lalsa(3)

NAME
lalsa - lalsa: SVD of coefficient matrix, step in gelsd

SYNOPSIS

  Functions

subroutine clalsa (icompq, smlsiz, n, nrhs, b, ldb, bx, ldbx, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, rwork, iwork, info)
CLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
subroutine dlalsa (icompq, smlsiz, n, nrhs, b, ldb, bx, ldbx, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, work, iwork, info)
DLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
subroutine slalsa (icompq, smlsiz, n, nrhs, b, ldb, bx, ldbx, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, work, iwork, info)
SLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
subroutine zlalsa (icompq, smlsiz, n, nrhs, b, ldb, bx, ldbx, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, rwork, iwork, info)
ZLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.

Detailed Description

Function Documentation

  subroutine clalsa (integer icompq, integer smlsiz, integer n, integer nrhs, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldbx, * ) bx, integer ldbx, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldu, * ) vt, integer, dimension( * ) k, real, dimension( ldu, * ) difl, real, dimension( ldu, * ) difr, real, dimension( ldu, * ) z, real, dimension( ldu, * ) poles, integer, dimension( * ) givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, integer, dimension( ldgcol, * ) perm, real, dimension( ldu, * ) givnum, real, dimension( * ) c, real, dimension( * ) s, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

CLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.

Purpose:

    !>
    !> CLALSA is an intermediate step in solving the least squares problem
    !> by computing the SVD of the coefficient matrix in compact form (The
    !> singular vectors are computed as products of simple orthogonal
    !> matrices.).
    !>
    !> If ICOMPQ = 0, CLALSA applies the inverse of the left singular vector
    !> matrix of an upper bidiagonal matrix to the right hand side; and if
    !> ICOMPQ = 1, CLALSA applies the right singular vector matrix to the
    !> right hand side. The singular vector matrices were generated in
    !> compact form by CLALSA.
    !>

Parameters
    ICOMPQ

    !>          ICOMPQ is INTEGER
    !>         Specifies whether the left or the right singular vector
    !>         matrix is involved.
    !>         = 0: Left singular vector matrix
    !>         = 1: Right singular vector matrix
    !>

    SMLSIZ

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

    N

    !>          N is INTEGER
    !>         The row and column dimensions of the upper bidiagonal matrix.
    !>

    NRHS

    !>          NRHS is INTEGER
    !>         The number of columns of B and BX. NRHS must be at least 1.
    !>

    B

    !>          B is COMPLEX array, dimension ( LDB, NRHS )
    !>         On input, B contains the right hand sides of the least
    !>         squares problem in rows 1 through M.
    !>         On output, B contains the solution X in rows 1 through N.
    !>

    LDB

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

    BX

    !>          BX is COMPLEX array, dimension ( LDBX, NRHS )
    !>         On exit, the result of applying the left or right singular
    !>         vector matrix to B.
    !>

    LDBX

    !>          LDBX is INTEGER
    !>         The leading dimension of BX.
    !>

    U

    !>          U is REAL array, dimension ( LDU, SMLSIZ ).
    !>         On entry, U contains the left singular vector matrices of all
    !>         subproblems at the bottom level.
    !>

    LDU

    !>          LDU is INTEGER, LDU = > N.
    !>         The leading dimension of arrays U, VT, DIFL, DIFR,
    !>         POLES, GIVNUM, and Z.
    !>

    VT

    !>          VT is REAL array, dimension ( LDU, SMLSIZ+1 ).
    !>         On entry, VT**H contains the right singular vector matrices of
    !>         all subproblems at the bottom level.
    !>

    K

    !>          K is INTEGER array, dimension ( N ).
    !>

    DIFL

    !>          DIFL is REAL array, dimension ( LDU, NLVL ).
    !>         where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
    !>

    DIFR

    !>          DIFR is REAL array, dimension ( LDU, 2 * NLVL ).
    !>         On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
    !>         distances between singular values on the I-th level and
    !>         singular values on the (I -1)-th level, and DIFR(*, 2 * I)
    !>         record the normalizing factors of the right singular vectors
    !>         matrices of subproblems on I-th level.
    !>

    Z

    !>          Z is REAL array, dimension ( LDU, NLVL ).
    !>         On entry, Z(1, I) contains the components of the deflation-
    !>         adjusted updating row vector for subproblems on the I-th
    !>         level.
    !>

