LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
ztplqt2.f
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1 *> \brief \b ZTPLQT2 computes a LQ 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 *
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZTPLQT2( 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 *> ZTPLQT2 computes a LQ a 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 lower 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,M)
69 *> On entry, the lower triangular M-by-M matrix A.
70 *> On exit, the elements on and below the diagonal of the array
71 *> contain the lower triangular matrix L.
72 *> \endverbatim
73 *>
74 *> \param[in] LDA
75 *> \verbatim
76 *> LDA is INTEGER
77 *> The leading dimension of the array A. LDA >= max(1,M).
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 N-L columns
84 *> are rectangular, and the last L columns are lower 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,M)
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,M)
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 *> \date June 2017
123 *
124 *> \ingroup doubleOTHERcomputational
125 *
126 *> \par Further Details:
127 * =====================
128 *>
129 *> \verbatim
130 *>
131 *> The input matrix C is a M-by-(M+N) matrix
132 *>
133 *> C = [ A ][ B ]
134 *>
135 *>
136 *> where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
137 *> matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
138 *> upper trapezoidal matrix B2:
139 *>
140 *> B = [ B1 ][ B2 ]
141 *> [ B1 ] <- M-by-(N-L) rectangular
142 *> [ B2 ] <- M-by-L lower trapezoidal.
143 *>
144 *> The lower trapezoidal matrix B2 consists of the first L columns of a
145 *> N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
146 *> B is rectangular M-by-N; if M=L=N, B is lower triangular.
147 *>
148 *> The matrix W stores the elementary reflectors H(i) in the i-th row
149 *> above the diagonal (of A) in the M-by-(M+N) input matrix C
150 *>
151 *> C = [ A ][ B ]
152 *> [ A ] <- lower triangular M-by-M
153 *> [ B ] <- M-by-N pentagonal
154 *>
155 *> so that W can be represented as
156 *>
157 *> W = [ I ][ V ]
158 *> [ I ] <- identity, M-by-M
159 *> [ V ] <- M-by-N, same form as B.
160 *>
161 *> Thus, all of information needed for W is contained on exit in B, which
162 *> we call V above. Note that V has the same form as B; that is,
163 *>
164 *> W = [ V1 ][ V2 ]
165 *> [ V1 ] <- M-by-(N-L) rectangular
166 *> [ V2 ] <- M-by-L lower trapezoidal.
167 *>
168 *> The rows of V represent the vectors which define the H(i)'s.
169 *> The (M+N)-by-(M+N) block reflector H is then given by
170 *>
171 *> H = I - W**T * T * W
172 *>
173 *> where W^H is the conjugate transpose of W and T is the upper triangular
174 *> factor of the block reflector.
175 *> \endverbatim
176 *>
177 * =====================================================================
178  SUBROUTINE ztplqt2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
179 *
180 * -- LAPACK computational routine (version 3.7.1) --
181 * -- LAPACK is a software package provided by Univ. of Tennessee, --
182 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
183 * June 2017
184 *
185 * .. Scalar Arguments ..
186  INTEGER INFO, LDA, LDB, LDT, N, M, L
187 * ..
188 * .. Array Arguments ..
189  COMPLEX*16 A( lda, * ), B( ldb, * ), T( ldt, * )
190 * ..
191 *
192 * =====================================================================
193 *
194 * .. Parameters ..
195  COMPLEX*16 ONE, ZERO
196  parameter( zero = ( 0.0d+0, 0.0d+0 ),one = ( 1.0d+0, 0.0d+0 ) )
197 * ..
198 * .. Local Scalars ..
199  INTEGER I, J, P, MP, NP
200  COMPLEX*16 ALPHA
201 * ..
202 * .. External Subroutines ..
203  EXTERNAL zlarfg, zgemv, zgerc, ztrmv, xerbla
204 * ..
205 * .. Intrinsic Functions ..
206  INTRINSIC max, min
207 * ..
208 * .. Executable Statements ..
