lals0(3)

NAME
lals0 - lals0: back multiplying factors, step in gelsd

SYNOPSIS

  Functions

subroutine clals0 (icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, rwork, info)
CLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.
subroutine dlals0 (icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, work, info)
DLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.
subroutine slals0 (icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, work, info)
SLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.
subroutine zlals0 (icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, rwork, info)
ZLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.

Detailed Description

Function Documentation

  subroutine clals0 (integer icompq, integer nl, integer nr, integer sqre, integer nrhs, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldbx, * ) bx, integer ldbx, integer, dimension( * ) perm, integer givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, real, dimension( ldgnum, * ) givnum, integer ldgnum, real, dimension( ldgnum, * ) poles, real, dimension( * ) difl, real, dimension( ldgnum, * ) difr, real, dimension( * ) z, integer k, real c, real s, real, dimension( * ) rwork, integer info)

CLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.

Purpose:

    !>
    !> CLALS0 applies back the multiplying factors of either the left or the
    !> right singular vector matrix of a diagonal matrix appended by a row
    !> to the right hand side matrix B in solving the least squares problem
    !> using the divide-and-conquer SVD approach.
    !>
    !> For the left singular vector matrix, three types of orthogonal
    !> matrices are involved:
    !>
    !> (1L) Givens rotations: the number of such rotations is GIVPTR; the
    !>      pairs of columns/rows they were applied to are stored in GIVCOL;
    !>      and the C- and S-values of these rotations are stored in GIVNUM.
    !>
    !> (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
    !>      row, and for J=2:N, PERM(J)-th row of B is to be moved to the
    !>      J-th row.
    !>
    !> (3L) The left singular vector matrix of the remaining matrix.
    !>
    !> For the right singular vector matrix, four types of orthogonal
    !> matrices are involved:
    !>
    !> (1R) The right singular vector matrix of the remaining matrix.
    !>
    !> (2R) If SQRE = 1, one extra Givens rotation to generate the right
    !>      null space.
    !>
    !> (3R) The inverse transformation of (2L).
    !>
    !> (4R) The inverse transformation of (1L).
    !>

Parameters
    ICOMPQ

    !>          ICOMPQ is INTEGER
    !>         Specifies whether singular vectors are to be computed in
    !>         factored form:
    !>         = 0: Left singular vector matrix.
    !>         = 1: Right singular vector matrix.
    !>

    NL

    !>          NL is INTEGER
    !>         The row dimension of the upper block. NL >= 1.
    !>

    NR

    !>          NR is INTEGER
    !>         The row dimension of the lower block. NR >= 1.
    !>

    SQRE

    !>          SQRE is INTEGER
    !>         = 0: the lower block is an NR-by-NR square matrix.
    !>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
    !>
    !>         The bidiagonal matrix has row dimension N = NL + NR + 1,
    !>         and column dimension M = N + SQRE.
    !>

    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. LDB must be at least
    !>         max(1,MAX( M, N ) ).
    !>

    BX

    !>          BX is COMPLEX array, dimension ( LDBX, NRHS )
    !>

    LDBX

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

    PERM

    !>          PERM is INTEGER array, dimension ( N )
    !>         The permutations (from deflation and sorting) applied
    !>         to the two blocks.
    !>

    GIVPTR

    !>          GIVPTR is INTEGER
    !>         The number of Givens rotations which took place in this
    !>         subproblem.
    !>

    GIVCOL

    !>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
    !>         Each pair of numbers indicates a pair of rows/columns
    !>         involved in a Givens rotation.
    !>

    LDGCOL

    !>          LDGCOL is INTEGER
    !>         The leading dimension of GIVCOL, must be at least N.
    !>

    GIVNUM

    !>          GIVNUM is REAL array, dimension ( LDGNUM, 2 )
    !>         Each number indicates the C or S value used in the
    !>         corresponding Givens rotation.
    !>

    LDGNUM

    !>          LDGNUM is INTEGER
    !>         The leading dimension of arrays DIFR, POLES and
    !>         GIVNUM, must be at least K.
    !>

    POLES

    !>          POLES is REAL array, dimension ( LDGNUM, 2 )
    !>         On entry, POLES(1:K, 1) contains the new singular
    !>         values obtained from solving the secular equation, and
    !>         POLES(1:K, 2) is an array containing the poles in the secular
    !>         equation.
    !>