    POLES

    !>          POLES is REAL array, dimension ( LDU, 2 * NLVL ).
    !>         On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
    !>         singular values involved in the secular equations on the I-th
    !>         level.
    !>

    GIVPTR

    !>          GIVPTR is INTEGER array, dimension ( N ).
    !>         On entry, GIVPTR( I ) records the number of Givens
    !>         rotations performed on the I-th problem on the computation
    !>         tree.
    !>

    GIVCOL

    !>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
    !>         On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
    !>         locations of Givens rotations performed on the I-th level on
    !>         the computation tree.
    !>

    LDGCOL

    !>          LDGCOL is INTEGER, LDGCOL = > N.
    !>         The leading dimension of arrays GIVCOL and PERM.
    !>

    PERM

    !>          PERM is INTEGER array, dimension ( LDGCOL, NLVL ).
    !>         On entry, PERM(*, I) records permutations done on the I-th
    !>         level of the computation tree.
    !>

    GIVNUM

    !>          GIVNUM is REAL array, dimension ( LDU, 2 * NLVL ).
    !>         On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
    !>         values of Givens rotations performed on the I-th level on the
    !>         computation tree.
    !>

    C

    !>          C is REAL array, dimension ( N ).
    !>         On entry, if the I-th subproblem is not square,
    !>         C( I ) contains the C-value of a Givens rotation related to
    !>         the right null space of the I-th subproblem.
    !>

    S

    !>          S is REAL array, dimension ( N ).
    !>         On entry, if the I-th subproblem is not square,
    !>         S( I ) contains the S-value of a Givens rotation related to
    !>         the right null space of the I-th subproblem.
    !>

    RWORK

    !>          RWORK is REAL array, dimension at least
    !>         MAX( (SMLSZ+1)*NRHS*3, N*(1+NRHS) + 2*NRHS ).
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (3*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>

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 dlalsa (integer icompq, integer smlsiz, integer n, integer nrhs, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldbx, * ) bx, integer ldbx, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldu, * ) vt, integer, dimension( * ) k, double precision, dimension( ldu, * ) difl, double precision, dimension( ldu, * ) difr, double precision, dimension( ldu, * ) z, double precision, dimension( ldu, * ) poles, integer, dimension( * ) givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, integer, dimension( ldgcol, * ) perm, double precision, dimension( ldu, * ) givnum, double precision, dimension( * ) c, double precision, dimension( * ) s, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.

Purpose:

    !>
    !> DLALSA is an intermediate step in solving the least squares problem
    !> by computing the SVD of the coefficient matrix in compact form (The
    !> singular vectors are computed as products of simple orthogonal
    !> matrices.).
    !>
    !> If ICOMPQ = 0, DLALSA applies the inverse of the left singular vector
    !> matrix of an upper bidiagonal matrix to the right hand side; and if
    !> ICOMPQ = 1, DLALSA applies the right singular vector matrix to the
    !> right hand side. The singular vector matrices were generated in
    !> compact form by DLALSA.
    !>

Parameters
    ICOMPQ

    !>          ICOMPQ is INTEGER
    !>         Specifies whether the left or the right singular vector
    !>         matrix is involved.
    !>         = 0: Left singular vector matrix
    !>         = 1: Right singular vector matrix
    !>

    SMLSIZ

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

    N

    !>          N is INTEGER
    !>         The row and column dimensions of the upper bidiagonal matrix.
    !>

    NRHS

    !>          NRHS is INTEGER
    !>         The number of columns of B and BX. NRHS must be at least 1.
    !>

    B

    !>          B is DOUBLE PRECISION array, dimension ( LDB, NRHS )
    !>         On input, B contains the right hand sides of the least
    !>         squares problem in rows 1 through M.
    !>         On output, B contains the solution X in rows 1 through N.
    !>

    LDB

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

    BX

    !>          BX is DOUBLE PRECISION array, dimension ( LDBX, NRHS )
    !>         On exit, the result of applying the left or right singular
    !>         vector matrix to B.
    !>

    LDBX

    !>          LDBX is INTEGER
    !>         The leading dimension of BX.
    !>

    U

    !>          U is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ).
    !>         On entry, U contains the left singular vector matrices of all
    !>         subproblems at the bottom level.
    !>

