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 triangular 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 Further 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 (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 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.