LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ 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

Download CGGRQF + dependencies [TGZ] [ZIP] [TXT]

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.
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 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 212 of file cggrqf.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 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 cgeqrf, cgerqf, cunmrq, xerbla
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, 'CGERQF', ' ', m, n, -1, -1 )
249 nb2 = ilaenv( 1, 'CGEQRF', ' ', p, n, -1, -1 )
250 nb3 = ilaenv( 1, 'CUNMRQ', ' ', 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( 'CGGRQF', -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 cgerqf( m, n, a, lda, taua, work, lwork, info )
278 lopt = int( work( 1 ) )
279*
280* Update B := B*Q**H
281*
282 CALL cunmrq( '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 cgeqrf( p, n, b, ldb, taub, work, lwork, info )
290 work( 1 ) = max( lopt, int( work( 1 ) ) )
291*
292 RETURN
293*
294* End of CGGRQF
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 cgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
CGEQRF
Definition: cgeqrf.f:146
subroutine cgerqf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
CGERQF
Definition: cgerqf.f:139
subroutine cunmrq(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMRQ
Definition: cunmrq.f:168
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