    LDU

    !>          LDU is INTEGER, LDU = > N.
    !>         The leading dimension of arrays U, VT, DIFL, DIFR,
    !>         POLES, GIVNUM, and Z.
    !>

    VT

    !>          VT is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ).
    !>         On entry, VT**T contains the right singular vector matrices of
    !>         all subproblems at the bottom level.
    !>

    K

    !>          K is INTEGER array, dimension ( N ).
    !>

    DIFL

    !>          DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
    !>         where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
    !>

    DIFR

    !>          DIFR is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
    !>         On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
    !>         distances between singular values on the I-th level and
    !>         singular values on the (I -1)-th level, and DIFR(*, 2 * I)
    !>         record the normalizing factors of the right singular vectors
    !>         matrices of subproblems on I-th level.
    !>

    Z

    !>          Z is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
    !>         On entry, Z(1, I) contains the components of the deflation-
    !>         adjusted updating row vector for subproblems on the I-th
    !>         level.
    !>

    POLES

    !>          POLES is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
    !>         On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
    !>         singular values involved in the secular equations on the I-th
    !>         level.
    !>

    GIVPTR

    !>          GIVPTR is INTEGER array, dimension ( N ).
    !>         On entry, GIVPTR( I ) records the number of Givens
    !>         rotations performed on the I-th problem on the computation
    !>         tree.
    !>

    GIVCOL

    !>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
    !>         On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
    !>         locations of Givens rotations performed on the I-th level on
    !>         the computation tree.
    !>

    LDGCOL

    !>          LDGCOL is INTEGER, LDGCOL = > N.
    !>         The leading dimension of arrays GIVCOL and PERM.
    !>

    PERM

    !>          PERM is INTEGER array, dimension ( LDGCOL, NLVL ).
    !>         On entry, PERM(*, I) records permutations done on the I-th
    !>         level of the computation tree.
    !>

    GIVNUM

    !>          GIVNUM is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
    !>         On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
    !>         values of Givens rotations performed on the I-th level on the
    !>         computation tree.
    !>

    C

    !>          C is DOUBLE PRECISION array, dimension ( N ).
    !>         On entry, if the I-th subproblem is not square,
    !>         C( I ) contains the C-value of a Givens rotation related to
    !>         the right null space of the I-th subproblem.
    !>

    S

    !>          S is DOUBLE PRECISION array, dimension ( N ).
    !>         On entry, if the I-th subproblem is not square,
    !>         S( I ) contains the S-value of a Givens rotation related to
    !>         the right null space of the I-th subproblem.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (N)
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (3*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>

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 slalsa (integer icompq, integer smlsiz, integer n, integer nrhs, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldbx, * ) bx, integer ldbx, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldu, * ) vt, integer, dimension( * ) k, real, dimension( ldu, * ) difl, real, dimension( ldu, * ) difr, real, dimension( ldu, * ) z, real, dimension( ldu, * ) poles, integer, dimension( * ) givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, integer, dimension( ldgcol, * ) perm, real, dimension( ldu, * ) givnum, real, dimension( * ) c, real, dimension( * ) s, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.

Purpose:

    !>
    !> SLALSA is an intermediate step in solving the least squares problem
    !> by computing the SVD of the coefficient matrix in compact form (The
    !> singular vectors are computed as products of simple orthogonal
    !> matrices.).
    !>
    !> If ICOMPQ = 0, SLALSA applies the inverse of the left singular vector
    !> matrix of an upper bidiagonal matrix to the right hand side; and if
    !> ICOMPQ = 1, SLALSA applies the right singular vector matrix to the
    !> right hand side. The singular vector matrices were generated in
    !> compact form by SLALSA.
    !>

Parameters
    ICOMPQ

    !>          ICOMPQ is INTEGER
    !>         Specifies whether the left or the right singular vector
    !>         matrix is involved.
    !>         = 0: Left singular vector matrix
    !>         = 1: Right singular vector matrix
    !>

    SMLSIZ

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

    N

    !>          N is INTEGER
    !>         The row and column dimensions of the upper bidiagonal matrix.
    !>

    NRHS

    !>          NRHS is INTEGER
    !>         The number of columns of B and BX. NRHS must be at least 1.
    !>

    B

    !>          B is REAL array, dimension ( LDB, NRHS )
    !>         On input, B contains the right hand sides of the least
    !>         squares problem in rows 1 through M.
    !>         On output, B contains the solution X in rows 1 through N.
    !>

