lalsd(3)

NAME
lalsd - lalsd: uses SVD for least squares, step in gelsd

SYNOPSIS

  Functions
subroutine clalsd (uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank, work, rwork, iwork, info)
CLALSD uses the singular value decomposition of A to solve the least squares problem.
subroutine dlalsd (uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank, work, iwork, info)
DLALSD uses the singular value decomposition of A to solve the least squares problem.
subroutine slalsd (uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank, work, iwork, info)
SLALSD uses the singular value decomposition of A to solve the least squares problem.
subroutine zlalsd (uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank, work, rwork, iwork, info)
ZLALSD uses the singular value decomposition of A to solve the least squares problem.

Detailed Description

Function Documentation

  subroutine clalsd (character uplo, integer smlsiz, integer n, integer nrhs, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( ldb, * ) b, integer ldb, real rcond, integer rank, complex, dimension( * ) work, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

CLALSD uses the singular value decomposition of A to solve the least squares problem.

Purpose:

     CLALSD uses the singular value decomposition of A to solve the least
     squares problem of finding X to minimize the Euclidean norm of each
     column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
     are N-by-NRHS. The solution X overwrites B.
     The singular values of A smaller than RCOND times the largest
     singular value are treated as zero in solving the least squares
     problem; in this case a minimum norm solution is returned.
     The actual singular values are returned in D in ascending order.

Parameters
    UPLO

              UPLO is CHARACTER*1
             = 'U': D and E define an upper bidiagonal matrix.
             = 'L': D and E define a  lower bidiagonal matrix.

    SMLSIZ

              SMLSIZ is INTEGER
             The maximum size of the subproblems at the bottom of the
             computation tree.

    N

              N is INTEGER
             The dimension of the  bidiagonal matrix.  N >= 0.

    NRHS

              NRHS is INTEGER
             The number of columns of B. NRHS must be at least 1.

    D

              D is REAL array, dimension (N)
             On entry D contains the main diagonal of the bidiagonal
             matrix. On exit, if INFO = 0, D contains its singular values.

    E

              E is REAL array, dimension (N-1)
             Contains the super-diagonal entries of the bidiagonal matrix.
             On exit, E has been destroyed.

    B

              B is COMPLEX array, dimension (LDB,NRHS)
             On input, B contains the right hand sides of the least
             squares problem. On output, B contains the solution X.

    LDB

              LDB is INTEGER
             The leading dimension of B in the calling subprogram.
             LDB must be at least max(1,N).

    RCOND

              RCOND is REAL
             The singular values of A less than or equal to RCOND times
             the largest singular value are treated as zero in solving
             the least squares problem. If RCOND is negative,
             machine precision is used instead.
             For example, if diag(S)*X=B were the least squares problem,
             where diag(S) is a diagonal matrix of singular values, the
             solution would be X(i) = B(i) / S(i) if S(i) is greater than
             RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
             RCOND*max(S).

    RANK

              RANK is INTEGER
             The number of singular values of A greater than RCOND times
             the largest singular value.

    WORK

              WORK is COMPLEX array, dimension (N * NRHS).

    RWORK

              RWORK is REAL array, dimension at least
             (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
             MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ),
             where
             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )

    IWORK

              IWORK is INTEGER array, dimension (3*N*NLVL + 11*N).

    INFO

              INFO is INTEGER
             = 0:  successful exit.
             < 0:  if INFO = -i, the i-th argument had an illegal value.
             > 0:  The algorithm failed to compute a singular value while
                   working on the submatrix lying in rows and columns
                   INFO/(N+1) through MOD(INFO,N+1).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

Definition at line 178 of file clalsd.f.

  subroutine dlalsd (character uplo, integer smlsiz, integer n, integer nrhs, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldb, * ) b, integer ldb, double precision rcond, integer rank, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLALSD uses the singular value decomposition of A to solve the least squares problem.

Purpose:

     DLALSD uses the singular value decomposition of A to solve the least
     squares problem of finding X to minimize the Euclidean norm of each
     column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
     are N-by-NRHS. The solution X overwrites B.
     The singular values of A smaller than RCOND times the largest
     singular value are treated as zero in solving the least squares
     problem; in this case a minimum norm solution is returned.
     The actual singular values are returned in D in ascending order.

