SGGRQF(1) LAPACK routine (version 3.2) SGGRQF(1)NAME
SGGRQF - computes a generalized RQ factorization of an M-by-N matrix A
and a P-by-N matrix B
SYNOPSIS
SUBROUTINE SGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK,
INFO )
INTEGER INFO, LDA, LDB, LWORK, M, N, P
REAL A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
WORK( * )
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,
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:
if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
N-M M ( R21 ) N
N
where R12 or R21 is upper triangular, and
if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
( 0 ) P-N P N-P
N
where T11 is upper triangular.
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'
where inv(B) denotes the inverse of the matrix B, and Z' denotes the
transpose of the matrix Z.
ARGUMENTS
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
P (input) INTEGER
The number of rows of the matrix B. P >= 0.
N (input) INTEGER
The number of columns of the matrices A and B. N >= 0.
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, repre‐
sent the orthogonal matrix Q as a product of elementary reflec‐
tors (see Further Details).
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
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).
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.
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 factoriza‐
tion 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.
INFO (output) INTEGER
= 0: successful exit
< 0: if INF0= -i, the i-th argument had an illegal value.
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).
Each H(i) has the form
H(i) = I - taua * v * v'
where taua is a real scalar, and v is a real vector with
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).
To form Q explicitly, use LAPACK subroutine SORGRQ.
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).
Each H(i) has the form
H(i) = I - taub * v * v'
where taub is a real scalar, and v is a real vector with
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).
To form Z explicitly, use LAPACK subroutine SORGQR.
To use Z to update another matrix, use LAPACK subroutine SORMQR.
LAPACK routine (version 3.2) November 2008 SGGRQF(1)
