SGGQRF(3F)SGGQRF(3F)NAME
SGGQRF - compute a generalized QR factorization of an N-by-M matrix A and
an N-by-P matrix B
SYNOPSIS
SUBROUTINE SGGQRF( N, M, P, 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
SGGQRF computes a generalized QR factorization of an N-by-M matrix A and
an N-by-P matrix B:
A = Q*R, B = Q*T*Z,
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 N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N,
( 0 ) N-M N M-N
M
where R11 is upper triangular, and
if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P,
P-N N ( T21 ) P
P
where T12 or T21 is upper triangular.
In particular, if B is square and nonsingular, the GQR factorization of A
and B implicitly gives the QR factorization of inv(B)*A:
inv(B)*A = Z'*(inv(T)*R)
where inv(B) denotes the inverse of the matrix B, and Z' denotes the
transpose of the matrix Z.
ARGUMENTS
N (input) INTEGER
The number of rows of the matrices A and B. N >= 0.
M (input) INTEGER
The number of columns of the matrix A. M >= 0.
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SGGQRF(3F)SGGQRF(3F)
P (input) INTEGER
The number of columns of the matrix B. P >= 0.
A (input/output) REAL array, dimension (LDA,M)
On entry, the N-by-M matrix A. On exit, the elements on and
above the diagonal of the array contain the min(N,M)-by-M upper
trapezoidal matrix R (R is upper triangular if N >= M); the
elements below the diagonal, with the array TAUA, represent the
orthogonal matrix Q as a product of min(N,M) elementary
reflectors (see Further Details).
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
TAUA (output) REAL array, dimension (min(N,M))
The scalar factors of the elementary reflectors which represent
the orthogonal matrix Q (see Further Details). B
(input/output) 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; the remaining elements,
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,N).
TAUB (output) REAL array, dimension (min(N,P))
The scalar factors of the elementary reflectors which represent
the orthogonal matrix Z (see Further 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 QR factorization of an N-
by-M matrix, NB2 is the optimal blocksize for the RQ
factorization of an N-by-P matrix, and NB3 is the optimal
blocksize for a call of SORMQR.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -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(n,m).
Each H(i) has the form
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SGGQRF(3F)SGGQRF(3F)H(i) = I - taua * v * v'
where taua is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and
taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine SORGQR.
To use Q to update another matrix, use LAPACK subroutine SORMQR.
The matrix Z is represented as a product of elementary reflectors
Z = H(1)H(2) . . . H(k), where k = min(n,p).
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(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine SORGRQ.
To use Z to update another matrix, use LAPACK subroutine SORMRQ.
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