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 factor
ization 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.
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 ele
ments on and above the diagonal of the array con
tain 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 Fur
ther 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 sub
array 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 ele
ments, with the array TAUB, represent the orthogo
nal 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 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 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.
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
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -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(n,m).
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(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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