Man page - gglse(3)

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Manual

gglse

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine cgglse (integer m, integer n, integer p, complex, dimension(lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb,complex, dimension( * ) c, complex, dimension( * ) d, complex,dimension( * ) x, complex, dimension( * ) work, integer lwork, integerinfo)
subroutine dgglse (integer m, integer n, integer p, double precision,dimension( lda, * ) a, integer lda, double precision, dimension( ldb, *) b, integer ldb, double precision, dimension( * ) c, double precision,dimension( * ) d, double precision, dimension( * ) x, double precision,dimension( * ) work, integer lwork, integer info)
subroutine sgglse (integer m, integer n, integer p, real, dimension( lda, *) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real,dimension( * ) c, real, dimension( * ) d, real, dimension( * ) x, real,dimension( * ) work, integer lwork, integer info)
subroutine zgglse (integer m, integer n, integer p, complex*16, dimension(lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integerldb, complex*16, dimension( * ) c, complex*16, dimension( * ) d,complex*16, dimension( * ) x, complex*16, dimension( * ) work, integerlwork, integer info)
Author

NAME

gglse - gglse: equality-constrained least squares

SYNOPSIS

Functions

subroutine cgglse (m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)
CGGLSE solves overdetermined or underdetermined systems for OTHER matrices
subroutine dgglse (m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)
DGGLSE solves overdetermined or underdetermined systems for OTHER matrices
subroutine sgglse (m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)
SGGLSE solves overdetermined or underdetermined systems for OTHER matrices
subroutine zgglse (m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)
ZGGLSE solves overdetermined or underdetermined systems for OTHER matrices

Detailed Description

Function Documentation

subroutine cgglse (integer m, integer n, integer p, complex, dimension(lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb,complex, dimension( * ) c, complex, dimension( * ) d, complex,dimension( * ) x, complex, dimension( * ) work, integer lwork, integerinfo)

CGGLSE solves overdetermined or underdetermined systems for OTHER matrices

Purpose:

CGGLSE solves the linear equality-constrained least squares (LSE)
problem:

minimize || c - A*x ||_2 subject to B*x = d

where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
M-vector, and d is a given P-vector. It is assumed that
P <= N <= M+P, and

rank(B) = P and rank( (A) ) = N.
( (B) )

These conditions ensure that the LSE problem has a unique solution,
which is obtained using a generalized RQ factorization of the
matrices (B, A) given by

B = (0 R)*Q, A = Z*T*Q.

Callers of this subroutine should note that the singularity/rank-deficiency checks
implemented in this subroutine are rudimentary. The CTRTRS subroutine called by this
subroutine only signals a failure due to singularity if the problem is exactly singular.

It is conceivable for one (or more) of the factors involved in the generalized RQ
factorization of the pair (B, A) to be subnormally close to singularity without this
subroutine signalling an error. The solutions computed for such almost-rank-deficient
problems may be less accurate due to a loss of numerical precision.

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 matrices A and B. N >= 0.

P

P is INTEGER
The number of rows of the matrix B. 0 <= P <= N <= M+P.

A

A is COMPLEX array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix T.

LDA

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

B

B is COMPLEX array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, the upper triangle of the subarray B(1:P,N-P+1:N)
contains the P-by-P upper triangular matrix R.

LDB

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

C

C is COMPLEX array, dimension (M)
On entry, C contains the right hand side vector for the
least squares part of the LSE problem.
On exit, the residual sum of squares for the solution
is given by the sum of squares of elements N-P+1 to M of
vector C.

D

D is COMPLEX array, dimension (P)
On entry, D contains the right hand side vector for the
constrained equation.
On exit, D is destroyed.

X

X is COMPLEX array, dimension (N)
On exit, X is the solution of the LSE problem.

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,M+N+P).
For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,
where NB is an upper bound for the optimal blocksizes for
CGEQRF, CGERQF, CUNMQR and CUNMRQ.

