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zggqrf


 NAME
      ZGGQRF - compute a generalized QR factorization of an N-by-M
      matrix A and an N-by-P matrix B

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

          INTEGER        INFO, LDA, LDB, LWORK, M, N, P

          COMPLEX*16     A( LDA, * ), B( LDB, * ), TAUA( * ),
                         TAUB( * ), WORK( * )

 PURPOSE
      ZGGQRF 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 unitary matrix, Z is a P-by-P unitary
      matrix, and R and T assumes 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 an upper triangular matrix, 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 a P-by-P upper triangular matrix.

      In particular, if B is square and nonsingular, the GQR fac-
      torization of A and B implicitly gives the QR factorization
      of inv(B)*A:

                   inv(B)*A = Z'*(inv(T)*R)

      where inv(B) denotes the inverse of the matrix B, Z' denotes
      the conjugate transpose of 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) COMPLEX*16 array, dimension (LDA,M)
              On entry, the N-by-M matrix A.  On exit, the ele-
              ments 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 orthog-
              onal 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) COMPLEX*16 array, dimension (MIN(N,M))
              The scalar factors of the elementary reflectors (see
              Further Details).

      B       (input/output) COMPLEX*16 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 subdiag-
              onal 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) COMPLEX*16 array, dimension (MIN(N,P))
              The scalar factors of the elementary reflectors (see
              Further Details).

      WORK    (workspace) COMPLEX*16 array, dimension (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(1,N,M,P)*MAX(NB1,NB2,NB3), where NB1 is the
              optimal blocksize for the QR factorization of an N-
              by-M matrix A.  NB2 is the optimal blocksize for the
              RQ factorization of an N-by-P matrix B.  NB3 is the
              optimal block size for calling ZUNMQR.

      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'

      where taua is a complex scalar, and v is a complex 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 ZUNGQR.
      To use Q to update another matrix, use LAPACK subroutine
      ZUNMQR.

      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'

      where taub is a complex scalar, and v is a complex 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 ZUNGRQ.
      To use Z to update another matrix, use LAPACK subroutine
      ZUNMRQ.