SRC/sggqrf.f(3)

NAME
SRC/sggqrf.f

SYNOPSIS

  Functions/Subroutines
subroutine sggqrf (n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)
SGGQRF

Function/Subroutine Documentation

  subroutine sggqrf (integer n, integer m, integer p, real, dimension( lda, * ) a, integer lda, real, dimension( * ) taua, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) taub, real, dimension( * ) work, integer lwork, integer info)

SGGQRF

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.

Parameters
    N

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

    M

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

    P

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

    A

              A is 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

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

    TAUA

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

    B

              B is 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

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

    TAUB

              TAUB is REAL array, dimension (min(N,P))
              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 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

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.

Definition at line 213 of file sggqrf.f.

Author

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