sggrqf.f(3)

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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