Man page - gelsd(3)

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Manual

gelsd

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine cgelsd (integer m, integer n, integer nrhs, complex, dimension(lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb,real, dimension( * ) s, real rcond, integer rank, complex, dimension( *) work, integer lwork, real, dimension( * ) rwork, integer, dimension(* ) iwork, integer info)
subroutine dgelsd (integer m, integer n, integer nrhs, double precision,dimension( lda, * ) a, integer lda, double precision, dimension( ldb, *) b, integer ldb, double precision, dimension( * ) s, double precisionrcond, integer rank, double precision, dimension( * ) work, integerlwork, integer, dimension( * ) iwork, integer info)
subroutine sgelsd (integer m, integer n, integer nrhs, real, dimension(lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb,real, dimension( * ) s, real rcond, integer rank, real, dimension( * )work, integer lwork, integer, dimension( * ) iwork, integer info)
subroutine zgelsd (integer m, integer n, integer nrhs, complex*16,dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b,integer ldb, double precision, dimension( * ) s, double precisionrcond, integer rank, complex*16, dimension( * ) work, integer lwork,double precision, dimension( * ) rwork, integer, dimension( * ) iwork,integer info)
Author

NAME

gelsd - gelsd: least squares using SVD, divide and conquer

SYNOPSIS

Functions

subroutine cgelsd (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, iwork, info)
CGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices
subroutine dgelsd (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, iwork, info)
DGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices
subroutine sgelsd (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, iwork, info)
SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices
subroutine zgelsd (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, iwork, info)
ZGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

Detailed Description

Function Documentation

subroutine cgelsd (integer m, integer n, integer nrhs, complex, dimension(lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb,real, dimension( * ) s, real rcond, integer rank, complex, dimension( *) work, integer lwork, real, dimension( * ) rwork, integer, dimension(* ) iwork, integer info)

CGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

Purpose:

CGELSD computes the minimum-norm solution to a real linear least
squares problem:
minimize 2-norm(| b - A*x |)
using the singular value decomposition (SVD) of A. A is an M-by-N
matrix which may be rank-deficient.

Several right hand side vectors b and solution vectors x can be
handled in a single call; they are stored as the columns of the
M-by-NRHS right hand side matrix B and the N-by-NRHS solution
matrix X.

The problem is solved in three steps:
(1) Reduce the coefficient matrix A to bidiagonal form with
Householder transformations, reducing the original problem
into a ’bidiagonal least squares problem’ (BLS)
(2) Solve the BLS using a divide and conquer approach.
(3) Apply back all the Householder transformations to solve
the original least squares problem.

The effective rank of A is determined by treating as zero those
singular values which are less than RCOND times the largest singular
value.

Parameters

M

M is INTEGER
The number of rows of the matrix A. M >= 0.

N

N is INTEGER
The number of columns of the matrix A. N >= 0.

NRHS

NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.

A

A is COMPLEX array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A has been destroyed.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

B

B is COMPLEX array, dimension (LDB,NRHS)
On entry, the M-by-NRHS right hand side matrix B.
On exit, B is overwritten by the N-by-NRHS solution matrix X.
If m >= n and RANK = n, the residual sum-of-squares for
the solution in the i-th column is given by the sum of
squares of the modulus of elements n+1:m in that column.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M,N).

S

S is REAL array, dimension (min(M,N))
The singular values of A in decreasing order.
The condition number of A in the 2-norm = S(1)/S(min(m,n)).

RCOND

RCOND is REAL
RCOND is used to determine the effective rank of A.
Singular values S(i) <= RCOND*S(1) are treated as zero.
If RCOND < 0, machine precision is used instead.

RANK

RANK is INTEGER
The effective rank of A, i.e., the number of singular values
which are greater than RCOND*S(1).

WORK

WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK must be at least 1.
The exact minimum amount of workspace needed depends on M,
N and NRHS. As long as LWORK is at least
2 * N + N * NRHS
if M is greater than or equal to N or
2 * M + M * NRHS
if M is less than N, the code will execute correctly.
For good performance, LWORK should generally be larger.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the array WORK and the
minimum sizes of the arrays RWORK and IWORK, and returns
these values as the first entries of the WORK, RWORK and
IWORK arrays, and no error message related to LWORK is issued
by XERBLA.

RWORK

RWORK is REAL array, dimension (MAX(1,LRWORK))
LRWORK >=
10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
if M is greater than or equal to N or
10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
if M is less than N, the code will execute correctly.
SMLSIZ is returned by ILAENV and is equal to the maximum
size of the subproblems at the bottom of the computation
tree (usually about 25), and
NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.

