LAPACK 3.12.1
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

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.
!> 

Definition at line 210 of file zggrqf.f.

212*
213* -- LAPACK computational routine --
214* -- LAPACK is a software package provided by Univ. of Tennessee, --
215* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
216*
217* .. Scalar Arguments ..
218 INTEGER INFO, LDA, LDB, LWORK, M, N, P
219* ..
220* .. Array Arguments ..
221 COMPLEX*16 A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
222 $ WORK( * )
223* ..
224*
225* =====================================================================
226*
227* .. Local Scalars ..
228 LOGICAL LQUERY
229 INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3
230* ..
231* .. External Subroutines ..
232 EXTERNAL xerbla, zgeqrf, zgerqf, zunmrq
233* ..
234* .. External Functions ..
235 INTEGER ILAENV
236 EXTERNAL ilaenv
237* ..
238* .. Intrinsic Functions ..
239 INTRINSIC int, max, min
240* ..
241* .. Executable Statements ..
242*
243* Test the input parameters
244*
245 info = 0
246 nb1 = ilaenv( 1, 'ZGERQF', ' ', m, n, -1, -1 )
247 nb2 = ilaenv( 1, 'ZGEQRF', ' ', p, n, -1, -1 )
248 nb3 = ilaenv( 1, 'ZUNMRQ', ' ', m, n, p, -1 )
249 nb = max( nb1, nb2, nb3 )
250 lwkopt = max( 1, max( n, m, p )*nb )
251 work( 1 ) = lwkopt
252 lquery = ( lwork.EQ.-1 )
253 IF( m.LT.0 ) THEN
254 info = -1
255 ELSE IF( p.LT.0 ) THEN
256 info = -2
257 ELSE IF( n.LT.0 ) THEN
258 info = -3
259 ELSE IF( lda.LT.max( 1, m ) ) THEN
260 info = -5
261 ELSE IF( ldb.LT.max( 1, p ) ) THEN
262 info = -8
263 ELSE IF( lwork.LT.max( 1, m, p, n ) .AND. .NOT.lquery ) THEN
264 info = -11
265 END IF
266 IF( info.NE.0 ) THEN
267 CALL xerbla( 'ZGGRQF', -info )
268 RETURN
269 ELSE IF( lquery ) THEN
270 RETURN
271 END IF
272*
273* RQ factorization of M-by-N matrix A: A = R*Q
274*
275 CALL zgerqf( m, n, a, lda, taua, work, lwork, info )
276 lopt = int( work( 1 ) )
277*
278* Update B := B*Q**H
279*
280 CALL zunmrq( 'Right', 'Conjugate Transpose', p, n, min( m, n ),
281 $ a( max( 1, m-n+1 ), 1 ), lda, taua, b, ldb, work,
282 $ lwork, info )
283 lopt = max( lopt, int( work( 1 ) ) )
284*
285* QR factorization of P-by-N matrix B: B = Z*T
286*
287 CALL zgeqrf( p, n, b, ldb, taub, work, lwork, info )
288 work( 1 ) = max( lopt, int( work( 1 ) ) )
289*
290 RETURN
291*
292* End of ZGGRQF
293*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zgeqrf(m, n, a, lda, tau, work, lwork, info)
ZGEQRF
Definition zgeqrf.f:144
subroutine zgerqf(m, n, a, lda, tau, work, lwork, info)
ZGERQF
Definition zgerqf.f:137
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:160
subroutine zunmrq(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
ZUNMRQ
Definition zunmrq.f:165
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