LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
ztpqrt2.f
Go to the documentation of this file.
1 *> \brief \b ZTPQRT2 computes a QR factorization of a real or complex "triangular-pentagonal" matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download ZTPQRT2 + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ztpqrt2.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ztpqrt2.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ztpqrt2.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZTPQRT2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, LDA, LDB, LDT, N, M, L
25 * ..
26 * .. Array Arguments ..
27 * COMPLEX*16 A( LDA, * ), B( LDB, * ), T( LDT, * )
28 * ..
29 *
30 *
31 *> \par Purpose:
32 * =============
33 *>
34 *> \verbatim
35 *>
36 *> ZTPQRT2 computes a QR factorization of a complex "triangular-pentagonal"
37 *> matrix C, which is composed of a triangular block A and pentagonal block B,
38 *> using the compact WY representation for Q.
39 *> \endverbatim
40 *
41 * Arguments:
42 * ==========
43 *
44 *> \param[in] M
45 *> \verbatim
46 *> M is INTEGER
47 *> The total number of rows of the matrix B.
48 *> M >= 0.
49 *> \endverbatim
50 *>
51 *> \param[in] N
52 *> \verbatim
53 *> N is INTEGER
54 *> The number of columns of the matrix B, and the order of
55 *> the triangular matrix A.
56 *> N >= 0.
57 *> \endverbatim
58 *>
59 *> \param[in] L
60 *> \verbatim
61 *> L is INTEGER
62 *> The number of rows of the upper trapezoidal part of B.
63 *> MIN(M,N) >= L >= 0. See Further Details.
64 *> \endverbatim
65 *>
66 *> \param[in,out] A
67 *> \verbatim
68 *> A is COMPLEX*16 array, dimension (LDA,N)
69 *> On entry, the upper triangular N-by-N matrix A.
70 *> On exit, the elements on and above the diagonal of the array
71 *> contain the upper triangular matrix R.
72 *> \endverbatim
73 *>
74 *> \param[in] LDA
75 *> \verbatim
76 *> LDA is INTEGER
77 *> The leading dimension of the array A. LDA >= max(1,N).
78 *> \endverbatim
79 *>
80 *> \param[in,out] B
81 *> \verbatim
82 *> B is COMPLEX*16 array, dimension (LDB,N)
83 *> On entry, the pentagonal M-by-N matrix B. The first M-L rows
84 *> are rectangular, and the last L rows are upper trapezoidal.
85 *> On exit, B contains the pentagonal matrix V. See Further Details.
86 *> \endverbatim
87 *>
88 *> \param[in] LDB
89 *> \verbatim
90 *> LDB is INTEGER
91 *> The leading dimension of the array B. LDB >= max(1,M).
92 *> \endverbatim
93 *>
94 *> \param[out] T
95 *> \verbatim
96 *> T is COMPLEX*16 array, dimension (LDT,N)
97 *> The N-by-N upper triangular factor T of the block reflector.
98 *> See Further Details.
99 *> \endverbatim
100 *>
101 *> \param[in] LDT
102 *> \verbatim
103 *> LDT is INTEGER
104 *> The leading dimension of the array T. LDT >= max(1,N)
105 *> \endverbatim
106 *>
107 *> \param[out] INFO
108 *> \verbatim
109 *> INFO is INTEGER
110 *> = 0: successful exit
111 *> < 0: if INFO = -i, the i-th argument had an illegal value
112 *> \endverbatim
113 *
114 * Authors:
115 * ========
116 *
117 *> \author Univ. of Tennessee
118 *> \author Univ. of California Berkeley
119 *> \author Univ. of Colorado Denver
120 *> \author NAG Ltd.
121 *
122 *> \ingroup complex16OTHERcomputational
123 *
124 *> \par Further Details:
125 * =====================
126 *>
127 *> \verbatim
128 *>
129 *> The input matrix C is a (N+M)-by-N matrix
130 *>
131 *> C = [ A ]
132 *> [ B ]
133 *>
134 *> where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
135 *> matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
136 *> upper trapezoidal matrix B2:
137 *>
138 *> B = [ B1 ] <- (M-L)-by-N rectangular
139 *> [ B2 ] <- L-by-N upper trapezoidal.
140 *>
141 *> The upper trapezoidal matrix B2 consists of the first L rows of a
142 *> N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
143 *> B is rectangular M-by-N; if M=L=N, B is upper triangular.
144 *>
145 *> The matrix W stores the elementary reflectors H(i) in the i-th column
146 *> below the diagonal (of A) in the (N+M)-by-N input matrix C
147 *>
148 *> C = [ A ] <- upper triangular N-by-N
149 *> [ B ] <- M-by-N pentagonal
150 *>
151 *> so that W can be represented as
152 *>
153 *> W = [ I ] <- identity, N-by-N
154 *> [ V ] <- M-by-N, same form as B.
155 *>
156 *> Thus, all of information needed for W is contained on exit in B, which
157 *> we call V above. Note that V has the same form as B; that is,
158 *>
159 *> V = [ V1 ] <- (M-L)-by-N rectangular
160 *> [ V2 ] <- L-by-N upper trapezoidal.
161 *>
162 *> The columns of V represent the vectors which define the H(i)'s.
163 *> The (M+N)-by-(M+N) block reflector H is then given by
164 *>
165 *> H = I - W * T * W**H
166 *>
167 *> where W**H is the conjugate transpose of W and T is the upper triangular
168 *> factor of the block reflector.
