ggglm(3)

NAME
ggglm - ggglm: Gauss-Markov linear model

SYNOPSIS

  Functions

subroutine cggglm (n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info)
CGGGLM
subroutine dggglm (n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info)
DGGGLM
subroutine sggglm (n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info)
SGGGLM
subroutine zggglm (n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info)
ZGGGLM

Detailed Description

Function Documentation

  subroutine cggglm (integer n, integer m, integer p, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) d, complex, dimension( * ) x, complex, dimension( * ) y, complex, dimension( * ) work, integer lwork, integer info)

CGGGLM

Purpose:

    !>
    !> CGGGLM solves a general Gauss-Markov linear model (GLM) problem:
    !>
    !>         minimize || y ||_2   subject to   d = A*x + B*y
    !>             x
    !>
    !> where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
    !> given N-vector. It is assumed that M <= N <= M+P, and
    !>
    !>            rank(A) = M    and    rank( A B ) = N.
    !>
    !> Under these assumptions, the constrained equation is always
    !> consistent, and there is a unique solution x and a minimal 2-norm
    !> solution y, which is obtained using a generalized QR factorization
    !> of the matrices (A, B) given by
    !>
    !>    A = Q*(R),   B = Q*T*Z.
    !>          (0)
    !>
    !> In particular, if matrix B is square nonsingular, then the problem
    !> GLM is equivalent to the following weighted linear least squares
    !> problem
    !>
    !>              minimize || inv(B)*(d-A*x) ||_2
    !>                  x
    !>
    !> where inv(B) denotes the inverse of B.
    !>
    !> 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 QR
    !> factorization of the pair (A, B) 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
    N

    !>          N is INTEGER
    !>          The number of rows of the matrices A and B.  N >= 0.
    !>

    M

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

    P

    !>          P is INTEGER
    !>          The number of columns of the matrix B.  P >= N-M.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA,M)
    !>          On entry, the N-by-M matrix A.
    !>          On exit, the upper triangular part of the array A contains
    !>          the M-by-M upper triangular matrix R.
    !>

    LDA

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

    B

    !>          B is COMPLEX array, dimension (LDB,P)
    !>          On entry, the N-by-P matrix B.
    !>          On exit, if N <= P, the upper triangle of the subarray
    !>          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
    !>          if N > P, the elements on and above the (N-P)th subdiagonal
    !>          contain the N-by-P upper trapezoidal matrix T.
    !>

    LDB

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

    D

    !>          D is COMPLEX array, dimension (N)
    !>          On entry, D is the left hand side of the GLM equation.
    !>          On exit, D is destroyed.
    !>

    X

    !>          X is COMPLEX array, dimension (M)
    !>

    Y

    !>          Y is COMPLEX array, dimension (P)
    !>
    !>          On exit, X and Y are the solutions of the GLM 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,N+M+P).
    !>          For optimum performance, LWORK >= M+min(N,P)+max(N,P)*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 A in the
    !>                generalized QR factorization of the pair (A, B) is exactly
    !>                singular, so that rank(A) < M; the least squares
    !>                solution could not be computed.
    !>          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
    !>                factor T associated with B in the generalized QR
    !>                factorization of the pair (A, B) is exactly singular, so that
    !>                rank( A B ) < N; the least squares solution could not
    !>                be computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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

DGGGLM

Purpose:

    !>
    !> DGGGLM solves a general Gauss-Markov linear model (GLM) problem:
    !>
    !>         minimize || y ||_2   subject to   d = A*x + B*y
    !>             x
    !>
    !> where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
    !> given N-vector. It is assumed that M <= N <= M+P, and
    !>
    !>            rank(A) = M    and    rank( A B ) = N.
    !>
    !> Under these assumptions, the constrained equation is always
    !> consistent, and there is a unique solution x and a minimal 2-norm
    !> solution y, which is obtained using a generalized QR factorization
    !> of the matrices (A, B) given by
    !>
    !>    A = Q*(R),   B = Q*T*Z.
    !>          (0)
    !>
    !> In particular, if matrix B is square nonsingular, then the problem
    !> GLM is equivalent to the following weighted linear least squares
    !> problem
    !>
    !>              minimize || inv(B)*(d-A*x) ||_2
    !>                  x
    !>
    !> where inv(B) denotes the inverse of B.
    !>
    !> 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 QR
    !> factorization of the pair (A, B) 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
    N

