SGGRQF - compute 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 factor
ization 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.
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 trape
zoidal matrix R; the remaining elements, with the
array TAUA, represent the orthogonal matrix Q as a
product of elementary reflectors (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 Fur
ther 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 trape
zoidal 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 prod
uct 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 Fur
ther Details). WORK (workspace/output) REAL
array, dimension (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 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.
= 0: successful exit
< 0: if INF0= -i, the i-th argument had an ille
gal 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.
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