zggrqf.f (3) - Linux Manuals

NAME

zggrqf.f -

SYNOPSIS


Functions/Subroutines


subroutine zggrqf (M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
ZGGRQF

Function/Subroutine Documentation

subroutine zggrqf (integerM, integerP, integerN, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( * )TAUA, complex*16, dimension( ldb, * )B, integerLDB, complex*16, dimension( * )TAUB, complex*16, dimension( * )WORK, integerLWORK, integerINFO)

ZGGRQF

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:

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 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).


LDA 

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


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).


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).


LDB 

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


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).


WORK 

          WORK is COMPLEX*16 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 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.


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.

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**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.


 

Definition at line 214 of file zggrqf.f.

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