.TH "ggglm" 3 "Tue Jan 28 2025 00:54:31" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
ggglm \- ggglm: Gauss-Markov linear model
.SH SYNOPSIS
.br
.PP
.SS "Functions"
.in +1c
.ti -1c
.RI "subroutine \fBcggglm\fP (n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info)"
.br
.RI "\fBCGGGLM\fP "
.ti -1c
.RI "subroutine \fBdggglm\fP (n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info)"
.br
.RI "\fBDGGGLM\fP "
.ti -1c
.RI "subroutine \fBsggglm\fP (n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info)"
.br
.RI "\fBSGGGLM\fP "
.ti -1c
.RI "subroutine \fBzggglm\fP (n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info)"
.br
.RI "\fBZGGGLM\fP "
.in -1c
.SH "Detailed Description"
.PP
.SH "Function Documentation"
.PP
.SS "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)"
.PP
\fBCGGGLM\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP
.PP
.nf
N is INTEGER
The number of rows of the matrices A and B\&. N >= 0\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
The number of columns of the matrix A\&. 0 <= M <= N\&.
.fi
.PP
.br
\fIP\fP
.PP
.nf
P is INTEGER
The number of columns of the matrix B\&. P >= N-M\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
LDB is INTEGER
The leading dimension of the array B\&. LDB >= max(1,N)\&.
.fi
.PP
.br
\fID\fP
.PP
.nf
D is COMPLEX array, dimension (N)
On entry, D is the left hand side of the GLM equation\&.
On exit, D is destroyed\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX array, dimension (M)
.fi
.PP
.br
\fIY\fP
.PP
.nf
Y is COMPLEX array, dimension (P)
On exit, X and Y are the solutions of the GLM problem\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
.SS "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)"
.PP
\fBDGGGLM\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP
.PP
.nf
N is INTEGER
The number of rows of the matrices A and B\&. N >= 0\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
The number of columns of the matrix A\&. 0 <= M <= N\&.
.fi
.PP
.br
\fIP\fP
.PP
.nf
P is INTEGER
The number of columns of the matrix B\&. P >= N-M\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
LDB is INTEGER
The leading dimension of the array B\&. LDB >= max(1,N)\&.
.fi
.PP
.br
\fID\fP
.PP
.nf
D is DOUBLE PRECISION array, dimension (N)
On entry, D is the left hand side of the GLM equation\&.
On exit, D is destroyed\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is DOUBLE PRECISION array, dimension (M)
.fi
.PP
.br
\fIY\fP
.PP
.nf
Y is DOUBLE PRECISION array, dimension (P)
On exit, X and Y are the solutions of the GLM problem\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
.SS "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)"
.PP
\fBSGGGLM\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP
.PP
.nf
N is INTEGER
The number of rows of the matrices A and B\&. N >= 0\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
The number of columns of the matrix A\&. 0 <= M <= N\&.
.fi
.PP
.br
\fIP\fP
.PP
.nf
P is INTEGER
The number of columns of the matrix B\&. P >= N-M\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
LDB is INTEGER
The leading dimension of the array B\&. LDB >= max(1,N)\&.
.fi
.PP
.br
\fID\fP
.PP
.nf
D is REAL array, dimension (N)
On entry, D is the left hand side of the GLM equation\&.
On exit, D is destroyed\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is REAL array, dimension (M)
.fi
.PP
.br
\fIY\fP
.PP
.nf
Y is REAL array, dimension (P)
On exit, X and Y are the solutions of the GLM problem\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
.SS "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)"
.PP
\fBZGGGLM\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP
.PP
.nf
N is INTEGER
The number of rows of the matrices A and B\&. N >= 0\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
The number of columns of the matrix A\&. 0 <= M <= N\&.
.fi
.PP
.br
\fIP\fP
.PP
.nf
P is INTEGER
The number of columns of the matrix B\&. P >= N-M\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
LDB is INTEGER
The leading dimension of the array B\&. LDB >= max(1,N)\&.
.fi
.PP
.br
\fID\fP
.PP
.nf
D is COMPLEX*16 array, dimension (N)
On entry, D is the left hand side of the GLM equation\&.
On exit, D is destroyed\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 array, dimension (M)
.fi
.PP
.br
\fIY\fP
.PP
.nf
Y is COMPLEX*16 array, dimension (P)
On exit, X and Y are the solutions of the GLM problem\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
.SH "Author"
.PP
Generated automatically by Doxygen for LAPACK from the source code\&.