 ZGGQRF(3) computes a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B

## SYNOPSIS

SUBROUTINE ZGGQRF(
N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO )

INTEGER INFO, LDA, LDB, LWORK, M, N, P

COMPLEX*16 A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), WORK( * )

## PURPOSE

ZGGQRF 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 unitary matrix, Z is a P-by-P 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  ) 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 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*16 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 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*16 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*16 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 unitary 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*16 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*16 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 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 ZUNMQR. 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 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

H(i) = I - taua * v * v'
where taua is a complex scalar, and v is a complex 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 ZUNGQR.
To use Q to update another matrix, use LAPACK subroutine ZUNMQR. 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(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 ZUNGRQ.
To use Z to update another matrix, use LAPACK subroutine ZUNMRQ.