LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
ctplqt2.f
Go to the documentation of this file.
1 *> \brief \b CTPLQT2
2 *
3 * Definition:
4 * ===========
5 *
6 * SUBROUTINE CTPLQT2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
7 *
8 * .. Scalar Arguments ..
9 * INTEGER INFO, LDA, LDB, LDT, N, M, L
10 * ..
11 * .. Array Arguments ..
12 * COMPLEX A( LDA, * ), B( LDB, * ), T( LDT, * )
13 * ..
14 *
15 *
16 *> \par Purpose:
17 * =============
18 *>
19 *> \verbatim
20 *>
21 *> CTPLQT2 computes a LQ a factorization of a complex "triangular-pentagonal"
22 *> matrix C, which is composed of a triangular block A and pentagonal block B,
23 *> using the compact WY representation for Q.
24 *> \endverbatim
25 *
26 * Arguments:
27 * ==========
28 *
29 *> \param[in] M
30 *> \verbatim
31 *> M is INTEGER
32 *> The total number of rows of the matrix B.
33 *> M >= 0.
34 *> \endverbatim
35 *>
36 *> \param[in] N
37 *> \verbatim
38 *> N is INTEGER
39 *> The number of columns of the matrix B, and the order of
40 *> the triangular matrix A.
41 *> N >= 0.
42 *> \endverbatim
43 *>
44 *> \param[in] L
45 *> \verbatim
46 *> L is INTEGER
47 *> The number of rows of the lower trapezoidal part of B.
48 *> MIN(M,N) >= L >= 0. See Further Details.
49 *> \endverbatim
50 *>
51 *> \param[in,out] A
52 *> \verbatim
53 *> A is COMPLEX array, dimension (LDA,M)
54 *> On entry, the lower triangular M-by-M matrix A.
55 *> On exit, the elements on and below the diagonal of the array
56 *> contain the lower triangular matrix L.
57 *> \endverbatim
58 *>
59 *> \param[in] LDA
60 *> \verbatim
61 *> LDA is INTEGER
62 *> The leading dimension of the array A. LDA >= max(1,M).
63 *> \endverbatim
64 *>
65 *> \param[in,out] B
66 *> \verbatim
67 *> B is COMPLEX array, dimension (LDB,N)
68 *> On entry, the pentagonal M-by-N matrix B. The first N-L columns
69 *> are rectangular, and the last L columns are lower trapezoidal.
70 *> On exit, B contains the pentagonal matrix V. See Further Details.
71 *> \endverbatim
72 *>
73 *> \param[in] LDB
74 *> \verbatim
75 *> LDB is INTEGER
76 *> The leading dimension of the array B. LDB >= max(1,M).
77 *> \endverbatim
78 *>
79 *> \param[out] T
80 *> \verbatim
81 *> T is COMPLEX array, dimension (LDT,M)
82 *> The N-by-N upper triangular factor T of the block reflector.
83 *> See Further Details.
84 *> \endverbatim
85 *>
86 *> \param[in] LDT
87 *> \verbatim
88 *> LDT is INTEGER
89 *> The leading dimension of the array T. LDT >= max(1,M)
90 *> \endverbatim
91 *>
92 *> \param[out] INFO
93 *> \verbatim
94 *> INFO is INTEGER
95 *> = 0: successful exit
96 *> < 0: if INFO = -i, the i-th argument had an illegal value
97 *> \endverbatim
98 *
99 * Authors:
100 * ========
101 *
102 *> \author Univ. of Tennessee
103 *> \author Univ. of California Berkeley
104 *> \author Univ. of Colorado Denver
105 *> \author NAG Ltd.
