Man page - gelst(3)

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Manual

gelst

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine cgelst (character trans, integer m, integer n, integer nrhs,complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, *) b, integer ldb, complex, dimension( * ) work, integer lwork, integerinfo)
subroutine dgelst (character trans, integer m, integer n, integer nrhs,double precision, dimension( lda, * ) a, integer lda, double precision,dimension( ldb, * ) b, integer ldb, double precision, dimension( * )work, integer lwork, integer info)
subroutine sgelst (character trans, integer m, integer n, integer nrhs,real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b,integer ldb, real, dimension( * ) work, integer lwork, integer info)
subroutine zgelst (character trans, integer m, integer n, integer nrhs,complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(ldb, * ) b, integer ldb, complex*16, dimension( * ) work, integerlwork, integer info)
Author

NAME

gelst - gelst: least squares using QR/LQ with T matrix

SYNOPSIS

Functions

subroutine cgelst (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
CGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.
subroutine dgelst (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
DGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.
subroutine sgelst (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
SGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.
subroutine zgelst (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
ZGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.

Detailed Description

Function Documentation

subroutine cgelst (character trans, integer m, integer n, integer nrhs,complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, *) b, integer ldb, complex, dimension( * ) work, integer lwork, integerinfo)

CGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.

Purpose:

CGELST solves overdetermined or underdetermined real linear systems
involving an M-by-N matrix A, or its conjugate-transpose, using a QR
or LQ factorization of A with compact WY representation of Q.

It is assumed that A has full rank, and only a rudimentary protection
against rank-deficient matrices is provided. This subroutine only detects
exact rank-deficiency, where a diagonal element of the triangular factor
of A is exactly zero.

It is conceivable for one (or more) of the diagonal elements of the triangular
factor of A to be subnormally tiny numbers without this subroutine signalling
an error. The solutions computed for such almost-rank-deficient matrices may
be less accurate due to a loss of numerical precision.

The following options are provided:

1. If TRANS = ’N’ and m >= n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A*X ||.

2. If TRANS = ’N’ and m < n: find the minimum norm solution of
an underdetermined system A * X = B.

3. If TRANS = ’C’ and m >= n: find the minimum norm solution of
an underdetermined system A**T * X = B.

4. If TRANS = ’C’ and m < n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A**T * X ||.

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.

Parameters

TRANS

TRANS is CHARACTER*1
= ’N’: the linear system involves A;
= ’C’: the linear system involves A**H.

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,
if M >= N, A is overwritten by details of its QR
factorization as returned by CGEQRT;
if M < N, A is overwritten by details of its LQ
factorization as returned by CGELQT.

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 matrix B of right hand side vectors, stored
columnwise; B is M-by-NRHS if TRANS = ’N’, or N-by-NRHS
if TRANS = ’C’.
On exit, if INFO = 0, B is overwritten by the solution
vectors, stored columnwise:
if TRANS = ’N’ and m >= n, rows 1 to n of B contain the least
squares solution vectors; the residual sum of squares for the
solution in each column is given by the sum of squares of
modulus of elements N+1 to M in that column;
if TRANS = ’N’ and m < n, rows 1 to N of B contain the
minimum norm solution vectors;
if TRANS = ’C’ and m >= n, rows 1 to M of B contain the
minimum norm solution vectors;
if TRANS = ’C’ and m < n, rows 1 to M of B contain the
least squares solution vectors; the residual sum of squares
for the solution in each column is given by the sum of
squares of the modulus of elements M+1 to N in that column.

LDB

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

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 >= max( 1, MN + max( MN, NRHS ) ).
For optimal performance,
LWORK >= max( 1, (MN + max( MN, NRHS ))*NB ).
where MN = min(M,N) and NB is the optimum block size.

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.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of the
triangular factor of A is exactly zero, so that A does not have
full rank; the least squares solution could not be
computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

November 2022, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine dgelst (character trans, integer m, integer n, integer nrhs,double precision, dimension( lda, * ) a, integer lda, double precision,dimension( ldb, * ) b, integer ldb, double precision, dimension( * )work, integer lwork, integer info)

DGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.

