CGGQRF man page on Scientific

```CGGQRF(1)		 LAPACK routine (version 3.2)		     CGGQRF(1)

NAME
CGGQRF  - computes a generalized QR factorization of an N-by-M matrix A
and an N-by-P 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  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 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 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
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 sub‐
diagonal	 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 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  (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 factoriza‐
tion  of an N-by-P 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 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 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(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 CUNGRQ.
To use Z to update another matrix, use LAPACK subroutine CUNMRQ.

LAPACK routine (version 3.2)	 November 2008			     CGGQRF(1)
```
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