106 *
107 *> \ingroup doubleOTHERcomputational
108 *
109 *> \par Further Details:
110 * =====================
111 *>
112 *> \verbatim
113 *>
114 *> The input matrix C is a M-by-(M+N) matrix
115 *>
116 *> C = [ A ][ B ]
117 *>
118 *>
119 *> where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
120 *> matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
121 *> upper trapezoidal matrix B2:
122 *>
123 *> B = [ B1 ][ B2 ]
124 *> [ B1 ] <- M-by-(N-L) rectangular
125 *> [ B2 ] <- M-by-L lower trapezoidal.
126 *>
127 *> The lower trapezoidal matrix B2 consists of the first L columns of a
128 *> N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
129 *> B is rectangular M-by-N; if M=L=N, B is lower triangular.
130 *>
131 *> The matrix W stores the elementary reflectors H(i) in the i-th row
132 *> above the diagonal (of A) in the M-by-(M+N) input matrix C
133 *>
134 *> C = [ A ][ B ]
135 *> [ A ] <- lower triangular M-by-M
136 *> [ B ] <- M-by-N pentagonal
137 *>
138 *> so that W can be represented as
139 *>
140 *> W = [ I ][ V ]
141 *> [ I ] <- identity, M-by-M
142 *> [ V ] <- M-by-N, same form as B.
143 *>
144 *> Thus, all of information needed for W is contained on exit in B, which
145 *> we call V above. Note that V has the same form as B; that is,
146 *>
147 *> W = [ V1 ][ V2 ]
148 *> [ V1 ] <- M-by-(N-L) rectangular
149 *> [ V2 ] <- M-by-L lower trapezoidal.
150 *>
151 *> The rows of V represent the vectors which define the H(i)'s.
152 *> The (M+N)-by-(M+N) block reflector H is then given by
153 *>
154 *> H = I - W**T * T * W
155 *>
156 *> where W^H is the conjugate transpose of W and T is the upper triangular
157 *> factor of the block reflector.
158 *> \endverbatim
159 *>
160 * =====================================================================
161  SUBROUTINE ctplqt2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
162 *
163 * -- LAPACK computational routine --
164 * -- LAPACK is a software package provided by Univ. of Tennessee, --
165 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
166 *
167 * .. Scalar Arguments ..
168  INTEGER INFO, LDA, LDB, LDT, N, M, L
169 * ..
170 * .. Array Arguments ..
171  COMPLEX A( LDA, * ), B( LDB, * ), T( LDT, * )
172 * ..
173 *
174 * =====================================================================
175 *
176 * .. Parameters ..
177  COMPLEX ONE, ZERO
178  parameter( zero = ( 0.0e+0, 0.0e+0 ),one = ( 1.0e+0, 0.0e+0 ) )
179 * ..
180 * .. Local Scalars ..
181  INTEGER I, J, P, MP, NP
182  COMPLEX ALPHA
183 * ..
184 * .. External Subroutines ..
185  EXTERNAL clarfg, cgemv, cgerc, ctrmv, xerbla
186 * ..
187 * .. Intrinsic Functions ..
188  INTRINSIC max, min
189 * ..
190 * .. Executable Statements ..