    DIFL

    !>          DIFL is REAL array, dimension ( K ).
    !>         On entry, DIFL(I) is the distance between I-th updated
    !>         (undeflated) singular value and the I-th (undeflated) old
    !>         singular value.
    !>

    DIFR

    !>          DIFR is REAL array, dimension ( LDGNUM, 2 ).
    !>         On entry, DIFR(I, 1) contains the distances between I-th
    !>         updated (undeflated) singular value and the I+1-th
    !>         (undeflated) old singular value. And DIFR(I, 2) is the
    !>         normalizing factor for the I-th right singular vector.
    !>

    Z

    !>          Z is REAL array, dimension ( K )
    !>         Contain the components of the deflation-adjusted updating row
    !>         vector.
    !>

    K

    !>          K is INTEGER
    !>         Contains the dimension of the non-deflated matrix,
    !>         This is the order of the related secular equation. 1 <= K <=N.
    !>

    C

    !>          C is REAL
    !>         C contains garbage if SQRE =0 and the C-value of a Givens
    !>         rotation related to the right null space if SQRE = 1.
    !>

    S

    !>          S is REAL
    !>         S contains garbage if SQRE =0 and the S-value of a Givens
    !>         rotation related to the right null space if SQRE = 1.
    !>

    RWORK

    !>          RWORK is REAL array, dimension
    !>         ( K*(1+NRHS) + 2*NRHS )
    !>

    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 dlals0 (integer icompq, integer nl, integer nr, integer sqre, integer nrhs, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldbx, * ) bx, integer ldbx, integer, dimension( * ) perm, integer givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, double precision, dimension( ldgnum, * ) givnum, integer ldgnum, double precision, dimension( ldgnum, * ) poles, double precision, dimension( * ) difl, double precision, dimension( ldgnum, * ) difr, double precision, dimension( * ) z, integer k, double precision c, double precision s, double precision, dimension( * ) work, integer info)

DLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.

Purpose:

    !>
    !> DLALS0 applies back the multiplying factors of either the left or the
    !> right singular vector matrix of a diagonal matrix appended by a row
    !> to the right hand side matrix B in solving the least squares problem
    !> using the divide-and-conquer SVD approach.
    !>
    !> For the left singular vector matrix, three types of orthogonal
    !> matrices are involved:
    !>
    !> (1L) Givens rotations: the number of such rotations is GIVPTR; the
    !>      pairs of columns/rows they were applied to are stored in GIVCOL;
    !>      and the C- and S-values of these rotations are stored in GIVNUM.
    !>
    !> (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
    !>      row, and for J=2:N, PERM(J)-th row of B is to be moved to the
    !>      J-th row.
    !>
    !> (3L) The left singular vector matrix of the remaining matrix.
    !>
    !> For the right singular vector matrix, four types of orthogonal
    !> matrices are involved:
    !>
    !> (1R) The right singular vector matrix of the remaining matrix.
    !>
    !> (2R) If SQRE = 1, one extra Givens rotation to generate the right
    !>      null space.
    !>
    !> (3R) The inverse transformation of (2L).
    !>
    !> (4R) The inverse transformation of (1L).
    !>

Parameters
    ICOMPQ

    !>          ICOMPQ is INTEGER
    !>         Specifies whether singular vectors are to be computed in
    !>         factored form:
    !>         = 0: Left singular vector matrix.
    !>         = 1: Right singular vector matrix.
    !>

    NL

    !>          NL is INTEGER
    !>         The row dimension of the upper block. NL >= 1.
    !>

    NR

    !>          NR is INTEGER
    !>         The row dimension of the lower block. NR >= 1.
    !>

    SQRE

    !>          SQRE is INTEGER
    !>         = 0: the lower block is an NR-by-NR square matrix.
    !>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
    !>
    !>         The bidiagonal matrix has row dimension N = NL + NR + 1,
    !>         and column dimension M = N + SQRE.
    !>

    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. LDB must be at least
    !>         max(1,MAX( M, N ) ).
    !>

    BX

    !>          BX is DOUBLE PRECISION array, dimension ( LDBX, NRHS )
    !>

    LDBX

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

    PERM

    !>          PERM is INTEGER array, dimension ( N )
    !>         The permutations (from deflation and sorting) applied
    !>         to the two blocks.
    !>

