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

  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

  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

  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

Author

Generated automatically by Doxygen for LAPACK from the source code.