209 *
210 * Test the input arguments
211 *
212  info = 0
213  IF( m.LT.0 ) THEN
214  info = -1
215  ELSE IF( n.LT.0 ) THEN
216  info = -2
217  ELSE IF( l.LT.0 .OR. l.GT.min(m,n) ) THEN
218  info = -3
219  ELSE IF( lda.LT.max( 1, m ) ) THEN
220  info = -5
221  ELSE IF( ldb.LT.max( 1, m ) ) THEN
222  info = -7
223  ELSE IF( ldt.LT.max( 1, m ) ) THEN
224  info = -9
225  END IF
226  IF( info.NE.0 ) THEN
227  CALL xerbla( 'ZTPLQT2', -info )
228  RETURN
229  END IF
230 *
231 * Quick return if possible
232 *
233  IF( n.EQ.0 .OR. m.EQ.0 ) RETURN
234 *
235  DO i = 1, m
236 *
237 * Generate elementary reflector H(I) to annihilate B(I,:)
238 *
239  p = n-l+min( l, i )
240  CALL zlarfg( p+1, a( i, i ), b( i, 1 ), ldb, t( 1, i ) )
241  t(1,i)=conjg(t(1,i))
242  IF( i.LT.m ) THEN
243  DO j = 1, p
244  b( i, j ) = conjg(b(i,j))
245  END DO
246 *
247 * W(M-I:1) := C(I+1:M,I:N) * C(I,I:N) [use W = T(M,:)]
248 *
249  DO j = 1, m-i
250  t( m, j ) = (a( i+j, i ))
251  END DO
252  CALL zgemv( 'N', m-i, p, one, b( i+1, 1 ), ldb,
253  $ b( i, 1 ), ldb, one, t( m, 1 ), ldt )
254 *
255 * C(I+1:M,I:N) = C(I+1:M,I:N) + alpha * C(I,I:N)*W(M-1:1)^H
256 *
257  alpha = -(t( 1, i ))
258  DO j = 1, m-i
259  a( i+j, i ) = a( i+j, i ) + alpha*(t( m, j ))
260  END DO
261  CALL zgerc( m-i, p, (alpha), t( m, 1 ), ldt,
262  $ b( i, 1 ), ldb, b( i+1, 1 ), ldb )
263  DO j = 1, p
264  b( i, j ) = conjg(b(i,j))
265  END DO
266  END IF
267  END DO
268 *
269  DO i = 2, m
270 *
271 * T(I,1:I-1) := C(I:I-1,1:N)**H * (alpha * C(I,I:N))
272 *
273  alpha = -(t( 1, i ))
274  DO j = 1, i-1
275  t( i, j ) = zero
276  END DO
277  p = min( i-1, l )
278  np = min( n-l+1, n )
279  mp = min( p+1, m )
280  DO j = 1, n-l+p
281  b(i,j)=conjg(b(i,j))
282  END DO
283 *
284 * Triangular part of B2
285 *
286  DO j = 1, p
287  t( i, j ) = (alpha*b( i, n-l+j ))
288  END DO
289  CALL ztrmv( 'L', 'N', 'N', p, b( 1, np ), ldb,
290  $ t( i, 1 ), ldt )
291 *
292 * Rectangular part of B2
293 *
294  CALL zgemv( 'N', i-1-p, l, alpha, b( mp, np ), ldb,
295  $ b( i, np ), ldb, zero, t( i,mp ), ldt )
296 *
297 * B1
298 
299 *
300  CALL zgemv( 'N', i-1, n-l, alpha, b, ldb, b( i, 1 ), ldb,
301  $ one, t( i, 1 ), ldt )
302 *
303 
304 *
305 * T(1:I-1,I) := T(1:I-1,1:I-1) * T(I,1:I-1)
306 *
307  DO j = 1, i-1
308  t(i,j)=conjg(t(i,j))
309  END DO
310  CALL ztrmv( 'L', 'C', 'N', i-1, t, ldt, t( i, 1 ), ldt )
311  DO j = 1, i-1
312  t(i,j)=conjg(t(i,j))
313  END DO
314  DO j = 1, n-l+p
315  b(i,j)=conjg(b(i,j))
316  END DO
317 *
318 * T(I,I) = tau(I)
319 *
320  t( i, i ) = t( 1, i )
321  t( 1, i ) = zero
322  END DO
323  DO i=1,m
324  DO j= i+1,m
325  t(i,j)=(t(j,i))
326  t(j,i)=zero
327  END DO
328  END DO
329 
330 *
331 * End of ZTPLQT2
332 *
333  END
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:160
subroutine zgerc(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
ZGERC
Definition: zgerc.f:132
subroutine ztplqt2(M, N, L, A, LDA, B, LDB, T, LDT, INFO)
ZTPLQT2 computes a LQ 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.
Definition: ztplqt2.f:179
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine ztrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
ZTRMV
Definition: ztrmv.f:149
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:108