    GIVPTR

    !>          GIVPTR is INTEGER
    !>         The number of Givens rotations which took place in this
    !>         subproblem.
    !>

    GIVCOL

    !>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
    !>         Each pair of numbers indicates a pair of rows/columns
    !>         involved in a Givens rotation.
    !>

    LDGCOL

    !>          LDGCOL is INTEGER
    !>         The leading dimension of GIVCOL, must be at least N.
    !>

    GIVNUM

    !>          GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
    !>         Each number indicates the C or S value used in the
    !>         corresponding Givens rotation.
    !>

    LDGNUM

    !>          LDGNUM is INTEGER
    !>         The leading dimension of arrays DIFR, POLES and
    !>         GIVNUM, must be at least K.
    !>

    POLES

    !>          POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
    !>         On entry, POLES(1:K, 1) contains the new singular
    !>         values obtained from solving the secular equation, and
    !>         POLES(1:K, 2) is an array containing the poles in the secular
    !>         equation.
    !>

    DIFL

    !>          DIFL is DOUBLE PRECISION array, dimension ( K ).
    !>         On entry, DIFL(I) is the distance between I-th updated
    !>         (undeflated) singular value and the I-th (undeflated) old
    !>         singular value.
    !>

    DIFR

    !>          DIFR is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ).
    !>         On entry, DIFR(I, 1) contains the distances between I-th
    !>         updated (undeflated) singular value and the I+1-th
    !>         (undeflated) old singular value. And DIFR(I, 2) is the
    !>         normalizing factor for the I-th right singular vector.
    !>

    Z

    !>          Z is DOUBLE PRECISION array, dimension ( K )
    !>         Contain the components of the deflation-adjusted updating row
    !>         vector.
    !>

    K

    !>          K is INTEGER
    !>         Contains the dimension of the non-deflated matrix,
    !>         This is the order of the related secular equation. 1 <= K <=N.
    !>

    C

    !>          C is DOUBLE PRECISION
    !>         C contains garbage if SQRE =0 and the C-value of a Givens
    !>         rotation related to the right null space if SQRE = 1.
    !>

    S

    !>          S is DOUBLE PRECISION
    !>         S contains garbage if SQRE =0 and the S-value of a Givens
    !>         rotation related to the right null space if SQRE = 1.
    !>

    WORK

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

    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 slals0 (integer icompq, integer nl, integer nr, integer sqre, integer nrhs, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldbx, * ) bx, integer ldbx, integer, dimension( * ) perm, integer givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, real, dimension( ldgnum, * ) givnum, integer ldgnum, real, dimension( ldgnum, * ) poles, real, dimension( * ) difl, real, dimension( ldgnum, * ) difr, real, dimension( * ) z, integer k, real c, real s, real, dimension( * ) work, integer info)

SLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.

Purpose:

    !>
    !> SLALS0 applies back the multiplying factors of either the left or the
    !> right singular vector matrix of a diagonal matrix appended by a row
    !> to the right hand side matrix B in solving the least squares problem
    !> using the divide-and-conquer SVD approach.
    !>
    !> For the left singular vector matrix, three types of orthogonal
    !> matrices are involved:
    !>
    !> (1L) Givens rotations: the number of such rotations is GIVPTR; the
    !>      pairs of columns/rows they were applied to are stored in GIVCOL;
    !>      and the C- and S-values of these rotations are stored in GIVNUM.
    !>
    !> (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
    !>      row, and for J=2:N, PERM(J)-th row of B is to be moved to the
    !>      J-th row.
    !>
    !> (3L) The left singular vector matrix of the remaining matrix.
    !>
    !> For the right singular vector matrix, four types of orthogonal
    !> matrices are involved:
    !>
    !> (1R) The right singular vector matrix of the remaining matrix.
    !>
    !> (2R) If SQRE = 1, one extra Givens rotation to generate the right
    !>      null space.
    !>
    !> (3R) The inverse transformation of (2L).
    !>
    !> (4R) The inverse transformation of (1L).
    !>

Parameters
    ICOMPQ

    !>          ICOMPQ is INTEGER
    !>         Specifies whether singular vectors are to be computed in
    !>         factored form:
    !>         = 0: Left singular vector matrix.
    !>         = 1: Right singular vector matrix.
    !>