Parameters
    UPLO

              UPLO is CHARACTER*1
             = 'U': D and E define an upper bidiagonal matrix.
             = 'L': D and E define a  lower bidiagonal matrix.

    SMLSIZ

              SMLSIZ is INTEGER
             The maximum size of the subproblems at the bottom of the
             computation tree.

    N

              N is INTEGER
             The dimension of the  bidiagonal matrix.  N >= 0.

    NRHS

              NRHS is INTEGER
             The number of columns of B. NRHS must be at least 1.

    D

              D is DOUBLE PRECISION array, dimension (N)
             On entry D contains the main diagonal of the bidiagonal
             matrix. On exit, if INFO = 0, D contains its singular values.

    E

              E is DOUBLE PRECISION array, dimension (N-1)
             Contains the super-diagonal entries of the bidiagonal matrix.
             On exit, E has been destroyed.

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
             On input, B contains the right hand sides of the least
             squares problem. On output, B contains the solution X.

    LDB

              LDB is INTEGER
             The leading dimension of B in the calling subprogram.
             LDB must be at least max(1,N).

    RCOND

              RCOND is DOUBLE PRECISION
             The singular values of A less than or equal to RCOND times
             the largest singular value are treated as zero in solving
             the least squares problem. If RCOND is negative,
             machine precision is used instead.
             For example, if diag(S)*X=B were the least squares problem,
             where diag(S) is a diagonal matrix of singular values, the
             solution would be X(i) = B(i) / S(i) if S(i) is greater than
             RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
             RCOND*max(S).

    RANK

              RANK is INTEGER
             The number of singular values of A greater than RCOND times
             the largest singular value.

    WORK

              WORK is DOUBLE PRECISION array, dimension at least
             (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
             where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).

    IWORK

              IWORK is INTEGER array, dimension at least
             (3*N*NLVL + 11*N)

    INFO

              INFO is INTEGER
             = 0:  successful exit.
             < 0:  if INFO = -i, the i-th argument had an illegal value.
             > 0:  The algorithm failed to compute a singular value while
                   working on the submatrix lying in rows and columns
                   INFO/(N+1) through MOD(INFO,N+1).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

Definition at line 171 of file dlalsd.f.

  subroutine slalsd (character uplo, integer smlsiz, integer n, integer nrhs, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldb, * ) b, integer ldb, real rcond, integer rank, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SLALSD uses the singular value decomposition of A to solve the least squares problem.

Purpose:

     SLALSD uses the singular value decomposition of A to solve the least
     squares problem of finding X to minimize the Euclidean norm of each
     column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
     are N-by-NRHS. The solution X overwrites B.
     The singular values of A smaller than RCOND times the largest
     singular value are treated as zero in solving the least squares
     problem; in this case a minimum norm solution is returned.
     The actual singular values are returned in D in ascending order.

Parameters
    UPLO

              UPLO is CHARACTER*1
             = 'U': D and E define an upper bidiagonal matrix.
             = 'L': D and E define a  lower bidiagonal matrix.

    SMLSIZ

              SMLSIZ is INTEGER
             The maximum size of the subproblems at the bottom of the
             computation tree.

    N

              N is INTEGER
             The dimension of the  bidiagonal matrix.  N >= 0.

    NRHS

              NRHS is INTEGER
             The number of columns of B. NRHS must be at least 1.

    D

              D is REAL array, dimension (N)
             On entry D contains the main diagonal of the bidiagonal
             matrix. On exit, if INFO = 0, D contains its singular values.

    E

              E is REAL array, dimension (N-1)
             Contains the super-diagonal entries of the bidiagonal matrix.
             On exit, E has been destroyed.

    B

              B is REAL array, dimension (LDB,NRHS)
             On input, B contains the right hand sides of the least
             squares problem. On output, B contains the solution X.

    LDB

              LDB is INTEGER
             The leading dimension of B in the calling subprogram.
             LDB must be at least max(1,N).