Purpose:

DGELST solves overdetermined or underdetermined real linear systems
involving an M-by-N matrix A, or its transpose, using a QR or LQ
factorization of A with compact WY representation of Q.

It is assumed that A has full rank, and only a rudimentary protection
against rank-deficient matrices is provided. This subroutine only detects
exact rank-deficiency, where a diagonal element of the triangular factor
of A is exactly zero.

It is conceivable for one (or more) of the diagonal elements of the triangular
factor of A to be subnormally tiny numbers without this subroutine signalling
an error. The solutions computed for such almost-rank-deficient matrices may
be less accurate due to a loss of numerical precision.

The following options are provided:

1. If TRANS = ’N’ and m >= n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A*X ||.

2. If TRANS = ’N’ and m < n: find the minimum norm solution of
an underdetermined system A * X = B.

3. If TRANS = ’T’ and m >= n: find the minimum norm solution of
an underdetermined system A**T * X = B.

4. If TRANS = ’T’ and m < n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A**T * X ||.

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.

Parameters

TRANS

TRANS is CHARACTER*1
= ’N’: the linear system involves A;
= ’T’: the linear system involves A**T.

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 DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
if M >= N, A is overwritten by details of its QR
factorization as returned by DGEQRT;
if M < N, A is overwritten by details of its LQ
factorization as returned by DGELQT.

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 matrix B of right hand side vectors, stored
columnwise; B is M-by-NRHS if TRANS = ’N’, or N-by-NRHS
if TRANS = ’T’.
On exit, if INFO = 0, B is overwritten by the solution
vectors, stored columnwise:
if TRANS = ’N’ and m >= n, rows 1 to n of B contain the least
squares solution vectors; the residual sum of squares for the
solution in each column is given by the sum of squares of
elements N+1 to M in that column;
if TRANS = ’N’ and m < n, rows 1 to N of B contain the
minimum norm solution vectors;
if TRANS = ’T’ and m >= n, rows 1 to M of B contain the
minimum norm solution vectors;
if TRANS = ’T’ and m < n, rows 1 to M of B contain the
least squares solution vectors; the residual sum of squares
for the solution in each column is given by the sum of
squares of elements M+1 to N in that column.

LDB

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

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 >= max( 1, MN + max( MN, NRHS ) ).
For optimal performance,
LWORK >= max( 1, (MN + max( MN, NRHS ))*NB ).
where MN = min(M,N) and NB is the optimum block size.

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.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of the
triangular factor of A is exactly zero, so that A does not have
full rank; the least squares solution could not be
computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

November 2022, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine sgelst (character trans, integer m, integer n, integer nrhs,real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b,integer ldb, real, dimension( * ) work, integer lwork, integer info)

SGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.

Purpose:

SGELST solves overdetermined or underdetermined real linear systems
involving an M-by-N matrix A, or its transpose, using a QR or LQ
factorization of A with compact WY representation of Q.

It is assumed that A has full rank, and only a rudimentary protection
against rank-deficient matrices is provided. This subroutine only detects
exact rank-deficiency, where a diagonal element of the triangular factor
of A is exactly zero.

It is conceivable for one (or more) of the diagonal elements of the triangular
factor of A to be subnormally tiny numbers without this subroutine signalling
an error. The solutions computed for such almost-rank-deficient matrices may
be less accurate due to a loss of numerical precision.

The following options are provided:

1. If TRANS = ’N’ and m >= n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A*X ||.

2. If TRANS = ’N’ and m < n: find the minimum norm solution of
an underdetermined system A * X = B.

3. If TRANS = ’T’ and m >= n: find the minimum norm solution of
an underdetermined system A**T * X = B.

4. If TRANS = ’T’ and m < n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A**T * X ||.

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.

Parameters

TRANS

TRANS is CHARACTER*1
= ’N’: the linear system involves A;
= ’T’: the linear system involves A**T.

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 REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
if M >= N, A is overwritten by details of its QR
factorization as returned by SGEQRT;
if M < N, A is overwritten by details of its LQ
factorization as returned by SGELQT.

