gglse(3)

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, integer info)

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, integer ldb, complex*16, dimension( * ) c, complex*16, dimension( * ) d, complex*16, dimension( * ) x, complex*16, dimension( * ) work, integer lwork, 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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