LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ cggrqf()

 subroutine cggrqf ( integer M, integer P, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAUA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) TAUB, complex, dimension( * ) WORK, integer LWORK, integer INFO )

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
 [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 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 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 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 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 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 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.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO=-i, the i-th argument had an illegal value.```
Date
December 2016
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 216 of file cggrqf.f.

216 *
217 * -- LAPACK computational routine (version 3.7.0) --
218 * -- LAPACK is a software package provided by Univ. of Tennessee, --
219 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
220 * December 2016
221 *
222 * .. Scalar Arguments ..
223  INTEGER info, lda, ldb, lwork, m, n, p
224 * ..
225 * .. Array Arguments ..
226  COMPLEX a( lda, * ), b( ldb, * ), taua( * ), taub( * ),
227  \$ work( * )
228 * ..
229 *
230 * =====================================================================
231 *
232 * .. Local Scalars ..
233  LOGICAL lquery
234  INTEGER lopt, lwkopt, nb, nb1, nb2, nb3
235 * ..
236 * .. External Subroutines ..
237  EXTERNAL cgeqrf, cgerqf, cunmrq, xerbla
238 * ..
239 * .. External Functions ..
240  INTEGER ilaenv
241  EXTERNAL ilaenv
242 * ..
243 * .. Intrinsic Functions ..
244  INTRINSIC int, max, min
245 * ..
246 * .. Executable Statements ..
247 *
248 * Test the input parameters
249 *
250  info = 0
251  nb1 = ilaenv( 1, 'CGERQF', ' ', m, n, -1, -1 )
252  nb2 = ilaenv( 1, 'CGEQRF', ' ', p, n, -1, -1 )
253  nb3 = ilaenv( 1, 'CUNMRQ', ' ', m, n, p, -1 )
254  nb = max( nb1, nb2, nb3 )
255  lwkopt = max( n, m, p)*nb
256  work( 1 ) = lwkopt
257  lquery = ( lwork.EQ.-1 )
258  IF( m.LT.0 ) THEN
259  info = -1
260  ELSE IF( p.LT.0 ) THEN
261  info = -2
262  ELSE IF( n.LT.0 ) THEN
263  info = -3
264  ELSE IF( lda.LT.max( 1, m ) ) THEN
265  info = -5
266  ELSE IF( ldb.LT.max( 1, p ) ) THEN
267  info = -8
268  ELSE IF( lwork.LT.max( 1, m, p, n ) .AND. .NOT.lquery ) THEN
269  info = -11
270  END IF
271  IF( info.NE.0 ) THEN
272  CALL xerbla( 'CGGRQF', -info )
273  RETURN
274  ELSE IF( lquery ) THEN
275  RETURN
276  END IF
277 *
278 * RQ factorization of M-by-N matrix A: A = R*Q
279 *
280  CALL cgerqf( m, n, a, lda, taua, work, lwork, info )
281  lopt = work( 1 )
282 *
283 * Update B := B*Q**H
284 *
285  CALL cunmrq( 'Right', 'Conjugate Transpose', p, n, min( m, n ),
286  \$ a( max( 1, m-n+1 ), 1 ), lda, taua, b, ldb, work,
287  \$ lwork, info )
288  lopt = max( lopt, int( work( 1 ) ) )
289 *
290 * QR factorization of P-by-N matrix B: B = Z*T
291 *
292  CALL cgeqrf( p, n, b, ldb, taub, work, lwork, info )
293  work( 1 ) = max( lopt, int( work( 1 ) ) )
294 *
295  RETURN
296 *
297 * End of CGGRQF
298 *
subroutine cunmrq(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMRQ
Definition: cunmrq.f:170
subroutine cgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
CGEQRF
Definition: cgeqrf.f:138
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: tstiee.f:83
subroutine cgerqf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
CGERQF
Definition: cgerqf.f:140
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