table of contents
| lals0(3) | LAPACK | 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
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
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
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
Osni Marques, LBNL/NERSC, USA
Author¶
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