    !>          N is INTEGER
    !>          The number of rows of the matrices A and B.  N >= 0.
    !>

    M

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

    P

    !>          P is INTEGER
    !>          The number of columns of the matrix B.  P >= N-M.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA,M)
    !>          On entry, the N-by-M matrix A.
    !>          On exit, the upper triangular part of the array A contains
    !>          the M-by-M upper triangular matrix R.
    !>

    LDA

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

    B

    !>          B is DOUBLE PRECISION array, dimension (LDB,P)
    !>          On entry, the N-by-P matrix B.
    !>          On exit, if N <= P, the upper triangle of the subarray
    !>          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
    !>          if N > P, the elements on and above the (N-P)th subdiagonal
    !>          contain the N-by-P upper trapezoidal matrix T.
    !>

    LDB

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

    D

    !>          D is DOUBLE PRECISION array, dimension (N)
    !>          On entry, D is the left hand side of the GLM equation.
    !>          On exit, D is destroyed.
    !>

    X

    !>          X is DOUBLE PRECISION array, dimension (M)
    !>

    Y

    !>          Y is DOUBLE PRECISION array, dimension (P)
    !>
    !>          On exit, X and Y are the solutions of the GLM 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,N+M+P).
    !>          For optimum performance, LWORK >= M+min(N,P)+max(N,P)*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 A in the
    !>                generalized QR factorization of the pair (A, B) is exactly
    !>                singular, so that rank(A) < M; the least squares
    !>                solution could not be computed.
    !>          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
    !>                factor T associated with B in the generalized QR
    !>                factorization of the pair (A, B) is exactly singular, so that
    !>                rank( A B ) < N; the least squares solution could not
    !>                be computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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

SGGGLM

Purpose:

    !>
    !> SGGGLM solves a general Gauss-Markov linear model (GLM) problem:
    !>
    !>         minimize || y ||_2   subject to   d = A*x + B*y
    !>             x
    !>
    !> where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
    !> given N-vector. It is assumed that M <= N <= M+P, and
    !>
    !>            rank(A) = M    and    rank( A B ) = N.
    !>
    !> Under these assumptions, the constrained equation is always
    !> consistent, and there is a unique solution x and a minimal 2-norm
    !> solution y, which is obtained using a generalized QR factorization
    !> of the matrices (A, B) given by
    !>
    !>    A = Q*(R),   B = Q*T*Z.
    !>          (0)
    !>
    !> In particular, if matrix B is square nonsingular, then the problem
    !> GLM is equivalent to the following weighted linear least squares
    !> problem
    !>
    !>              minimize || inv(B)*(d-A*x) ||_2
    !>                  x
    !>
    !> where inv(B) denotes the inverse of B.
    !>
    !> 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 QR
    !> factorization of the pair (A, B) 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
    N

    !>          N is INTEGER
    !>          The number of rows of the matrices A and B.  N >= 0.
    !>

    M

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

    P

    !>          P is INTEGER
    !>          The number of columns of the matrix B.  P >= N-M.
    !>

    A

    !>          A is REAL array, dimension (LDA,M)
    !>          On entry, the N-by-M matrix A.
    !>          On exit, the upper triangular part of the array A contains
    !>          the M-by-M upper triangular matrix R.
    !>

    LDA

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

    B

    !>          B is REAL array, dimension (LDB,P)
    !>          On entry, the N-by-P matrix B.
    !>          On exit, if N <= P, the upper triangle of the subarray
    !>          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
    !>          if N > P, the elements on and above the (N-P)th subdiagonal
    !>          contain the N-by-P upper trapezoidal matrix T.
    !>

