Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  computes  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**T*(inv(T)*R)
        where inv(B) denotes the inverse of the matrix B, and Z**T 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**T
        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**T
        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 routine (version 3.3.1)             April 2011                            SGGQRF(3lapack)