
CGGQRF(l) ) CGGQRF(l)
NAME
CGGQRF  compute a generalized QR factorization of an NbyM matrix A and an NbyP matrix
B
SYNOPSIS
SUBROUTINE CGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO )
INTEGER INFO, LDA, LDB, LWORK, M, N, P
COMPLEX A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), WORK( * )
PURPOSE
CGGQRF computes a generalized QR factorization of an NbyM matrix A and an NbyP matrix
B:
A = Q*R, B = Q*T*Z,
where Q is an NbyN unitary matrix, Z is a PbyP unitary 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 ) NM N MN
M
where R11 is upper triangular, and
if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) NP,
PN 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 conjugate transpose
of 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.
P (input) INTEGER
The number of columns of the matrix B. P >= 0.
A (input/output) COMPLEX array, dimension (LDA,M)
On entry, the NbyM matrix A. On exit, the elements on and above the diagonal of
the array contain the min(N,M)byM upper trapezoidal matrix R (R is upper trian
gular if N >= M); the elements below the diagonal, with the array TAUA, represent
the unitary 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) COMPLEX array, dimension (min(N,M))
The scalar factors of the elementary reflectors which represent the unitary matrix
Q (see Further Details). B (input/output) COMPLEX array, dimension (LDB,P)
On entry, the NbyP matrix B. On exit, if N <= P, the upper triangle of the sub
array B(1:N,PN+1:P) contains the NbyN upper triangular matrix T; if N > P, the
elements on and above the (NP)th subdiagonal contain the NbyP upper trape
zoidal matrix T; the remaining elements, with the array TAUB, represent the uni
tary 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) COMPLEX array, dimension (min(N,P))
The scalar factors of the elementary reflectors which represent the unitary matrix
Z (see Further Details). WORK (workspace/output) COMPLEX 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 NbyM matrix, NB2 is the optimal blocksize for the RQ fac
torization of an NbyP matrix, and NB3 is the optimal blocksize for a call of
CUNMQR.
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 INFO = i, the ith 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
H(i) = I  taua * v * v'
where taua is a complex scalar, and v is a complex vector with v(1:i1) = 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 CUNGQR.
To use Q to update another matrix, use LAPACK subroutine CUNMQR.
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 complex scalar, and v is a complex vector with v(pk+i+1:p) = 0 and v(p
k+i) = 1; v(1:pk+i1) is stored on exit in B(nk+i,1:pk+i1), and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine CUNGRQ.
To use Z to update another matrix, use LAPACK subroutine CUNMRQ.
LAPACK version 3.0 15 June 2000 CGGQRF(l) 
