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## zggrqf()

subroutine zggrqf | ( | integer | m,
|  | integer | p,
|  | integer | n,
|  | complex*16, dimension( lda, * ) | a,
|  | integer | lda,
|  | complex*16, dimension( * ) | taua,
|  | complex*16, dimension( ldb, * ) | b,
|  | integer | ldb,
|  | complex*16, dimension( * ) | taub,
|  | complex*16, dimension( * ) | work,
|  | integer | lwork,
|  | integer | info )

ZGGRQF

Download ZGGRQF + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> ZGGRQF 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 unitary matrix, Z is a P-by-P unitary
      !> 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**H
      !>
      !> where inv(B) denotes the inverse of the matrix B, and Z**H denotes the
      !> conjugate transpose of the matrix Z.
      !>

Parameters

  [in]

  M

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

  [in]

  P

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

  [in]

  N

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

  [in,out]

  A

      !>          A is COMPLEX*16 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 unitary
      !>          matrix Q as a product of elementary reflectors (see Further
      !>          Details).
      !>

  [in]

  LDA

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

  [out]

  TAUA

      !>          TAUA is COMPLEX*16 array, dimension (min(M,N))
      !>          The scalar factors of the elementary reflectors which
      !>          represent the unitary matrix Q (see Further Details).
      !>

  [in,out]

  B

      !>          B is COMPLEX*16 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 unitary matrix Z as a
      !>          product of elementary reflectors (see Further Details).
      !>

  [in]

  LDB

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

  [out]

  TAUB

      !>          TAUB is COMPLEX*16 array, dimension (min(P,N))
      !>          The scalar factors of the elementary reflectors which
      !>          represent the unitary matrix Z (see Further Details).
      !>

  [out]

  WORK

      !>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
      !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
      !>

  [in]

  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 ZUNMRQ.
      !>
      !>          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.
      !>

  [out]

  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(m,n).
      !>
      !>  Each H(i) has the form
      !>
      !>     H(i) = I - taua * v * v**H
      !>
      !>  where taua is a complex scalar, and v is a complex 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 ZUNGRQ.
      !>  To use Q to update another matrix, use LAPACK subroutine ZUNMRQ.
      !>
      !>  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**H
      !>
      !>  where taub is a complex scalar, and v is a complex 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 ZUNGQR.
      !>  To use Z to update another matrix, use LAPACK subroutine ZUNMQR.
      !>