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 matrix B of right hand side vectors, stored
columnwise; B is M-by-NRHS if TRANS = ’N’, or N-by-NRHS
if TRANS = ’T’.
On exit, if INFO = 0, B is overwritten by the solution
vectors, stored columnwise:
if TRANS = ’N’ and m >= n, rows 1 to n of B contain the least
squares solution vectors; the residual sum of squares for the
solution in each column is given by the sum of squares of
elements N+1 to M in that column;
if TRANS = ’N’ and m < n, rows 1 to N of B contain the
minimum norm solution vectors;
if TRANS = ’T’ and m >= n, rows 1 to M of B contain the
minimum norm solution vectors;
if TRANS = ’T’ and m < n, rows 1 to M of B contain the
least squares solution vectors; the residual sum of squares
for the solution in each column is given by the sum of
squares of elements M+1 to N in that column.

LDB

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

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 >= max( 1, MN + max( MN, NRHS ) ).
For optimal performance,
LWORK >= max( 1, (MN + max( MN, NRHS ))*NB ).
where MN = min(M,N) and NB is the optimum block size.

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.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of the
triangular factor of A is exactly zero, so that A does not have
full rank; the least squares solution could not be
computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

November 2022, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine zgelst (character trans, integer m, integer n, integer nrhs,complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(ldb, * ) b, integer ldb, complex*16, dimension( * ) work, integerlwork, integer info)

ZGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.

Purpose:

ZGELST solves overdetermined or underdetermined real linear systems
involving an M-by-N matrix A, or its conjugate-transpose, using a QR
or LQ factorization of A with compact WY representation of Q.

It is assumed that A has full rank, and only a rudimentary protection
against rank-deficient matrices is provided. This subroutine only detects
exact rank-deficiency, where a diagonal element of the triangular factor
of A is exactly zero.

It is conceivable for one (or more) of the diagonal elements of the triangular
factor of A to be subnormally tiny numbers without this subroutine signalling
an error. The solutions computed for such almost-rank-deficient matrices may
be less accurate due to a loss of numerical precision.

The following options are provided:

1. If TRANS = ’N’ and m >= n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A*X ||.

2. If TRANS = ’N’ and m < n: find the minimum norm solution of
an underdetermined system A * X = B.

3. If TRANS = ’C’ and m >= n: find the minimum norm solution of
an underdetermined system A**T * X = B.

4. If TRANS = ’C’ and m < n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A**T * X ||.

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.

Parameters

TRANS

TRANS is CHARACTER*1
= ’N’: the linear system involves A;
= ’C’: the linear system involves A**H.

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,
if M >= N, A is overwritten by details of its QR
factorization as returned by ZGEQRT;
if M < N, A is overwritten by details of its LQ
factorization as returned by ZGELQT.

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 matrix B of right hand side vectors, stored
columnwise; B is M-by-NRHS if TRANS = ’N’, or N-by-NRHS
if TRANS = ’C’.
On exit, if INFO = 0, B is overwritten by the solution
vectors, stored columnwise:
if TRANS = ’N’ and m >= n, rows 1 to n of B contain the least
squares solution vectors; the residual sum of squares for the
solution in each column is given by the sum of squares of
modulus of elements N+1 to M in that column;
if TRANS = ’N’ and m < n, rows 1 to N of B contain the
minimum norm solution vectors;
if TRANS = ’C’ and m >= n, rows 1 to M of B contain the
minimum norm solution vectors;
if TRANS = ’C’ and m < n, rows 1 to M of B contain the
least squares solution vectors; the residual sum of squares
for the solution in each column is given by the sum of
squares of the modulus of elements M+1 to N in that column.

LDB

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

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 >= max( 1, MN + max( MN, NRHS ) ).
For optimal performance,
LWORK >= max( 1, (MN + max( MN, NRHS ))*NB ).
where MN = min(M,N) and NB is the optimum block size.

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.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of the
triangular factor of A is exactly zero, so that A does not have
full rank; the least squares solution could not be
computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

November 2022, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

Author

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