    LDB

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

    D

    !>          D is REAL array, dimension (N)
    !>          On entry, D is the left hand side of the GLM equation.
    !>          On exit, D is destroyed.
    !>

    X

    !>          X is REAL array, dimension (M)
    !>

    Y

    !>          Y is REAL array, dimension (P)
    !>
    !>          On exit, X and Y are the solutions of the GLM 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,N+M+P).
    !>          For optimum performance, LWORK >= M+min(N,P)+max(N,P)*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 A in the
    !>                generalized QR factorization of the pair (A, B) is exactly
    !>                singular, so that rank(A) < M; the least squares
    !>                solution could not be computed.
    !>          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
    !>                factor T associated with B in the generalized QR
    !>                factorization of the pair (A, B) is exactly singular, so that
    !>                rank( A B ) < N; the least squares solution could not
    !>                be computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine zggglm (integer n, integer m, integer p, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) d, complex*16, dimension( * ) x, complex*16, dimension( * ) y, complex*16, dimension( * ) work, integer lwork, integer info)

ZGGGLM

Purpose:

    !>
    !> ZGGGLM solves a general Gauss-Markov linear model (GLM) problem:
    !>
    !>         minimize || y ||_2   subject to   d = A*x + B*y
    !>             x
    !>
    !> where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
    !> given N-vector. It is assumed that M <= N <= M+P, and
    !>
    !>            rank(A) = M    and    rank( A B ) = N.
    !>
    !> Under these assumptions, the constrained equation is always
    !> consistent, and there is a unique solution x and a minimal 2-norm
    !> solution y, which is obtained using a generalized QR factorization
    !> of the matrices (A, B) given by
    !>
    !>    A = Q*(R),   B = Q*T*Z.
    !>          (0)
    !>
    !> In particular, if matrix B is square nonsingular, then the problem
    !> GLM is equivalent to the following weighted linear least squares
    !> problem
    !>
    !>              minimize || inv(B)*(d-A*x) ||_2
    !>                  x
    !>
    !> where inv(B) denotes the inverse of B.
    !>
    !> 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 QR
    !> factorization of the pair (A, B) 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
    N

    !>          N is INTEGER
    !>          The number of rows of the matrices A and B.  N >= 0.
    !>

    M

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

    P

    !>          P is INTEGER
    !>          The number of columns of the matrix B.  P >= N-M.
    !>

    A

    !>          A is COMPLEX*16 array, dimension (LDA,M)
    !>          On entry, the N-by-M matrix A.
    !>          On exit, the upper triangular part of the array A contains
    !>          the M-by-M upper triangular matrix R.
    !>

    LDA

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

    B

    !>          B is COMPLEX*16 array, dimension (LDB,P)
    !>          On entry, the N-by-P matrix B.
    !>          On exit, if N <= P, the upper triangle of the subarray
    !>          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
    !>          if N > P, the elements on and above the (N-P)th subdiagonal
    !>          contain the N-by-P upper trapezoidal matrix T.
    !>

    LDB

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

    D

    !>          D is COMPLEX*16 array, dimension (N)
    !>          On entry, D is the left hand side of the GLM equation.
    !>          On exit, D is destroyed.
    !>

    X

    !>          X is COMPLEX*16 array, dimension (M)
    !>

    Y

    !>          Y is COMPLEX*16 array, dimension (P)
    !>
    !>          On exit, X and Y are the solutions of the GLM 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,N+M+P).
    !>          For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
    !>          where NB is an upper bound for the optimal blocksizes for
    !>          ZGEQRF, ZGERQF, ZUNMQR and ZUNMRQ.
    !>
    !>          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 A in the
    !>                generalized QR factorization of the pair (A, B) is exactly
    !>                singular, so that rank(A) < M; the least squares
    !>                solution could not be computed.
    !>          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
    !>                factor T associated with B in the generalized QR
    !>                factorization of the pair (A, B) is exactly singular, so that
    !>                rank( A B ) < N; the least squares solution could not
    !>                be computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Author

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