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