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NAME

       sggrqf.f -

SYNOPSIS

   Functions/Subroutines
       subroutine sggrqf (M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
           SGGRQF

Function/Subroutine Documentation

   subroutine sggrqf (integerM, integerP, integerN, real, dimension( lda, * )A, integerLDA, real,
       dimension( * )TAUA, real, dimension( ldb, * )B, integerLDB, real, dimension( * )TAUB,
       real, dimension( * )WORK, integerLWORK, integerINFO)
       SGGRQF

       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**T

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

       Parameters:
           M

                     M is INTEGER
                     The number of rows of the matrix A.  M >= 0.

           P

                     P is INTEGER
                     The number of rows of the matrix B.  P >= 0.

           N

                     N is INTEGER
                     The number of columns of the matrices A and B. N >= 0.

           A

                     A is REAL array, dimension (LDA,N)
                     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

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

           TAUA

                     TAUA is REAL array, dimension (min(M,N))
                     The scalar factors of the elementary reflectors which
                     represent the orthogonal matrix Q (see Further Details).

           B

                     B is REAL array, dimension (LDB,N)
                     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

                     LDB is INTEGER
                     The leading dimension of the array B. LDB >= max(1,P).

           TAUB

                     TAUB is REAL array, dimension (min(P,N))
                     The scalar factors of the elementary reflectors which
                     represent the orthogonal matrix Z (see Further Details).

           WORK

                     WORK is REAL array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is 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 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

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INF0= -i, the i-th argument had an illegal value.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2011

       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**T

             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**T

             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.

       Definition at line 214 of file sggrqf.f.

Author

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