169 *> \endverbatim
170 *>
171 * =====================================================================
172  SUBROUTINE ztpqrt2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
173 *
174 * -- LAPACK computational routine --
175 * -- LAPACK is a software package provided by Univ. of Tennessee, --
176 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
177 *
178 * .. Scalar Arguments ..
179  INTEGER INFO, LDA, LDB, LDT, N, M, L
180 * ..
181 * .. Array Arguments ..
182  COMPLEX*16 A( LDA, * ), B( LDB, * ), T( LDT, * )
183 * ..
184 *
185 * =====================================================================
186 *
187 * .. Parameters ..
188  COMPLEX*16 ONE, ZERO
189  parameter( one = (1.0,0.0), zero = (0.0,0.0) )
190 * ..
191 * .. Local Scalars ..
192  INTEGER I, J, P, MP, NP
193  COMPLEX*16 ALPHA
194 * ..
195 * .. External Subroutines ..
196  EXTERNAL zlarfg, zgemv, zgerc, ztrmv, xerbla
197 * ..
198 * .. Intrinsic Functions ..
199  INTRINSIC max, min
200 * ..
201 * .. Executable Statements ..
202 *
203 * Test the input arguments
204 *
205  info = 0
206  IF( m.LT.0 ) THEN
207  info = -1
208  ELSE IF( n.LT.0 ) THEN
209  info = -2
210  ELSE IF( l.LT.0 .OR. l.GT.min(m,n) ) THEN
211  info = -3
212  ELSE IF( lda.LT.max( 1, n ) ) THEN
213  info = -5
214  ELSE IF( ldb.LT.max( 1, m ) ) THEN
215  info = -7
216  ELSE IF( ldt.LT.max( 1, n ) ) THEN
217  info = -9
218  END IF
219  IF( info.NE.0 ) THEN
220  CALL xerbla( 'ZTPQRT2', -info )
221  RETURN
222  END IF
223 *
224 * Quick return if possible
225 *
226  IF( n.EQ.0 .OR. m.EQ.0 ) RETURN
227 *
228  DO i = 1, n
229 *
230 * Generate elementary reflector H(I) to annihilate B(:,I)
231 *
232  p = m-l+min( l, i )
233  CALL zlarfg( p+1, a( i, i ), b( 1, i ), 1, t( i, 1 ) )
234  IF( i.LT.n ) THEN
235 *
236 * W(1:N-I) := C(I:M,I+1:N)**H * C(I:M,I) [use W = T(:,N)]
237 *
238  DO j = 1, n-i
239  t( j, n ) = conjg(a( i, i+j ))
240  END DO
241  CALL zgemv( 'C', p, n-i, one, b( 1, i+1 ), ldb,
242  $ b( 1, i ), 1, one, t( 1, n ), 1 )
243 *
244 * C(I:M,I+1:N) = C(I:m,I+1:N) + alpha*C(I:M,I)*W(1:N-1)**H
245 *
246  alpha = -conjg(t( i, 1 ))
247  DO j = 1, n-i
248  a( i, i+j ) = a( i, i+j ) + alpha*conjg(t( j, n ))
249  END DO
250  CALL zgerc( p, n-i, alpha, b( 1, i ), 1,
251  $ t( 1, n ), 1, b( 1, i+1 ), ldb )
252  END IF
253  END DO
254 *
255  DO i = 2, n
256 *
257 * T(1:I-1,I) := C(I:M,1:I-1)**H * (alpha * C(I:M,I))
258 *
259  alpha = -t( i, 1 )
260 
261  DO j = 1, i-1
262  t( j, i ) = zero
263  END DO
264  p = min( i-1, l )
265  mp = min( m-l+1, m )
266  np = min( p+1, n )
267 *
268 * Triangular part of B2
269 *
270  DO j = 1, p
271  t( j, i ) = alpha*b( m-l+j, i )
272  END DO
273  CALL ztrmv( 'U', 'C', 'N', p, b( mp, 1 ), ldb,
274  $ t( 1, i ), 1 )
275 *
276 * Rectangular part of B2
277 *
278  CALL zgemv( 'C', l, i-1-p, alpha, b( mp, np ), ldb,
279  $ b( mp, i ), 1, zero, t( np, i ), 1 )
280 *
281 * B1
282 *
283  CALL zgemv( 'C', m-l, i-1, alpha, b, ldb, b( 1, i ), 1,
284  $ one, t( 1, i ), 1 )
285 *
286 * T(1:I-1,I) := T(1:I-1,1:I-1) * T(1:I-1,I)
287 *
288  CALL ztrmv( 'U', 'N', 'N', i-1, t, ldt, t( 1, i ), 1 )
289 *
290 * T(I,I) = tau(I)
291 *
292  t( i, i ) = t( i, 1 )
293  t( i, 1 ) = zero
294  END DO
295 
296 *
297 * End of ZTPQRT2
298 *
299  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine ztrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
ZTRMV
Definition: ztrmv.f:147
subroutine zgerc(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
ZGERC
Definition: zgerc.f:130
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:158
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:106
subroutine ztpqrt2(M, N, L, A, LDA, B, LDB, T, LDT, INFO)
ZTPQRT2 computes a QR factorization of a real or complex "triangular-pentagonal" matrix,...
Definition: ztpqrt2.f:173