.SH NAME
SGGRQF - a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B
.SH SYNOPSIS
.TP 19
SUBROUTINE SGGRQF(
M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
LWORK, INFO )
.TP 19
.ti +4
INTEGER
INFO, LDA, LDB, LWORK, M, N, P
.TP 19
.ti +4
REAL
A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
WORK( * )
.SH PURPOSE
SGGRQF computes a generalized RQ factorization of an M-by-N matrix A
and a P-by-N matrix B:
A = R*Q, B = Z*T*Q,
.br
where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
matrix, and R and T assume one of the forms:
.br
if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
N-M M ( R21 ) N
N
.br
where R12 or R21 is upper triangular, and
.br
if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
( 0 ) P-N P N-P
N
.br
where T11 is upper triangular.
.br
In particular, if B is square and nonsingular, the GRQ factorization
of A and B implicitly gives the RQ factorization of A*inv(B):
A*inv(B) = (R*inv(T))*Z\(aq
.br
where inv(B) denotes the inverse of the matrix B, and Z\(aq denotes the
transpose of the matrix Z.
.br
.SH ARGUMENTS
.TP 8
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
.TP 8
P (input) INTEGER
The number of rows of the matrix B. P >= 0.
.TP 8
N (input) INTEGER
The number of columns of the matrices A and B. N >= 0.
.TP 8
A (input/output) REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, if M <= N, the upper triangle of the subarray
A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
if M > N, the elements on and above the (M-N)-th subdiagonal
contain the M-by-N upper trapezoidal matrix R; the remaining
elements, with the array TAUA, represent the orthogonal
matrix Q as a product of elementary reflectors (see Further
Details).
.TP 8
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
.TP 8
TAUA (output) REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q (see Further Details).
B (input/output) REAL array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, the elements on and above the diagonal of the array
contain the min(P,N)-by-N upper trapezoidal matrix T (T is
upper triangular if P >= N); the elements below the diagonal,
with the array TAUB, represent the orthogonal matrix Z as a
product of elementary reflectors (see Further Details).
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,P).
.TP 8
TAUB (output) REAL array, dimension (min(P,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Z (see Further Details).
WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
.TP 8
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,N,M,P).
For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
where NB1 is the optimal blocksize for the RQ factorization
of an M-by-N matrix, NB2 is the optimal blocksize for the
QR factorization of a P-by-N matrix, and NB3 is the optimal
blocksize for a call of 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.
.TP 8
INFO (output) INTEGER
= 0: successful exit
.br
< 0: if INF0= -i, the i-th argument had an illegal value.
.SH FURTHER DETAILS
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
.br
Each H(i) has the form
.br
H(i) = I - taua * v * v\(aq
.br
where taua is a real scalar, and v is a real vector with
.br
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
.br
To form Q explicitly, use LAPACK subroutine SORGRQ.
.br
To use Q to update another matrix, use LAPACK subroutine SORMRQ.
The matrix Z is represented as a product of elementary reflectors
Z = H(1) H(2) . . . H(k), where k = min(p,n).
.br
Each H(i) has the form
.br
H(i) = I - taub * v * v\(aq
.br
where taub is a real scalar, and v is a real vector with
.br
v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
and taub in TAUB(i).
.br
To form Z explicitly, use LAPACK subroutine SORGQR.
.br
To use Z to update another matrix, use LAPACK subroutine SORMQR.