LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ cggqrf()

 subroutine cggqrf ( integer n, integer m, integer p, 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 )

CGGQRF

Purpose:
``` CGGQRF computes a generalized QR factorization of an N-by-M matrix A
and an N-by-P matrix B:

A = Q*R,        B = Q*T*Z,

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 N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
(  0  ) N-M                         N   M-N
M

where R11 is upper triangular, and

if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
P-N  N                           ( T21 ) P
P

where T12 or T21 is upper triangular.

In particular, if B is square and nonsingular, the GQR factorization
of A and B implicitly gives the QR factorization of inv(B)*A:

inv(B)*A = Z**H * (inv(T)*R)

where inv(B) denotes the inverse of the matrix B, and Z' denotes the
conjugate transpose of matrix Z.```
Parameters
 [in] N ``` N is INTEGER The number of rows of the matrices A and B. N >= 0.``` [in] M ``` M is INTEGER The number of columns of the matrix A. M >= 0.``` [in] P ``` P is INTEGER The number of columns of the matrix B. P >= 0.``` [in,out] A ``` A is COMPLEX array, dimension (LDA,M) On entry, the N-by-M matrix A. On exit, the elements on and above the diagonal of the array contain the min(N,M)-by-M upper trapezoidal matrix R (R is upper triangular if N >= M); the elements below the diagonal, with the array TAUA, represent the unitary matrix Q as a product of min(N,M) elementary reflectors (see Further Details).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] TAUA ``` TAUA is COMPLEX array, dimension (min(N,M)) 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,P) On entry, the N-by-P matrix B. On exit, if N <= P, the upper triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; if N > P, the elements on and above the (N-P)-th subdiagonal contain the N-by-P upper trapezoidal matrix T; the remaining elements, 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,N).``` [out] TAUB ``` TAUB is COMPLEX array, dimension (min(N,P)) 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 QR factorization of an N-by-M matrix, NB2 is the optimal blocksize for the RQ factorization of an N-by-P matrix, and NB3 is the optimal blocksize for a call of CUNMQR. 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(n,m).

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(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine CUNGQR.
To use Q to update another matrix, use LAPACK subroutine CUNMQR.

The matrix Z is represented as a product of elementary reflectors

Z = H(1) H(2) . . . H(k), where k = min(n,p).

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(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine CUNGRQ.
To use Z to update another matrix, use LAPACK subroutine CUNMRQ.```

Definition at line 213 of file cggqrf.f.

215*
216* -- LAPACK computational routine --
217* -- LAPACK is a software package provided by Univ. of Tennessee, --
218* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
219*
220* .. Scalar Arguments ..
221 INTEGER INFO, LDA, LDB, LWORK, M, N, P
222* ..
223* .. Array Arguments ..
224 COMPLEX A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
225 \$ WORK( * )
226* ..
227*
228* =====================================================================
229*
230* .. Local Scalars ..
231 LOGICAL LQUERY
232 INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3
233* ..
234* .. External Subroutines ..
235 EXTERNAL cgeqrf, cgerqf, cunmqr, xerbla
236* ..
237* .. External Functions ..
238 INTEGER ILAENV
239 REAL SROUNDUP_LWORK
240 EXTERNAL ilaenv, sroundup_lwork
241* ..
242* .. Intrinsic Functions ..
243 INTRINSIC int, max, min
244* ..
245* .. Executable Statements ..
246*
247* Test the input parameters
248*
249 info = 0
250 nb1 = ilaenv( 1, 'CGEQRF', ' ', n, m, -1, -1 )
251 nb2 = ilaenv( 1, 'CGERQF', ' ', n, p, -1, -1 )
252 nb3 = ilaenv( 1, 'CUNMQR', ' ', n, m, p, -1 )
253 nb = max( nb1, nb2, nb3 )
254 lwkopt = max( n, m, p)*nb
255 work( 1 ) = sroundup_lwork(lwkopt)
256 lquery = ( lwork.EQ.-1 )
257 IF( n.LT.0 ) THEN
258 info = -1
259 ELSE IF( m.LT.0 ) THEN
260 info = -2
261 ELSE IF( p.LT.0 ) THEN
262 info = -3
263 ELSE IF( lda.LT.max( 1, n ) ) THEN
264 info = -5
265 ELSE IF( ldb.LT.max( 1, n ) ) THEN
266 info = -8
267 ELSE IF( lwork.LT.max( 1, n, m, p ) .AND. .NOT.lquery ) THEN
268 info = -11
269 END IF
270 IF( info.NE.0 ) THEN
271 CALL xerbla( 'CGGQRF', -info )
272 RETURN
273 ELSE IF( lquery ) THEN
274 RETURN
275 END IF
276*
277* QR factorization of N-by-M matrix A: A = Q*R
278*
279 CALL cgeqrf( n, m, a, lda, taua, work, lwork, info )
280 lopt = int( work( 1 ) )
281*
282* Update B := Q**H*B.
283*
284 CALL cunmqr( 'Left', 'Conjugate Transpose', n, p, min( n, m ), a,
285 \$ lda, taua, b, ldb, work, lwork, info )
286 lopt = max( lopt, int( work( 1 ) ) )
287*
288* RQ factorization of N-by-P matrix B: B = T*Z.
289*
290 CALL cgerqf( n, p, b, ldb, taub, work, lwork, info )
291 work( 1 ) = max( lopt, int( work( 1 ) ) )
292*
293 RETURN
294*
295* End of CGGQRF
296*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
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
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
real function sroundup_lwork(lwork)
SROUNDUP_LWORK
subroutine cunmqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
CUNMQR
Definition cunmqr.f:168
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