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RedHat 9 (Linux i386) - man page for sggqrf (redhat section l)

SGGQRF(l)					)					SGGQRF(l)

NAME
       SGGQRF - compute a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix
       B

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

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

	   REAL 	  A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), WORK( * )

PURPOSE
       SGGQRF 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  orthogonal  matrix, Z is a P-by-P orthogonal 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 transpose of the
       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) REAL 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 trian-
	       gular if N >= M); the elements below the diagonal, with the array TAUA,	represent
	       the  orthogonal	matrix Q as a product of min(N,M) elementary reflectors (see Fur-
	       ther Details).

       LDA     (input) INTEGER
	       The leading dimension of the array A. LDA >= max(1,N).

       TAUA    (output) REAL array, dimension (min(N,M))
	       The scalar factors of the elementary reflectors	which  represent  the  orthogonal
	       matrix  Q  (see	Further  Details).   B	     (input/output) REAL 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
	       orthogonal 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) REAL array, dimension (min(N,P))
	       The scalar factors of the elementary reflectors	which  represent  the  orthogonal
	       matrix  Z (see Further Details).  WORK	 (workspace/output) REAL 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 N-by-M matrix, NB2 is the optimal blocksize for the RQ fac-
	       torization  of  an  N-by-P  matrix, and NB3 is the optimal blocksize for a call of
	       SORMQR.

	       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 real scalar, and v is a real 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 SORGQR.
       To use Q to update another matrix, use LAPACK subroutine SORMQR.

       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 real scalar, and v is a real 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 SORGRQ.
       To use Z to update another matrix, use LAPACK subroutine SORMRQ.

LAPACK version 3.0			   15 June 2000 				SGGQRF(l)


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