    LDB

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

    BX

    !>          BX is REAL array, dimension ( LDBX, NRHS )
    !>         On exit, the result of applying the left or right singular
    !>         vector matrix to B.
    !>

    LDBX

    !>          LDBX is INTEGER
    !>         The leading dimension of BX.
    !>

    U

    !>          U is REAL array, dimension ( LDU, SMLSIZ ).
    !>         On entry, U contains the left singular vector matrices of all
    !>         subproblems at the bottom level.
    !>

    LDU

    !>          LDU is INTEGER, LDU = > N.
    !>         The leading dimension of arrays U, VT, DIFL, DIFR,
    !>         POLES, GIVNUM, and Z.
    !>

    VT

    !>          VT is REAL array, dimension ( LDU, SMLSIZ+1 ).
    !>         On entry, VT**T contains the right singular vector matrices of
    !>         all subproblems at the bottom level.
    !>

    K

    !>          K is INTEGER array, dimension ( N ).
    !>

    DIFL

    !>          DIFL is REAL array, dimension ( LDU, NLVL ).
    !>         where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
    !>

    DIFR

    !>          DIFR is REAL array, dimension ( LDU, 2 * NLVL ).
    !>         On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
    !>         distances between singular values on the I-th level and
    !>         singular values on the (I -1)-th level, and DIFR(*, 2 * I)
    !>         record the normalizing factors of the right singular vectors
    !>         matrices of subproblems on I-th level.
    !>

    Z

    !>          Z is REAL array, dimension ( LDU, NLVL ).
    !>         On entry, Z(1, I) contains the components of the deflation-
    !>         adjusted updating row vector for subproblems on the I-th
    !>         level.
    !>

    POLES

    !>          POLES is REAL array, dimension ( LDU, 2 * NLVL ).
    !>         On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
    !>         singular values involved in the secular equations on the I-th
    !>         level.
    !>

    GIVPTR

    !>          GIVPTR is INTEGER array, dimension ( N ).
    !>         On entry, GIVPTR( I ) records the number of Givens
    !>         rotations performed on the I-th problem on the computation
    !>         tree.
    !>

    GIVCOL

    !>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
    !>         On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
    !>         locations of Givens rotations performed on the I-th level on
    !>         the computation tree.
    !>

    LDGCOL

    !>          LDGCOL is INTEGER, LDGCOL = > N.
    !>         The leading dimension of arrays GIVCOL and PERM.
    !>

    PERM

    !>          PERM is INTEGER array, dimension ( LDGCOL, NLVL ).
    !>         On entry, PERM(*, I) records permutations done on the I-th
    !>         level of the computation tree.
    !>

    GIVNUM

    !>          GIVNUM is REAL array, dimension ( LDU, 2 * NLVL ).
    !>         On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
    !>         values of Givens rotations performed on the I-th level on the
    !>         computation tree.
    !>

    C

    !>          C is REAL array, dimension ( N ).
    !>         On entry, if the I-th subproblem is not square,
    !>         C( I ) contains the C-value of a Givens rotation related to
    !>         the right null space of the I-th subproblem.
    !>

    S

    !>          S is REAL array, dimension ( N ).
    !>         On entry, if the I-th subproblem is not square,
    !>         S( I ) contains the S-value of a Givens rotation related to
    !>         the right null space of the I-th subproblem.
    !>

    WORK

    !>          WORK is REAL array, dimension (N)
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (3*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>

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 zlalsa (integer icompq, integer smlsiz, integer n, integer nrhs, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldbx, * ) bx, integer ldbx, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldu, * ) vt, integer, dimension( * ) k, double precision, dimension( ldu, * ) difl, double precision, dimension( ldu, * ) difr, double precision, dimension( ldu, * ) z, double precision, dimension( ldu, * ) poles, integer, dimension( * ) givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, integer, dimension( ldgcol, * ) perm, double precision, dimension( ldu, * ) givnum, double precision, dimension( * ) c, double precision, dimension( * ) s, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

ZLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.

