LAPACK  3.7.0 LAPACK: Linear Algebra PACKage
clagsy.f
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1 *> \brief \b CLAGSY
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 * Definition:
9 * ===========
10 *
11 * SUBROUTINE CLAGSY( N, K, D, A, LDA, ISEED, WORK, INFO )
12 *
13 * .. Scalar Arguments ..
14 * INTEGER INFO, K, LDA, N
15 * ..
16 * .. Array Arguments ..
17 * INTEGER ISEED( 4 )
18 * REAL D( * )
19 * COMPLEX A( LDA, * ), WORK( * )
20 * ..
21 *
22 *
23 *> \par Purpose:
24 * =============
25 *>
26 *> \verbatim
27 *>
28 *> CLAGSY generates a complex symmetric matrix A, by pre- and post-
29 *> multiplying a real diagonal matrix D with a random unitary matrix:
30 *> A = U*D*U**T. The semi-bandwidth may then be reduced to k by
31 *> additional unitary transformations.
32 *> \endverbatim
33 *
34 * Arguments:
35 * ==========
36 *
37 *> \param[in] N
38 *> \verbatim
39 *> N is INTEGER
40 *> The order of the matrix A. N >= 0.
41 *> \endverbatim
42 *>
43 *> \param[in] K
44 *> \verbatim
45 *> K is INTEGER
46 *> The number of nonzero subdiagonals within the band of A.
47 *> 0 <= K <= N-1.
48 *> \endverbatim
49 *>
50 *> \param[in] D
51 *> \verbatim
52 *> D is REAL array, dimension (N)
53 *> The diagonal elements of the diagonal matrix D.
54 *> \endverbatim
55 *>
56 *> \param[out] A
57 *> \verbatim
58 *> A is COMPLEX array, dimension (LDA,N)
59 *> The generated n by n symmetric matrix A (the full matrix is
60 *> stored).
61 *> \endverbatim
62 *>
63 *> \param[in] LDA
64 *> \verbatim
65 *> LDA is INTEGER
66 *> The leading dimension of the array A. LDA >= N.
67 *> \endverbatim
68 *>
69 *> \param[in,out] ISEED
70 *> \verbatim
71 *> ISEED is INTEGER array, dimension (4)
72 *> On entry, the seed of the random number generator; the array
73 *> elements must be between 0 and 4095, and ISEED(4) must be
74 *> odd.
75 *> On exit, the seed is updated.
76 *> \endverbatim
77 *>
78 *> \param[out] WORK
79 *> \verbatim
80 *> WORK is COMPLEX array, dimension (2*N)
81 *> \endverbatim
82 *>
83 *> \param[out] INFO
84 *> \verbatim
85 *> INFO is INTEGER
86 *> = 0: successful exit
87 *> < 0: if INFO = -i, the i-th argument had an illegal value
88 *> \endverbatim
89 *
90 * Authors:
91 * ========
92 *
93 *> \author Univ. of Tennessee
94 *> \author Univ. of California Berkeley
95 *> \author Univ. of Colorado Denver
96 *> \author NAG Ltd.
97 *
98 *> \date December 2016
99 *
100 *> \ingroup complex_matgen
101 *
102 * =====================================================================
103  SUBROUTINE clagsy( N, K, D, A, LDA, ISEED, WORK, INFO )
104 *
105 * -- LAPACK auxiliary routine (version 3.7.0) --
106 * -- LAPACK is a software package provided by Univ. of Tennessee, --
107 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
108 * December 2016
109 *
110 * .. Scalar Arguments ..
111  INTEGER INFO, K, LDA, N
112 * ..
113 * .. Array Arguments ..
114  INTEGER ISEED( 4 )
115  REAL D( * )
116  COMPLEX A( lda, * ), WORK( * )
117 * ..
118 *
119 * =====================================================================
120 *
121 * .. Parameters ..
122  COMPLEX ZERO, ONE, HALF
123  parameter ( zero = ( 0.0e+0, 0.0e+0 ),
124  \$ one = ( 1.0e+0, 0.0e+0 ),
125  \$ half = ( 0.5e+0, 0.0e+0 ) )
126 * ..
127 * .. Local Scalars ..
128  INTEGER I, II, J, JJ
129  REAL WN
130  COMPLEX ALPHA, TAU, WA, WB
131 * ..
132 * .. External Subroutines ..
133  EXTERNAL caxpy, cgemv, cgerc, clacgv, clarnv, cscal,
134  \$ csymv, xerbla
135 * ..
136 * .. External Functions ..
137  REAL SCNRM2
138  COMPLEX CDOTC
139  EXTERNAL scnrm2, cdotc
140 * ..
141 * .. Intrinsic Functions ..
142  INTRINSIC abs, max, real
143 * ..
144 * .. Executable Statements ..