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.
= 1: the upper triangular factor R associated with B in the
generalized RQ factorization of the pair (B, A) is exactly
singular, so that rank(B) < P; the least squares
solution could not be computed.
= 2: the (N-P) by (N-P) part of the upper trapezoidal factor
T associated with A in the generalized RQ factorization
of the pair (B, A) is exactly singular, so that
rank( (A) ) < N; the least squares solution could not
( (B) )
be computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgglse (integer m, integer n, integer p, double precision,dimension( lda, * ) a, integer lda, double precision, dimension( ldb, *) b, integer ldb, double precision, dimension( * ) c, double precision,dimension( * ) d, double precision, dimension( * ) x, double precision,dimension( * ) work, integer lwork, integer info)

DGGLSE solves overdetermined or underdetermined systems for OTHER matrices

Purpose:

DGGLSE solves the linear equality-constrained least squares (LSE)
problem:

minimize || c - A*x ||_2 subject to B*x = d

where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
M-vector, and d is a given P-vector. It is assumed that
P <= N <= M+P, and

rank(B) = P and rank( (A) ) = N.
( (B) )

These conditions ensure that the LSE problem has a unique solution,
which is obtained using a generalized RQ factorization of the
matrices (B, A) given by

B = (0 R)*Q, A = Z*T*Q.

Callers of this subroutine should note that the singularity/rank-deficiency checks
implemented in this subroutine are rudimentary. The DTRTRS subroutine called by this
subroutine only signals a failure due to singularity if the problem is exactly singular.

It is conceivable for one (or more) of the factors involved in the generalized RQ
factorization of the pair (B, A) to be subnormally close to singularity without this
subroutine signalling an error. The solutions computed for such almost-rank-deficient
problems may be less accurate due to a loss of numerical precision.

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 matrices A and B. N >= 0.

P

P is INTEGER
The number of rows of the matrix B. 0 <= P <= N <= M+P.

A

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix T.

LDA

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

B

B is DOUBLE PRECISION array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, the upper triangle of the subarray B(1:P,N-P+1:N)
contains the P-by-P upper triangular matrix R.

LDB

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

C

C is DOUBLE PRECISION array, dimension (M)
On entry, C contains the right hand side vector for the
least squares part of the LSE problem.
On exit, the residual sum of squares for the solution
is given by the sum of squares of elements N-P+1 to M of
vector C.

D

D is DOUBLE PRECISION array, dimension (P)
On entry, D contains the right hand side vector for the
constrained equation.
On exit, D is destroyed.

X

X is DOUBLE PRECISION array, dimension (N)
On exit, X is the solution of the LSE problem.

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,M+N+P).
For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,
where NB is an upper bound for the optimal blocksizes for
DGEQRF, SGERQF, DORMQR and SORMRQ.

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.
= 1: the upper triangular factor R associated with B in the
generalized RQ factorization of the pair (B, A) is exactly
singular, so that rank(B) < P; the least squares
solution could not be computed.
= 2: the (N-P) by (N-P) part of the upper trapezoidal factor
T associated with A in the generalized RQ factorization
of the pair (B, A) is exactly singular, so that
rank( (A) ) < N; the least squares solution could not
( (B) )
be computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine sgglse (integer m, integer n, integer p, real, dimension( lda, *) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real,dimension( * ) c, real, dimension( * ) d, real, dimension( * ) x, real,dimension( * ) work, integer lwork, integer info)

SGGLSE solves overdetermined or underdetermined systems for OTHER matrices

Purpose:

SGGLSE solves the linear equality-constrained least squares (LSE)
problem:

minimize || c - A*x ||_2 subject to B*x = d

where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
M-vector, and d is a given P-vector. It is assumed that
P <= N <= M+P, and

rank(B) = P and rank( (A) ) = N.
( (B) )

These conditions ensure that the LSE problem has a unique solution,
which is obtained using a generalized RQ factorization of the
matrices (B, A) given by

B = (0 R)*Q, A = Z*T*Q.

Callers of this subroutine should note that the singularity/rank-deficiency checks
implemented in this subroutine are rudimentary. The STRTRS subroutine called by this
subroutine only signals a failure due to singularity if the problem is exactly singular.

It is conceivable for one (or more) of the factors involved in the generalized RQ
factorization of the pair (B, A) to be subnormally close to singularity without this
subroutine signalling an error. The solutions computed for such almost-rank-deficient
problems may be less accurate due to a loss of numerical precision.