Purpose:

    !>
    !> ZLALSA is an intermediate step in solving the least squares problem
    !> by computing the SVD of the coefficient matrix in compact form (The
    !> singular vectors are computed as products of simple orthogonal
    !> matrices.).
    !>
    !> If ICOMPQ = 0, ZLALSA applies the inverse of the left singular vector
    !> matrix of an upper bidiagonal matrix to the right hand side; and if
    !> ICOMPQ = 1, ZLALSA applies the right singular vector matrix to the
    !> right hand side. The singular vector matrices were generated in
    !> compact form by ZLALSA.
    !>

Parameters
    ICOMPQ

    !>          ICOMPQ is INTEGER
    !>         Specifies whether the left or the right singular vector
    !>         matrix is involved.
    !>         = 0: Left singular vector matrix
    !>         = 1: Right singular vector matrix
    !>

    SMLSIZ

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

    N

    !>          N is INTEGER
    !>         The row and column dimensions of the upper bidiagonal matrix.
    !>

    NRHS

    !>          NRHS is INTEGER
    !>         The number of columns of B and BX. NRHS must be at least 1.
    !>

    B

    !>          B is COMPLEX*16 array, dimension ( LDB, NRHS )
    !>         On input, B contains the right hand sides of the least
    !>         squares problem in rows 1 through M.
    !>         On output, B contains the solution X in rows 1 through N.
    !>

    LDB

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

    BX

    !>          BX is COMPLEX*16 array, dimension ( LDBX, NRHS )
    !>         On exit, the result of applying the left or right singular
    !>         vector matrix to B.
    !>

    LDBX

    !>          LDBX is INTEGER
    !>         The leading dimension of BX.
    !>

    U

    !>          U is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ).
    !>         On entry, U contains the left singular vector matrices of all
    !>         subproblems at the bottom level.
    !>

    LDU

    !>          LDU is INTEGER, LDU = > N.
    !>         The leading dimension of arrays U, VT, DIFL, DIFR,
    !>         POLES, GIVNUM, and Z.
    !>

    VT

    !>          VT is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ).
    !>         On entry, VT**H contains the right singular vector matrices of
    !>         all subproblems at the bottom level.
    !>

    K

    !>          K is INTEGER array, dimension ( N ).
    !>

    DIFL

    !>          DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
    !>         where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
    !>

    DIFR

    !>          DIFR is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
    !>         On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
    !>         distances between singular values on the I-th level and
    !>         singular values on the (I -1)-th level, and DIFR(*, 2 * I)
    !>         record the normalizing factors of the right singular vectors
    !>         matrices of subproblems on I-th level.
    !>

    Z

    !>          Z is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
    !>         On entry, Z(1, I) contains the components of the deflation-
    !>         adjusted updating row vector for subproblems on the I-th
    !>         level.
    !>

    POLES

    !>          POLES is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
    !>         On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
    !>         singular values involved in the secular equations on the I-th
    !>         level.
    !>

    GIVPTR

    !>          GIVPTR is INTEGER array, dimension ( N ).
    !>         On entry, GIVPTR( I ) records the number of Givens
    !>         rotations performed on the I-th problem on the computation
    !>         tree.
    !>

    GIVCOL

    !>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
    !>         On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
    !>         locations of Givens rotations performed on the I-th level on
    !>         the computation tree.
    !>

    LDGCOL

    !>          LDGCOL is INTEGER, LDGCOL = > N.
    !>         The leading dimension of arrays GIVCOL and PERM.
    !>

    PERM

    !>          PERM is INTEGER array, dimension ( LDGCOL, NLVL ).
    !>         On entry, PERM(*, I) records permutations done on the I-th
    !>         level of the computation tree.
    !>

    GIVNUM

    !>          GIVNUM is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
    !>         On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
    !>         values of Givens rotations performed on the I-th level on the
    !>         computation tree.
    !>

    C

    !>          C is DOUBLE PRECISION array, dimension ( N ).
    !>         On entry, if the I-th subproblem is not square,
    !>         C( I ) contains the C-value of a Givens rotation related to
    !>         the right null space of the I-th subproblem.
    !>

    S

    !>          S is DOUBLE PRECISION array, dimension ( N ).
    !>         On entry, if the I-th subproblem is not square,
    !>         S( I ) contains the S-value of a Givens rotation related to
    !>         the right null space of the I-th subproblem.
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension at least
    !>         MAX( (SMLSZ+1)*NRHS*3, N*(1+NRHS) + 2*NRHS ).
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (3*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>

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

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