145 *
146 * Test the input arguments
147 *
148  info = 0
149  IF( n.LT.0 ) THEN
150  info = -1
151  ELSE IF( k.LT.0 .OR. k.GT.n-1 ) THEN
152  info = -2
153  ELSE IF( lda.LT.max( 1, n ) ) THEN
154  info = -5
155  END IF
156  IF( info.LT.0 ) THEN
157  CALL xerbla( 'CLAGSY', -info )
158  RETURN
159  END IF
160 *
161 * initialize lower triangle of A to diagonal matrix
162 *
163  DO 20 j = 1, n
164  DO 10 i = j + 1, n
165  a( i, j ) = zero
166  10 CONTINUE
167  20 CONTINUE
168  DO 30 i = 1, n
169  a( i, i ) = d( i )
170  30 CONTINUE
171 *
172 * Generate lower triangle of symmetric matrix
173 *
174  DO 60 i = n - 1, 1, -1
175 *
176 * generate random reflection
177 *
178  CALL clarnv( 3, iseed, n-i+1, work )
179  wn = scnrm2( n-i+1, work, 1 )
180  wa = ( wn / abs( work( 1 ) ) )*work( 1 )
181  IF( wn.EQ.zero ) THEN
182  tau = zero
183  ELSE
184  wb = work( 1 ) + wa
185  CALL cscal( n-i, one / wb, work( 2 ), 1 )
186  work( 1 ) = one
187  tau = REAL( wb / wa )
188  END IF
189 *
190 * apply random reflection to A(i:n,i:n) from the left
191 * and the right
192 *
193 * compute y := tau * A * conjg(u)
194 *
195  CALL clacgv( n-i+1, work, 1 )
196  CALL csymv( 'Lower', n-i+1, tau, a( i, i ), lda, work, 1, zero,
197  \$ work( n+1 ), 1 )
198  CALL clacgv( n-i+1, work, 1 )
199 *
200 * compute v := y - 1/2 * tau * ( u, y ) * u
201 *
202  alpha = -half*tau*cdotc( n-i+1, work, 1, work( n+1 ), 1 )
203  CALL caxpy( n-i+1, alpha, work, 1, work( n+1 ), 1 )
204 *
205 * apply the transformation as a rank-2 update to A(i:n,i:n)
206 *
207 * CALL CSYR2( 'Lower', N-I+1, -ONE, WORK, 1, WORK( N+1 ), 1,
208 * \$ A( I, I ), LDA )
209 *
210  DO 50 jj = i, n
211  DO 40 ii = jj, n
212  a( ii, jj ) = a( ii, jj ) -
213  \$ work( ii-i+1 )*work( n+jj-i+1 ) -
214  \$ work( n+ii-i+1 )*work( jj-i+1 )
215  40 CONTINUE
216  50 CONTINUE
217  60 CONTINUE
218 *
219 * Reduce number of subdiagonals to K
220 *
221  DO 100 i = 1, n - 1 - k
222 *
223 * generate reflection to annihilate A(k+i+1:n,i)
224 *
225  wn = scnrm2( n-k-i+1, a( k+i, i ), 1 )
226  wa = ( wn / abs( a( k+i, i ) ) )*a( k+i, i )
227  IF( wn.EQ.zero ) THEN
228  tau = zero
229  ELSE
230  wb = a( k+i, i ) + wa
231  CALL cscal( n-k-i, one / wb, a( k+i+1, i ), 1 )
232  a( k+i, i ) = one
233  tau = REAL( wb / wa )
234  END IF
235 *
236 * apply reflection to A(k+i:n,i+1:k+i-1) from the left
237 *
238  CALL cgemv( 'Conjugate transpose', n-k-i+1, k-1, one,
239  \$ a( k+i, i+1 ), lda, a( k+i, i ), 1, zero, work, 1 )
240  CALL cgerc( n-k-i+1, k-1, -tau, a( k+i, i ), 1, work, 1,
241  \$ a( k+i, i+1 ), lda )
242 *
243 * apply reflection to A(k+i:n,k+i:n) from the left and the right
244 *
245 * compute y := tau * A * conjg(u)
246 *
247  CALL clacgv( n-k-i+1, a( k+i, i ), 1 )
248  CALL csymv( 'Lower', n-k-i+1, tau, a( k+i, k+i ), lda,
249  \$ a( k+i, i ), 1, zero, work, 1 )
250  CALL clacgv( n-k-i+1, a( k+i, i ), 1 )
251 *
252 * compute v := y - 1/2 * tau * ( u, y ) * u
253 *
254  alpha = -half*tau*cdotc( n-k-i+1, a( k+i, i ), 1, work, 1 )
255  CALL caxpy( n-k-i+1, alpha, a( k+i, i ), 1, work, 1 )
256 *
257 * apply symmetric rank-2 update to A(k+i:n,k+i:n)
258 *
259 * CALL CSYR2( 'Lower', N-K-I+1, -ONE, A( K+I, I ), 1, WORK, 1,
260 * \$ A( K+I, K+I ), LDA )
261 *
262  DO 80 jj = k + i, n
263  DO 70 ii = jj, n
264  a( ii, jj ) = a( ii, jj ) - a( ii, i )*work( jj-k-i+1 ) -
265  \$ work( ii-k-i+1 )*a( jj, i )
266  70 CONTINUE
267  80 CONTINUE
268 *
269  a( k+i, i ) = -wa
270  DO 90 j = k + i + 1, n
271  a( j, i ) = zero
272  90 CONTINUE
273  100 CONTINUE
274 *
275 * Store full symmetric matrix
276 *
277  DO 120 j = 1, n
278  DO 110 i = j + 1, n
279  a( j, i ) = a( i, j )
280  110 CONTINUE
281  120 CONTINUE
282  RETURN
283 *
284 * End of CLAGSY
285 *
286  END
subroutine csymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CSYMV computes a matrix-vector product for a complex symmetric matrix.
Definition: csymv.f:159
subroutine clarnv(IDIST, ISEED, N, X)
CLARNV returns a vector of random numbers from a uniform or normal distribution.
Definition: clarnv.f:101
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:54
subroutine cgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGEMV
Definition: cgemv.f:160
subroutine cgerc(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
CGERC
Definition: cgerc.f:132
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine clacgv(N, X, INCX)
CLACGV conjugates a complex vector.
Definition: clacgv.f:76
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:53
subroutine clagsy(N, K, D, A, LDA, ISEED, WORK, INFO)
CLAGSY
Definition: clagsy.f:104