LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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cggrqf.f
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1*> \brief \b CGGRQF
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cggrqf.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
22* LWORK, INFO )
23*
24* .. Scalar Arguments ..
25* INTEGER INFO, LDA, LDB, LWORK, M, N, P
26* ..
27* .. Array Arguments ..
28* COMPLEX A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
29* \$ WORK( * )
30* ..
31*
32*
33*> \par Purpose:
34* =============
35*>
36*> \verbatim
37*>
38*> CGGRQF computes a generalized RQ factorization of an M-by-N matrix A
39*> and a P-by-N matrix B:
40*>
41*> A = R*Q, B = Z*T*Q,
42*>
43*> where Q is an N-by-N unitary matrix, Z is a P-by-P unitary
44*> matrix, and R and T assume one of the forms:
45*>
46*> if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
47*> N-M M ( R21 ) N
48*> N
49*>
50*> where R12 or R21 is upper triangular, and
51*>
52*> if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
53*> ( 0 ) P-N P N-P
54*> N
55*>
56*> where T11 is upper triangular.
57*>
58*> In particular, if B is square and nonsingular, the GRQ factorization
59*> of A and B implicitly gives the RQ factorization of A*inv(B):
60*>
61*> A*inv(B) = (R*inv(T))*Z**H
62*>
63*> where inv(B) denotes the inverse of the matrix B, and Z**H denotes the
64*> conjugate transpose of the matrix Z.
65*> \endverbatim
66*
67* Arguments:
68* ==========
69*
70*> \param[in] M
71*> \verbatim
72*> M is INTEGER
73*> The number of rows of the matrix A. M >= 0.
74*> \endverbatim
75*>
76*> \param[in] P
77*> \verbatim
78*> P is INTEGER
79*> The number of rows of the matrix B. P >= 0.
80*> \endverbatim
81*>
82*> \param[in] N
83*> \verbatim
84*> N is INTEGER
85*> The number of columns of the matrices A and B. N >= 0.
86*> \endverbatim
87*>
88*> \param[in,out] A
89*> \verbatim
90*> A is COMPLEX array, dimension (LDA,N)
91*> On entry, the M-by-N matrix A.
92*> On exit, if M <= N, the upper triangle of the subarray
93*> A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
94*> if M > N, the elements on and above the (M-N)-th subdiagonal
95*> contain the M-by-N upper trapezoidal matrix R; the remaining
96*> elements, with the array TAUA, represent the unitary
97*> matrix Q as a product of elementary reflectors (see Further
98*> Details).
99*> \endverbatim
100*>
101*> \param[in] LDA
102*> \verbatim
103*> LDA is INTEGER
104*> The leading dimension of the array A. LDA >= max(1,M).
105*> \endverbatim
106*>
107*> \param[out] TAUA
108*> \verbatim
109*> TAUA is COMPLEX array, dimension (min(M,N))
110*> The scalar factors of the elementary reflectors which
111*> represent the unitary matrix Q (see Further Details).
112*> \endverbatim
113*>
114*> \param[in,out] B
115*> \verbatim
116*> B is COMPLEX array, dimension (LDB,N)
117*> On entry, the P-by-N matrix B.
118*> On exit, the elements on and above the diagonal of the array
119*> contain the min(P,N)-by-N upper trapezoidal matrix T (T is
120*> upper triangular if P >= N); the elements below the diagonal,
121*> with the array TAUB, represent the unitary matrix Z as a
122*> product of elementary reflectors (see Further Details).
123*> \endverbatim
124*>
125*> \param[in] LDB
126*> \verbatim
127*> LDB is INTEGER
128*> The leading dimension of the array B. LDB >= max(1,P).
129*> \endverbatim
130*>
131*> \param[out] TAUB
132*> \verbatim
133*> TAUB is COMPLEX array, dimension (min(P,N))
134*> The scalar factors of the elementary reflectors which
135*> represent the unitary matrix Z (see Further Details).
136*> \endverbatim
137*>
138*> \param[out] WORK
139*> \verbatim
140*> WORK is COMPLEX array, dimension (MAX(1,LWORK))
141*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
142*> \endverbatim
143*>
144*> \param[in] LWORK
145*> \verbatim
146*> LWORK is INTEGER
147*> The dimension of the array WORK. LWORK >= max(1,N,M,P).
148*> For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
149*> where NB1 is the optimal blocksize for the RQ factorization
150*> of an M-by-N matrix, NB2 is the optimal blocksize for the
151*> QR factorization of a P-by-N matrix, and NB3 is the optimal
152*> blocksize for a call of CUNMRQ.