    NL

    !>          NL is INTEGER
    !>         The row dimension of the upper block. NL >= 1.
    !>

    NR

    !>          NR is INTEGER
    !>         The row dimension of the lower block. NR >= 1.
    !>

    SQRE

    !>          SQRE is INTEGER
    !>         = 0: the lower block is an NR-by-NR square matrix.
    !>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
    !>
    !>         The bidiagonal matrix has row dimension N = NL + NR + 1,
    !>         and column dimension M = N + SQRE.
    !>

    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. LDB must be at least
    !>         max(1,MAX( M, N ) ).
    !>

    BX

    !>          BX is REAL array, dimension ( LDBX, NRHS )
    !>

    LDBX

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

    PERM

    !>          PERM is INTEGER array, dimension ( N )
    !>         The permutations (from deflation and sorting) applied
    !>         to the two blocks.
    !>

    GIVPTR

    !>          GIVPTR is INTEGER
    !>         The number of Givens rotations which took place in this
    !>         subproblem.
    !>

    GIVCOL

    !>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
    !>         Each pair of numbers indicates a pair of rows/columns
    !>         involved in a Givens rotation.
    !>

    LDGCOL

    !>          LDGCOL is INTEGER
    !>         The leading dimension of GIVCOL, must be at least N.
    !>

    GIVNUM

    !>          GIVNUM is REAL array, dimension ( LDGNUM, 2 )
    !>         Each number indicates the C or S value used in the
    !>         corresponding Givens rotation.
    !>

    LDGNUM

    !>          LDGNUM is INTEGER
    !>         The leading dimension of arrays DIFR, POLES and
    !>         GIVNUM, must be at least K.
    !>

    POLES

    !>          POLES is REAL array, dimension ( LDGNUM, 2 )
    !>         On entry, POLES(1:K, 1) contains the new singular
    !>         values obtained from solving the secular equation, and
    !>         POLES(1:K, 2) is an array containing the poles in the secular
    !>         equation.
    !>

    DIFL

    !>          DIFL is REAL array, dimension ( K ).
    !>         On entry, DIFL(I) is the distance between I-th updated
    !>         (undeflated) singular value and the I-th (undeflated) old
    !>         singular value.
    !>

    DIFR

    !>          DIFR is REAL array, dimension ( LDGNUM, 2 ).
    !>         On entry, DIFR(I, 1) contains the distances between I-th
    !>         updated (undeflated) singular value and the I+1-th
    !>         (undeflated) old singular value. And DIFR(I, 2) is the
    !>         normalizing factor for the I-th right singular vector.
    !>

    Z

    !>          Z is REAL array, dimension ( K )
    !>         Contain the components of the deflation-adjusted updating row
    !>         vector.
    !>

    K

    !>          K is INTEGER
    !>         Contains the dimension of the non-deflated matrix,
    !>         This is the order of the related secular equation. 1 <= K <=N.
    !>

    C

    !>          C is REAL
    !>         C contains garbage if SQRE =0 and the C-value of a Givens
    !>         rotation related to the right null space if SQRE = 1.
    !>

    S

    !>          S is REAL
    !>         S contains garbage if SQRE =0 and the S-value of a Givens
    !>         rotation related to the right null space if SQRE = 1.
    !>

    WORK

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

    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 zlals0 (integer icompq, integer nl, integer nr, integer sqre, integer nrhs, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldbx, * ) bx, integer ldbx, integer, dimension( * ) perm, integer givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, double precision, dimension( ldgnum, * ) givnum, integer ldgnum, double precision, dimension( ldgnum, * ) poles, double precision, dimension( * ) difl, double precision, dimension( ldgnum, * ) difr, double precision, dimension( * ) z, integer k, double precision c, double precision s, double precision, dimension( * ) rwork, integer info)

ZLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.

