NAME

sggrqf - compute a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B

SYNOPSIS

  SUBROUTINE SGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, 
 *      INFO)
  INTEGER M, P, N, LDA, LDB, LWORK, INFO
  REAL A(LDA,*), TAUA(*), B(LDB,*), TAUB(*), WORK(*)

  SUBROUTINE SGGRQF_64( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, 
 *      LWORK, INFO)
  INTEGER*8 M, P, N, LDA, LDB, LWORK, INFO
  REAL A(LDA,*), TAUA(*), B(LDB,*), TAUB(*), WORK(*)

F95 INTERFACE

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

  SUBROUTINE GGRQF_64( [M], [P], [N], A, [LDA], TAUA, B, [LDB], TAUB, 
 *       [WORK], [LWORK], [INFO])
  INTEGER(8) :: M, P, N, LDA, LDB, LWORK, INFO
  REAL, DIMENSION(:) :: TAUA, TAUB, WORK
  REAL, DIMENSION(:,:) :: A, B

C INTERFACE

#include <sunperf.h>

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

void sggrqf_64(long m, long p, long n, float *a, long lda, float *taua, float *b, long ldb, float *taub, long *info);

PURPOSE

sggrqf computes a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B:

            A = R*Q,        B = Z*T*Q,

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 M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N, N-M M ( R21 ) N N

where R12 or R21 is upper triangular, and

if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P, ( 0 ) P-N P N-P N

where T11 is upper triangular.

In particular, if B is square and nonsingular, the GRQ factorization of A and B implicitly gives the RQ factorization of A*inv(B):

             A*inv(B) = (R*inv(T))*Z'

where inv(B) denotes the inverse of the matrix B, and Z' denotes the transpose of the matrix Z.

ARGUMENTS

M (input)
The number of rows of the matrix A. M > = 0.
P (input)
The number of rows of the matrix B. P > = 0.
N (input)
The number of columns of the matrices A and B. N > = 0.
A (input/output)
On entry, the M-by-N matrix A. On exit, if M < = N, the upper triangle of the subarray A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R; if M > N, the elements on and above the (M-N)-th subdiagonal contain the M-by-N upper trapezoidal matrix R; the remaining elements, with the array TAUA, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details).
LDA (input)
The leading dimension of the array A. LDA > = max(1,M).
TAUA (output)
The scalar factors of the elementary reflectors which represent the orthogonal matrix Q (see Further Details).
B (input/output)
On entry, the P-by-N matrix B. On exit, the elements on and above the diagonal of the array contain the min(P,N)-by-N upper trapezoidal matrix T (T is upper triangular if P > = N); the elements below the diagonal, with the array TAUB, represent the orthogonal matrix Z as a product of elementary reflectors (see Further Details).
LDB (input)
The leading dimension of the array B. LDB > = max(1,P).
TAUB (output)
The scalar factors of the elementary reflectors which represent the orthogonal 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 RQ factorization of an M-by-N matrix, NB2 is the optimal blocksize for the QR factorization of a P-by-N matrix, and NB3 is the optimal blocksize for a call of SORMRQ.
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 INF0 = -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(m,n).

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(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in A(m-k+i,1:n-k+i-1), and taua in TAUA(i).

To form Q explicitly, use LAPACK subroutine SORGRQ.

To use Q to update another matrix, use LAPACK subroutine SORMRQ.

The matrix Z is represented as a product of elementary reflectors

   Z  = H(1) H(2) . . . H(k), where k  = min(p,n).

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(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i), and taub in TAUB(i).

To form Z explicitly, use LAPACK subroutine SORGQR.

To use Z to update another matrix, use LAPACK subroutine SORMQR.