 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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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

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.```
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 212 of file zggrqf.f.

214*
215* -- LAPACK computational routine --
216* -- LAPACK is a software package provided by Univ. of Tennessee, --
217* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
218*
219* .. Scalar Arguments ..
220 INTEGER INFO, LDA, LDB, LWORK, M, N, P
221* ..
222* .. Array Arguments ..
223 COMPLEX*16 A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
224 \$ WORK( * )
225* ..
226*
227* =====================================================================
228*
229* .. Local Scalars ..
230 LOGICAL LQUERY
231 INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3
232* ..
233* .. External Subroutines ..
234 EXTERNAL xerbla, zgeqrf, zgerqf, zunmrq
235* ..
236* .. External Functions ..
237 INTEGER ILAENV
238 EXTERNAL ilaenv
239* ..
240* .. Intrinsic Functions ..
241 INTRINSIC int, max, min
242* ..
243* .. Executable Statements ..
244*
245* Test the input parameters
246*
247 info = 0
248 nb1 = ilaenv( 1, 'ZGERQF', ' ', m, n, -1, -1 )
249 nb2 = ilaenv( 1, 'ZGEQRF', ' ', p, n, -1, -1 )
250 nb3 = ilaenv( 1, 'ZUNMRQ', ' ', m, n, p, -1 )
251 nb = max( nb1, nb2, nb3 )
252 lwkopt = max( n, m, p )*nb
253 work( 1 ) = lwkopt
254 lquery = ( lwork.EQ.-1 )
255 IF( m.LT.0 ) THEN
256 info = -1
257 ELSE IF( p.LT.0 ) THEN
258 info = -2
259 ELSE IF( n.LT.0 ) THEN
260 info = -3
261 ELSE IF( lda.LT.max( 1, m ) ) THEN
262 info = -5
263 ELSE IF( ldb.LT.max( 1, p ) ) THEN
264 info = -8
265 ELSE IF( lwork.LT.max( 1, m, p, n ) .AND. .NOT.lquery ) THEN
266 info = -11
267 END IF
268 IF( info.NE.0 ) THEN
269 CALL xerbla( 'ZGGRQF', -info )
270 RETURN
271 ELSE IF( lquery ) THEN
272 RETURN
273 END IF
274*
275* RQ factorization of M-by-N matrix A: A = R*Q
276*
277 CALL zgerqf( m, n, a, lda, taua, work, lwork, info )
278 lopt = int( work( 1 ) )
279*
280* Update B := B*Q**H
281*
282 CALL zunmrq( 'Right', 'Conjugate Transpose', p, n, min( m, n ),
283 \$ a( max( 1, m-n+1 ), 1 ), lda, taua, b, ldb, work,
284 \$ lwork, info )
285 lopt = max( lopt, int( work( 1 ) ) )
286*
287* QR factorization of P-by-N matrix B: B = Z*T
288*
289 CALL zgeqrf( p, n, b, ldb, taub, work, lwork, info )
290 work( 1 ) = max( lopt, int( work( 1 ) ) )
291*
292 RETURN
293*
294* End of ZGGRQF
295*
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zgerqf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
ZGERQF
Definition: zgerqf.f:139
subroutine zunmrq(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
ZUNMRQ
Definition: zunmrq.f:167
subroutine zgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
ZGEQRF VARIANT: left-looking Level 3 BLAS of the algorithm.
Definition: zgeqrf.f:152
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