cggrqf-f(3)

cggrqf.f –

Synopsis

Functions/Subroutines

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

Function/Subroutine Documentation

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

CGGRQF

Purpose:

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

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

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

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

 B is COMPLEX 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 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 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 CUNMRQ.
 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 CUNGRQ.
 To use Q to update another matrix, use LAPACK subroutine CUNMRQ.
 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 CUNGQR.
 To use Z to update another matrix, use LAPACK subroutine CUNMQR.

Definition at line 214 of file cggrqf.f.

Author

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