153*>
154*> If LWORK = -1, then a workspace query is assumed; the routine
155*> only calculates the optimal size of the WORK array, returns
156*> this value as the first entry of the WORK array, and no error
157*> message related to LWORK is issued by XERBLA.
158*> \endverbatim
159*>
160*> \param[out] INFO
161*> \verbatim
162*> INFO is INTEGER
163*> = 0: successful exit
164*> < 0: if INFO=-i, the i-th argument had an illegal value.
165*> \endverbatim
166*
167* Authors:
168* ========
169*
170*> \author Univ. of Tennessee
171*> \author Univ. of California Berkeley
172*> \author Univ. of Colorado Denver
173*> \author NAG Ltd.
174*
175*> \ingroup ggrqf
176*
177*> \par Further Details:
178* =====================
179*>
180*> \verbatim
181*>
182*> The matrix Q is represented as a product of elementary reflectors
183*>
184*> Q = H(1) H(2) . . . H(k), where k = min(m,n).
185*>
186*> Each H(i) has the form
187*>
188*> H(i) = I - taua * v * v**H
189*>
190*> where taua is a complex scalar, and v is a complex vector with
191*> v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
192*> A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
193*> To form Q explicitly, use LAPACK subroutine CUNGRQ.
194*> To use Q to update another matrix, use LAPACK subroutine CUNMRQ.
195*>
196*> The matrix Z is represented as a product of elementary reflectors
197*>
198*> Z = H(1) H(2) . . . H(k), where k = min(p,n).
199*>
200*> Each H(i) has the form
201*>
202*> H(i) = I - taub * v * v**H
203*>
204*> where taub is a complex scalar, and v is a complex vector with
205*> v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
206*> and taub in TAUB(i).
207*> To form Z explicitly, use LAPACK subroutine CUNGQR.
208*> To use Z to update another matrix, use LAPACK subroutine CUNMQR.
209*> \endverbatim
210*>
211* =====================================================================
212 SUBROUTINE cggrqf( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
213 \$ LWORK, INFO )
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 REAL SROUNDUP_LWORK
239 EXTERNAL ilaenv, sroundup_lwork
240* ..
241* .. Intrinsic Functions ..
242 INTRINSIC int, max, min
243* ..
244* .. Executable Statements ..
245*
246* Test the input parameters
247*
248 info = 0
249 nb1 = ilaenv( 1, 'CGERQF', ' ', m, n, -1, -1 )
250 nb2 = ilaenv( 1, 'CGEQRF', ' ', p, n, -1, -1 )
251 nb3 = ilaenv( 1, 'CUNMRQ', ' ', m, n, p, -1 )
252 nb = max( nb1, nb2, nb3 )
253 lwkopt = max( n, m, p)*nb
254 work( 1 ) = sroundup_lwork(lwkopt)
255 lquery = ( lwork.EQ.-1 )
256 IF( m.LT.0 ) THEN
257 info = -1
258 ELSE IF( p.LT.0 ) THEN
259 info = -2
260 ELSE IF( n.LT.0 ) THEN
261 info = -3
262 ELSE IF( lda.LT.max( 1, m ) ) THEN
263 info = -5
264 ELSE IF( ldb.LT.max( 1, p ) ) THEN
265 info = -8
266 ELSE IF( lwork.LT.max( 1, m, p, n ) .AND. .NOT.lquery ) THEN
267 info = -11
268 END IF
269 IF( info.NE.0 ) THEN
270 CALL xerbla( 'CGGRQF', -info )
271 RETURN
272 ELSE IF( lquery ) THEN
273 RETURN
274 END IF
275*
276* RQ factorization of M-by-N matrix A: A = R*Q
277*
278 CALL cgerqf( m, n, a, lda, taua, work, lwork, info )
279 lopt = int( work( 1 ) )
280*
281* Update B := B*Q**H
282*
283 CALL cunmrq( 'Right', 'Conjugate Transpose', p, n, min( m, n ),
284 \$ a( max( 1, m-n+1 ), 1 ), lda, taua, b, ldb, work,
285 \$ lwork, info )
286 lopt = max( lopt, int( work( 1 ) ) )
287*
288* QR factorization of P-by-N matrix B: B = Z*T
289*
290 CALL cgeqrf( p, n, b, ldb, taub, work, lwork, info )
291 work( 1 ) = max( lopt, int( work( 1 ) ) )
292*
293 RETURN
294*
295* End of CGGRQF
296*
297 END
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
subroutine cggrqf(m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info)
CGGRQF
Definition cggrqf.f:214
subroutine cunmrq(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
CUNMRQ
Definition cunmrq.f:168