191 *
192 * Test the input arguments
193 *
194  info = 0
195  IF( m.LT.0 ) THEN
196  info = -1
197  ELSE IF( n.LT.0 ) THEN
198  info = -2
199  ELSE IF( l.LT.0 .OR. l.GT.min(m,n) ) THEN
200  info = -3
201  ELSE IF( lda.LT.max( 1, m ) ) THEN
202  info = -5
203  ELSE IF( ldb.LT.max( 1, m ) ) THEN
204  info = -7
205  ELSE IF( ldt.LT.max( 1, m ) ) THEN
206  info = -9
207  END IF
208  IF( info.NE.0 ) THEN
209  CALL xerbla( 'CTPLQT2', -info )
210  RETURN
211  END IF
212 *
213 * Quick return if possible
214 *
215  IF( n.EQ.0 .OR. m.EQ.0 ) RETURN
216 *
217  DO i = 1, m
218 *
219 * Generate elementary reflector H(I) to annihilate B(I,:)
220 *
221  p = n-l+min( l, i )
222  CALL clarfg( p+1, a( i, i ), b( i, 1 ), ldb, t( 1, i ) )
223  t(1,i)=conjg(t(1,i))
224  IF( i.LT.m ) THEN
225  DO j = 1, p
226  b( i, j ) = conjg(b(i,j))
227  END DO
228 *
229 * W(M-I:1) := C(I+1:M,I:N) * C(I,I:N) [use W = T(M,:)]
230 *
231  DO j = 1, m-i
232  t( m, j ) = (a( i+j, i ))
233  END DO
234  CALL cgemv( 'N', m-i, p, one, b( i+1, 1 ), ldb,
235  $ b( i, 1 ), ldb, one, t( m, 1 ), ldt )
236 *
237 * C(I+1:M,I:N) = C(I+1:M,I:N) + alpha * C(I,I:N)*W(M-1:1)^H
238 *
239  alpha = -(t( 1, i ))
240  DO j = 1, m-i
241  a( i+j, i ) = a( i+j, i ) + alpha*(t( m, j ))
242  END DO
243  CALL cgerc( m-i, p, (alpha), t( m, 1 ), ldt,
244  $ b( i, 1 ), ldb, b( i+1, 1 ), ldb )
245  DO j = 1, p
246  b( i, j ) = conjg(b(i,j))
247  END DO
248  END IF
249  END DO
250 *
251  DO i = 2, m
252 *
253 * T(I,1:I-1) := C(I:I-1,1:N)**H * (alpha * C(I,I:N))
254 *
255  alpha = -(t( 1, i ))
256  DO j = 1, i-1
257  t( i, j ) = zero
258  END DO
259  p = min( i-1, l )
260  np = min( n-l+1, n )
261  mp = min( p+1, m )
262  DO j = 1, n-l+p
263  b(i,j)=conjg(b(i,j))
264  END DO
265 *
266 * Triangular part of B2
267 *
268  DO j = 1, p
269  t( i, j ) = (alpha*b( i, n-l+j ))
270  END DO
271  CALL ctrmv( 'L', 'N', 'N', p, b( 1, np ), ldb,
272  $ t( i, 1 ), ldt )
273 *
274 * Rectangular part of B2
275 *
276  CALL cgemv( 'N', i-1-p, l, alpha, b( mp, np ), ldb,
277  $ b( i, np ), ldb, zero, t( i,mp ), ldt )
278 *
279 * B1
280 
281 *
282  CALL cgemv( 'N', i-1, n-l, alpha, b, ldb, b( i, 1 ), ldb,
283  $ one, t( i, 1 ), ldt )
284 *
285 
286 *
287 * T(1:I-1,I) := T(1:I-1,1:I-1) * T(I,1:I-1)
288 *
289  DO j = 1, i-1
290  t(i,j)=conjg(t(i,j))
291  END DO
292  CALL ctrmv( 'L', 'C', 'N', i-1, t, ldt, t( i, 1 ), ldt )
293  DO j = 1, i-1
294  t(i,j)=conjg(t(i,j))
295  END DO
296  DO j = 1, n-l+p
297  b(i,j)=conjg(b(i,j))
298  END DO
299 *
300 * T(I,I) = tau(I)
301 *
302  t( i, i ) = t( 1, i )
303  t( 1, i ) = zero
304  END DO
305  DO i=1,m
306  DO j= i+1,m
307  t(i,j)=(t(j,i))
308  t(j,i)=zero
309  END DO
310  END DO
311 
312 *
313 * End of CTPLQT2
314 *
315  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGEMV
Definition: cgemv.f:158
subroutine cgerc(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
CGERC
Definition: cgerc.f:130
subroutine ctrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
CTRMV
Definition: ctrmv.f:147
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:106
subroutine ctplqt2(M, N, L, A, LDA, B, LDB, T, LDT, INFO)
CTPLQT2
Definition: ctplqt2.f:162