    RCOND

              RCOND is REAL
             The singular values of A less than or equal to RCOND times
             the largest singular value are treated as zero in solving
             the least squares problem. If RCOND is negative,
             machine precision is used instead.
             For example, if diag(S)*X=B were the least squares problem,
             where diag(S) is a diagonal matrix of singular values, the
             solution would be X(i) = B(i) / S(i) if S(i) is greater than
             RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
             RCOND*max(S).

    RANK

              RANK is INTEGER
             The number of singular values of A greater than RCOND times
             the largest singular value.

    WORK

              WORK is REAL array, dimension at least
             (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
             where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).

    IWORK

              IWORK is INTEGER array, dimension at least
             (3*N*NLVL + 11*N)

    INFO

              INFO is INTEGER
             = 0:  successful exit.
             < 0:  if INFO = -i, the i-th argument had an illegal value.
             > 0:  The algorithm failed to compute a singular value while
                   working on the submatrix lying in rows and columns
                   INFO/(N+1) through MOD(INFO,N+1).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

Definition at line 171 of file slalsd.f.

  subroutine zlalsd (character uplo, integer smlsiz, integer n, integer nrhs, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( ldb, * ) b, integer ldb, double precision rcond, integer rank, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

ZLALSD uses the singular value decomposition of A to solve the least squares problem.

Purpose:

     ZLALSD uses the singular value decomposition of A to solve the least
     squares problem of finding X to minimize the Euclidean norm of each
     column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
     are N-by-NRHS. The solution X overwrites B.
     The singular values of A smaller than RCOND times the largest
     singular value are treated as zero in solving the least squares
     problem; in this case a minimum norm solution is returned.
     The actual singular values are returned in D in ascending order.

Parameters
    UPLO

              UPLO is CHARACTER*1
             = 'U': D and E define an upper bidiagonal matrix.
             = 'L': D and E define a  lower bidiagonal matrix.

    SMLSIZ

              SMLSIZ is INTEGER
             The maximum size of the subproblems at the bottom of the
             computation tree.

    N

              N is INTEGER
             The dimension of the  bidiagonal matrix.  N >= 0.

    NRHS

              NRHS is INTEGER
             The number of columns of B. NRHS must be at least 1.

    D

              D is DOUBLE PRECISION array, dimension (N)
             On entry D contains the main diagonal of the bidiagonal
             matrix. On exit, if INFO = 0, D contains its singular values.

    E

              E is DOUBLE PRECISION array, dimension (N-1)
             Contains the super-diagonal entries of the bidiagonal matrix.
             On exit, E has been destroyed.

    B

              B is COMPLEX*16 array, dimension (LDB,NRHS)
             On input, B contains the right hand sides of the least
             squares problem. On output, B contains the solution X.

    LDB

              LDB is INTEGER
             The leading dimension of B in the calling subprogram.
             LDB must be at least max(1,N).

    RCOND

              RCOND is DOUBLE PRECISION
             The singular values of A less than or equal to RCOND times
             the largest singular value are treated as zero in solving
             the least squares problem. If RCOND is negative,
             machine precision is used instead.
             For example, if diag(S)*X=B were the least squares problem,
             where diag(S) is a diagonal matrix of singular values, the
             solution would be X(i) = B(i) / S(i) if S(i) is greater than
             RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
             RCOND*max(S).

    RANK

              RANK is INTEGER
             The number of singular values of A greater than RCOND times
             the largest singular value.

    WORK

              WORK is COMPLEX*16 array, dimension (N * NRHS)

    RWORK

              RWORK is DOUBLE PRECISION array, dimension at least
             (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
             MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ),
             where
             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )

    IWORK

              IWORK is INTEGER array, dimension at least
             (3*N*NLVL + 11*N).

    INFO

              INFO is INTEGER
             = 0:  successful exit.
             < 0:  if INFO = -i, the i-th argument had an illegal value.
             > 0:  The algorithm failed to compute a singular value while
                   working on the submatrix lying in rows and columns
                   INFO/(N+1) through MOD(INFO,N+1).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

Definition at line 179 of file zlalsd.f.

Author

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