -1 ) )
                  MAXWRK = MAX( MAXWRK, 3*M + M*ILAENV( 1, 'SORGBR',
     $                          'P', M, N, M, -1 ) )
                  MAXWRK = MAX( MAXWRK, BDSPAC )
                  MAXWRK = MAX( MAXWRK, N*NRHS )
               END IF
            END IF
            MAXWRK = MAX( MINWRK, MAXWRK )
         END IF
         WORK( 1 ) = MAXWRK
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
     $      INFO = -12
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'SGELSS', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         RANK = 0
         RETURN
      END IF
*
*     Get machine parameters
*
      EPS = SLAMCH( 'P' )
      SFMIN = SLAMCH( 'S' )
      SMLNUM = SFMIN / EPS
      BIGNUM = ONE / SMLNUM
      CALL SLABAD( SMLNUM, BIGNUM )
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = SLANGE( 'M', M, N, A, LDA, WORK )
      IASCL = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
         IASCL = 1
      ELSE IF( ANRM.GT.BIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
         IASCL = 2
      ELSE IF( ANRM.EQ.ZERO ) THEN
*
*        Matrix all zero. Return zero solution.
*
         CALL SLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
         CALL SLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
         RANK = 0
         GO TO 70
      END IF
*
*     Scale B if max element outside range [SMLNUM,BIGNUM]
*
      BNRM = SLANGE( 'M', M, NRHS, B, LDB, WORK )
      IBSCL = 0
      IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL SLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
         IBSCL = 1
      ELSE IF( BNRM.GT.BIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL SLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
         IBSCL = 2
      END IF
*
*     Overdetermined case
*
      IF( M.GE.N ) THEN