NAME

cggqrf - compute 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)
  COMPLEX A(LDA,*), TAUA(*), B(LDB,*), TAUB(*), WORK(*)
  INTEGER N, M, P, LDA, LDB, LWORK, INFO

  SUBROUTINE CGGQRF_64( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, 
 *      LWORK, INFO)
  COMPLEX A(LDA,*), TAUA(*), B(LDB,*), TAUB(*), WORK(*)
  INTEGER*8 N, M, P, LDA, LDB, LWORK, INFO

F95 INTERFACE

  SUBROUTINE GGQRF( [N], [M], [P], A, [LDA], TAUA, B, [LDB], TAUB, 
 *       [WORK], [LWORK], [INFO])
  COMPLEX, DIMENSION(:) :: TAUA, TAUB, WORK
  COMPLEX, DIMENSION(:,:) :: A, B
  INTEGER :: N, M, P, LDA, LDB, LWORK, INFO

  SUBROUTINE GGQRF_64( [N], [M], [P], A, [LDA], TAUA, B, [LDB], TAUB, 
 *       [WORK], [LWORK], [INFO])
  COMPLEX, DIMENSION(:) :: TAUA, TAUB, WORK
  COMPLEX, DIMENSION(:,:) :: A, B
  INTEGER(8) :: N, M, P, LDA, LDB, LWORK, INFO

C INTERFACE

#include <sunperf.h>

void cggqrf(int n, int m, int p, complex *a, int lda, complex *taua, complex *b, int ldb, complex *taub, int *info);

void cggqrf_64(long n, long m, long p, complex *a, long lda, complex *taua, complex *b, long ldb, complex *taub, long *info);

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)
The number of rows of the matrices A and B. N > = 0.
M (input)
The number of columns of the matrix A. M > = 0.
P (input)
The number of columns of the matrix B. P > = 0.
A (input)
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)
The leading dimension of the array A. LDA > = max(1,N).
TAUA (output)
The scalar factors of the elementary reflectors which represent the unitary matrix Q (see Further Details).
B (input)
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)
The leading dimension of the array B. LDB > = max(1,N).
TAUB (output)
The scalar factors of the elementary reflectors which represent the unitary matrix Z (see Further Details).
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
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 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)

 = 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.