Purpose:

    !>
    !> ZLALS0 applies back the multiplying factors of either the left or the
    !> right singular vector matrix of a diagonal matrix appended by a row
    !> to the right hand side matrix B in solving the least squares problem
    !> using the divide-and-conquer SVD approach.
    !>
    !> For the left singular vector matrix, three types of orthogonal
    !> matrices are involved:
    !>
    !> (1L) Givens rotations: the number of such rotations is GIVPTR; the
    !>      pairs of columns/rows they were applied to are stored in GIVCOL;
    !>      and the C- and S-values of these rotations are stored in GIVNUM.
    !>
    !> (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
    !>      row, and for J=2:N, PERM(J)-th row of B is to be moved to the
    !>      J-th row.
    !>
    !> (3L) The left singular vector matrix of the remaining matrix.
    !>
    !> For the right singular vector matrix, four types of orthogonal
    !> matrices are involved:
    !>
    !> (1R) The right singular vector matrix of the remaining matrix.
    !>
    !> (2R) If SQRE = 1, one extra Givens rotation to generate the right
    !>      null space.
    !>
    !> (3R) The inverse transformation of (2L).
    !>
    !> (4R) The inverse transformation of (1L).
    !>

Parameters
    ICOMPQ

    !>          ICOMPQ is INTEGER
    !>         Specifies whether singular vectors are to be computed in
    !>         factored form:
    !>         = 0: Left singular vector matrix.
    !>         = 1: Right singular vector matrix.
    !>

    NL

    !>          NL is INTEGER
    !>         The row dimension of the upper block. NL >= 1.
    !>

    NR

    !>          NR is INTEGER
    !>         The row dimension of the lower block. NR >= 1.
    !>

    SQRE

    !>          SQRE is INTEGER
    !>         = 0: the lower block is an NR-by-NR square matrix.
    !>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
    !>
    !>         The bidiagonal matrix has row dimension N = NL + NR + 1,
    !>         and column dimension M = N + SQRE.
    !>

    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. LDB must be at least
    !>         max(1,MAX( M, N ) ).
    !>

    BX

    !>          BX is COMPLEX*16 array, dimension ( LDBX, NRHS )
    !>

    LDBX

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

    PERM

    !>          PERM is INTEGER array, dimension ( N )
    !>         The permutations (from deflation and sorting) applied
    !>         to the two blocks.
    !>

    GIVPTR

    !>          GIVPTR is INTEGER
    !>         The number of Givens rotations which took place in this
    !>         subproblem.
    !>

    GIVCOL

    !>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
    !>         Each pair of numbers indicates a pair of rows/columns
    !>         involved in a Givens rotation.
    !>

    LDGCOL

    !>          LDGCOL is INTEGER
    !>         The leading dimension of GIVCOL, must be at least N.
    !>

    GIVNUM

    !>          GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
    !>         Each number indicates the C or S value used in the
    !>         corresponding Givens rotation.
    !>

    LDGNUM

    !>          LDGNUM is INTEGER
    !>         The leading dimension of arrays DIFR, POLES and
    !>         GIVNUM, must be at least K.
    !>

    POLES

    !>          POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
    !>         On entry, POLES(1:K, 1) contains the new singular
    !>         values obtained from solving the secular equation, and
    !>         POLES(1:K, 2) is an array containing the poles in the secular
    !>         equation.
    !>

    DIFL

    !>          DIFL is DOUBLE PRECISION array, dimension ( K ).
    !>         On entry, DIFL(I) is the distance between I-th updated
    !>         (undeflated) singular value and the I-th (undeflated) old
    !>         singular value.
    !>

    DIFR

    !>          DIFR is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ).
    !>         On entry, DIFR(I, 1) contains the distances between I-th
    !>         updated (undeflated) singular value and the I+1-th
    !>         (undeflated) old singular value. And DIFR(I, 2) is the
    !>         normalizing factor for the I-th right singular vector.
    !>

    Z

    !>          Z is DOUBLE PRECISION array, dimension ( K )
    !>         Contain the components of the deflation-adjusted updating row
    !>         vector.
    !>

    K

    !>          K is INTEGER
    !>         Contains the dimension of the non-deflated matrix,
    !>         This is the order of the related secular equation. 1 <= K <=N.
    !>

    C

    !>          C is DOUBLE PRECISION
    !>         C contains garbage if SQRE =0 and the C-value of a Givens
    !>         rotation related to the right null space if SQRE = 1.
    !>

    S

    !>          S is DOUBLE PRECISION
    !>         S contains garbage if SQRE =0 and the S-value of a Givens
    !>         rotation related to the right null space if SQRE = 1.
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension
    !>         ( K*(1+NRHS) + 2*NRHS )
    !>

    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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