IWORK

IWORK is INTEGER array, dimension (MAX(1,LIWORK))
LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
where MINMN = MIN( M,N ).
On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: the algorithm for computing the SVD failed to converge;
if INFO = i, i off-diagonal elements of an intermediate
bidiagonal form did not converge to zero.

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 dgelsd (integer m, integer n, integer nrhs, double precision,dimension( lda, * ) a, integer lda, double precision, dimension( ldb, *) b, integer ldb, double precision, dimension( * ) s, double precisionrcond, integer rank, double precision, dimension( * ) work, integerlwork, integer, dimension( * ) iwork, integer info)

DGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

Purpose:

DGELSD computes the minimum-norm solution to a real linear least
squares problem:
minimize 2-norm(| b - A*x |)
using the singular value decomposition (SVD) of A. A is an M-by-N
matrix which may be rank-deficient.

Several right hand side vectors b and solution vectors x can be
handled in a single call; they are stored as the columns of the
M-by-NRHS right hand side matrix B and the N-by-NRHS solution
matrix X.

The problem is solved in three steps:
(1) Reduce the coefficient matrix A to bidiagonal form with
Householder transformations, reducing the original problem
into a ’bidiagonal least squares problem’ (BLS)
(2) Solve the BLS using a divide and conquer approach.
(3) Apply back all the Householder transformations to solve
the original least squares problem.

The effective rank of A is determined by treating as zero those
singular values which are less than RCOND times the largest singular
value.

Parameters

M

M is INTEGER
The number of rows of A. M >= 0.

N

N is INTEGER
The number of columns of A. N >= 0.

NRHS

NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.

A

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A has been destroyed.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

B

B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the M-by-NRHS right hand side matrix B.
On exit, B is overwritten by the N-by-NRHS solution
matrix X. If m >= n and RANK = n, the residual
sum-of-squares for the solution in the i-th column is given
by the sum of squares of elements n+1:m in that column.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,max(M,N)).

S

S is DOUBLE PRECISION array, dimension (min(M,N))
The singular values of A in decreasing order.
The condition number of A in the 2-norm = S(1)/S(min(m,n)).

RCOND

RCOND is DOUBLE PRECISION
RCOND is used to determine the effective rank of A.
Singular values S(i) <= RCOND*S(1) are treated as zero.
If RCOND < 0, machine precision is used instead.

RANK

RANK is INTEGER
The effective rank of A, i.e., the number of singular values
which are greater than RCOND*S(1).

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK must be at least 1.
The exact minimum amount of workspace needed depends on M,
N and NRHS. As long as LWORK is at least
12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
if M is greater than or equal to N or
12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
if M is less than N, the code will execute correctly.
SMLSIZ is returned by ILAENV and is equal to the maximum
size of the subproblems at the bottom of the computation
tree (usually about 25), and
NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
For good performance, LWORK should generally be larger.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

IWORK

IWORK is INTEGER array, dimension (MAX(1,LIWORK))
LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),
where MINMN = MIN( M,N ).
On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: the algorithm for computing the SVD failed to converge;
if INFO = i, i off-diagonal elements of an intermediate
bidiagonal form did not converge to zero.

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 sgelsd (integer m, integer n, integer nrhs, real, dimension(lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb,real, dimension( * ) s, real rcond, integer rank, real, dimension( * )work, integer lwork, integer, dimension( * ) iwork, integer info)

SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

Purpose:

SGELSD computes the minimum-norm solution to a real linear least
squares problem:
minimize 2-norm(| b - A*x |)
using the singular value decomposition (SVD) of A. A is an M-by-N
matrix which may be rank-deficient.

Several right hand side vectors b and solution vectors x can be
handled in a single call; they are stored as the columns of the
M-by-NRHS right hand side matrix B and the N-by-NRHS solution
matrix X.

The problem is solved in three steps:
(1) Reduce the coefficient matrix A to bidiagonal form with
Householder transformations, reducing the original problem
into a ’bidiagonal least squares problem’ (BLS)
(2) Solve the BLS using a divide and conquer approach.
(3) Apply back all the Householder transformations to solve
the original least squares problem.

The effective rank of A is determined by treating as zero those
singular values which are less than RCOND times the largest singular
value.

Parameters

M

M is INTEGER
The number of rows of A. M >= 0.

N

N is INTEGER
The number of columns of A. N >= 0.

NRHS

NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.

A

A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A has been destroyed.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

B

B is REAL array, dimension (LDB,NRHS)
On entry, the M-by-NRHS right hand side matrix B.
On exit, B is overwritten by the N-by-NRHS solution
matrix X. If m >= n and RANK = n, the residual
sum-of-squares for the solution in the i-th column is given
by the sum of squares of elements n+1:m in that column.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,max(M,N)).