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 matrices A and B. N >= 0.

P

P is INTEGER
The number of rows of the matrix B. 0 <= P <= N <= M+P.

A

A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix T.

LDA

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

B

B is REAL array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, the upper triangle of the subarray B(1:P,N-P+1:N)
contains the P-by-P upper triangular matrix R.

LDB

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

C

C is REAL array, dimension (M)
On entry, C contains the right hand side vector for the
least squares part of the LSE problem.
On exit, the residual sum of squares for the solution
is given by the sum of squares of elements N-P+1 to M of
vector C.

D

D is REAL array, dimension (P)
On entry, D contains the right hand side vector for the
constrained equation.
On exit, D is destroyed.

X

X is REAL array, dimension (N)
On exit, X is the solution of the LSE problem.

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,M+N+P).
For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,
where NB is an upper bound for the optimal blocksizes for
SGEQRF, SGERQF, SORMQR and SORMRQ.

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.
= 1: the upper triangular factor R associated with B in the
generalized RQ factorization of the pair (B, A) is exactly
singular, so that rank(B) < P; the least squares
solution could not be computed.
= 2: the (N-P) by (N-P) part of the upper trapezoidal factor
T associated with A in the generalized RQ factorization
of the pair (B, A) is exactly singular, so that
rank( (A) ) < N; the least squares solution could not
( (B) )
be computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine zgglse (integer m, integer n, integer p, complex*16, dimension(lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integerldb, complex*16, dimension( * ) c, complex*16, dimension( * ) d,complex*16, dimension( * ) x, complex*16, dimension( * ) work, integerlwork, integer info)

ZGGLSE solves overdetermined or underdetermined systems for OTHER matrices

Purpose:

ZGGLSE solves the linear equality-constrained least squares (LSE)
problem:

minimize || c - A*x ||_2 subject to B*x = d

where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
M-vector, and d is a given P-vector. It is assumed that
P <= N <= M+P, and

rank(B) = P and rank( (A) ) = N.
( (B) )

These conditions ensure that the LSE problem has a unique solution,
which is obtained using a generalized RQ factorization of the
matrices (B, A) given by

B = (0 R)*Q, A = Z*T*Q.

Callers of this subroutine should note that the singularity/rank-deficiency checks
implemented in this subroutine are rudimentary. The ZTRTRS subroutine called by this
subroutine only signals a failure due to singularity if the problem is exactly singular.

It is conceivable for one (or more) of the factors involved in the generalized RQ
factorization of the pair (B, A) to be subnormally close to singularity without this
subroutine signalling an error. The solutions computed for such almost-rank-deficient
problems may be less accurate due to a loss of numerical precision.

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 matrices A and B. N >= 0.

P

P is INTEGER
The number of rows of the matrix B. 0 <= P <= N <= M+P.

A

A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix T.

LDA

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

B

B is COMPLEX*16 array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, the upper triangle of the subarray B(1:P,N-P+1:N)
contains the P-by-P upper triangular matrix R.

LDB

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

C

C is COMPLEX*16 array, dimension (M)
On entry, C contains the right hand side vector for the
least squares part of the LSE problem.
On exit, the residual sum of squares for the solution
is given by the sum of squares of elements N-P+1 to M of
vector C.

D

D is COMPLEX*16 array, dimension (P)
On entry, D contains the right hand side vector for the
constrained equation.
On exit, D is destroyed.

X

X is COMPLEX*16 array, dimension (N)
On exit, X is the solution of the LSE problem.

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,M+N+P).
For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,
where NB is an upper bound for the optimal blocksizes for
ZGEQRF, CGERQF, ZUNMQR and CUNMRQ.

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.
= 1: the upper triangular factor R associated with B in the
generalized RQ factorization of the pair (B, A) is exactly
singular, so that rank(B) < P; the least squares
solution could not be computed.
= 2: the (N-P) by (N-P) part of the upper trapezoidal factor
T associated with A in the generalized RQ factorization
of the pair (B, A) is exactly singular, so that
rank( (A) ) < N; the least squares solution could not
( (B) )
be computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Author

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