DGGQRF(3)      LAPACK routine of NEC Numeric Library Collection      DGGQRF(3)



NAME
       DGGQRF

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



PURPOSE
            DGGQRF 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)
                     N is INTEGER
                     The number of rows of the matrices A and B. N >= 0.

           M         (input)
                     M is INTEGER
                     The number of columns of the matrix A.  M >= 0.

           P         (input)
                     P is INTEGER
                     The number of columns of the matrix B.  P >= 0.

           A         (input/output)
                     A is DOUBLE PRECISION 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)
                     LDA is INTEGER
                     The leading dimension of the array A. LDA >= max(1,N).

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

           B         (input/output)
                     B is DOUBLE PRECISION 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)
                     LDB is INTEGER
                     The leading dimension of the array B. LDB >= max(1,N).

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

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

           LWORK     (input)
                     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 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 DORMQR.

                     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)
                     INFO is 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 DORGQR.
             To use Q to update another matrix, use LAPACK subroutine DORMQR.

             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 DORGRQ.
             To use Z to update another matrix, use LAPACK subroutine DORMRQ.



LAPACK routine                  31 October 2017                      DGGQRF(3)