S

S is REAL array, dimension (min(M,N))
The singular values of A in decreasing order.
The condition number of A in the 2-norm = S(1)/S(min(m,n)).

RCOND

RCOND is REAL
RCOND is used to determine the effective rank of A.
Singular values S(i) <= RCOND*S(1) are treated as zero.
If RCOND < 0, machine precision is used instead.

RANK

RANK is INTEGER
The effective rank of A, i.e., the number of singular values
which are greater than RCOND*S(1).

WORK

WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK must be at least 1.
The exact minimum amount of workspace needed depends on M,
N and NRHS. As long as LWORK is at least
12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
if M is greater than or equal to N or
12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
if M is less than N, the code will execute correctly.
SMLSIZ is returned by ILAENV and is equal to the maximum
size of the subproblems at the bottom of the computation
tree (usually about 25), and
NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
For good performance, LWORK should generally be larger.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the array WORK and the
minimum size of the array IWORK, and returns these values as
the first entries of the WORK and IWORK arrays, and no error
message related to LWORK is issued by XERBLA.

IWORK

IWORK is INTEGER array, dimension (MAX(1,LIWORK))
LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
where MINMN = MIN( M,N ).
On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: the algorithm for computing the SVD failed to converge;
if INFO = i, i off-diagonal elements of an intermediate
bidiagonal form did not converge to zero.

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 zgelsd (integer m, integer n, integer nrhs, complex*16,dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b,integer ldb, double precision, dimension( * ) s, double precisionrcond, integer rank, complex*16, dimension( * ) work, integer lwork,double precision, dimension( * ) rwork, integer, dimension( * ) iwork,integer info)

ZGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

Purpose:

ZGELSD computes the minimum-norm solution to a real linear least
squares problem:
minimize 2-norm(| b - A*x |)
using the singular value decomposition (SVD) of A. A is an M-by-N
matrix which may be rank-deficient.

Several right hand side vectors b and solution vectors x can be
handled in a single call; they are stored as the columns of the
M-by-NRHS right hand side matrix B and the N-by-NRHS solution
matrix X.

The problem is solved in three steps:
(1) Reduce the coefficient matrix A to bidiagonal form with
Householder transformations, reducing the original problem
into a ’bidiagonal least squares problem’ (BLS)
(2) Solve the BLS using a divide and conquer approach.
(3) Apply back all the Householder transformations to solve
the original least squares problem.

The effective rank of A is determined by treating as zero those
singular values which are less than RCOND times the largest singular
value.

Parameters

M

M is INTEGER
The number of rows of the matrix A. M >= 0.

N

N is INTEGER
The number of columns of the matrix A. N >= 0.

NRHS

NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.

A

A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A has been destroyed.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

B

B is COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the M-by-NRHS right hand side matrix B.
On exit, B is overwritten by the N-by-NRHS solution matrix X.
If m >= n and RANK = n, the residual sum-of-squares for
the solution in the i-th column is given by the sum of
squares of the modulus of elements n+1:m in that column.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M,N).

S

S is DOUBLE PRECISION array, dimension (min(M,N))
The singular values of A in decreasing order.
The condition number of A in the 2-norm = S(1)/S(min(m,n)).

RCOND

RCOND is DOUBLE PRECISION
RCOND is used to determine the effective rank of A.
Singular values S(i) <= RCOND*S(1) are treated as zero.
If RCOND < 0, machine precision is used instead.

RANK

RANK is INTEGER
The effective rank of A, i.e., the number of singular values
which are greater than RCOND*S(1).

WORK

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK must be at least 1.
The exact minimum amount of workspace needed depends on M,
N and NRHS. As long as LWORK is at least
2*N + N*NRHS
if M is greater than or equal to N or
2*M + M*NRHS
if M is less than N, the code will execute correctly.
For good performance, LWORK should generally be larger.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the array WORK and the
minimum sizes of the arrays RWORK and IWORK, and returns
these values as the first entries of the WORK, RWORK and
IWORK arrays, and no error message related to LWORK is issued
by XERBLA.

RWORK

RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
LRWORK >=
10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
if M is greater than or equal to N or
10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
if M is less than N, the code will execute correctly.
SMLSIZ is returned by ILAENV and is equal to the maximum
size of the subproblems at the bottom of the computation
tree (usually about 25), and
NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.

IWORK

IWORK is INTEGER array, dimension (MAX(1,LIWORK))
LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
where MINMN = MIN( M,N ).
On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: the algorithm for computing the SVD failed to converge;
if INFO = i, i off-diagonal elements of an intermediate
bidiagonal form did not converge to zero.

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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