LAPACK  3.9.0
LAPACK: Linear Algebra PACKage
zlaein.f
Go to the documentation of this file.
1 *> \brief \b ZLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download ZLAEIN + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaein.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaein.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaein.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZLAEIN( RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK,
22 * EPS3, SMLNUM, INFO )
23 *
24 * .. Scalar Arguments ..
25 * LOGICAL NOINIT, RIGHTV
26 * INTEGER INFO, LDB, LDH, N
27 * DOUBLE PRECISION EPS3, SMLNUM
28 * COMPLEX*16 W
29 * ..
30 * .. Array Arguments ..
31 * DOUBLE PRECISION RWORK( * )
32 * COMPLEX*16 B( LDB, * ), H( LDH, * ), V( * )
33 * ..
34 *
35 *
36 *> \par Purpose:
37 * =============
38 *>
39 *> \verbatim
40 *>
41 *> ZLAEIN uses inverse iteration to find a right or left eigenvector
42 *> corresponding to the eigenvalue W of a complex upper Hessenberg
43 *> matrix H.
44 *> \endverbatim
45 *
46 * Arguments:
47 * ==========
48 *
49 *> \param[in] RIGHTV
50 *> \verbatim
51 *> RIGHTV is LOGICAL
52 *> = .TRUE. : compute right eigenvector;
53 *> = .FALSE.: compute left eigenvector.
54 *> \endverbatim
55 *>
56 *> \param[in] NOINIT
57 *> \verbatim
58 *> NOINIT is LOGICAL
59 *> = .TRUE. : no initial vector supplied in V
60 *> = .FALSE.: initial vector supplied in V.
61 *> \endverbatim
62 *>
63 *> \param[in] N
64 *> \verbatim
65 *> N is INTEGER
66 *> The order of the matrix H. N >= 0.
67 *> \endverbatim
68 *>
69 *> \param[in] H
70 *> \verbatim
71 *> H is COMPLEX*16 array, dimension (LDH,N)
72 *> The upper Hessenberg matrix H.
73 *> \endverbatim
74 *>
75 *> \param[in] LDH
76 *> \verbatim
77 *> LDH is INTEGER
78 *> The leading dimension of the array H. LDH >= max(1,N).
79 *> \endverbatim
80 *>
81 *> \param[in] W
82 *> \verbatim
83 *> W is COMPLEX*16
84 *> The eigenvalue of H whose corresponding right or left
85 *> eigenvector is to be computed.
86 *> \endverbatim
87 *>
88 *> \param[in,out] V
89 *> \verbatim
90 *> V is COMPLEX*16 array, dimension (N)
91 *> On entry, if NOINIT = .FALSE., V must contain a starting
92 *> vector for inverse iteration; otherwise V need not be set.
93 *> On exit, V contains the computed eigenvector, normalized so
94 *> that the component of largest magnitude has magnitude 1; here
95 *> the magnitude of a complex number (x,y) is taken to be
96 *> |x| + |y|.
97 *> \endverbatim
98 *>
99 *> \param[out] B
100 *> \verbatim
101 *> B is COMPLEX*16 array, dimension (LDB,N)
102 *> \endverbatim
103 *>
104 *> \param[in] LDB
105 *> \verbatim
106 *> LDB is INTEGER
107 *> The leading dimension of the array B. LDB >= max(1,N).
108 *> \endverbatim
109 *>
110 *> \param[out] RWORK
111 *> \verbatim
112 *> RWORK is DOUBLE PRECISION array, dimension (N)
113 *> \endverbatim
114 *>
115 *> \param[in] EPS3
116 *> \verbatim
117 *> EPS3 is DOUBLE PRECISION
118 *> A small machine-dependent value which is used to perturb
119 *> close eigenvalues, and to replace zero pivots.
120 *> \endverbatim
121 *>
122 *> \param[in] SMLNUM
123 *> \verbatim
124 *> SMLNUM is DOUBLE PRECISION
125 *> A machine-dependent value close to the underflow threshold.
126 *> \endverbatim
127 *>
128 *> \param[out] INFO
129 *> \verbatim
130 *> INFO is INTEGER
131 *> = 0: successful exit
132 *> = 1: inverse iteration did not converge; V is set to the
133 *> last iterate.
134 *> \endverbatim
135 *
136 * Authors:
137 * ========
138 *
139 *> \author Univ. of Tennessee
140 *> \author Univ. of California Berkeley
141 *> \author Univ. of Colorado Denver
142 *> \author NAG Ltd.
143 *
144 *> \date December 2016
145 *
146 *> \ingroup complex16OTHERauxiliary
147 *
148 * =====================================================================
149  SUBROUTINE zlaein( RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK,
150  $ EPS3, SMLNUM, INFO )
151 *
152 * -- LAPACK auxiliary routine (version 3.7.0) --
153 * -- LAPACK is a software package provided by Univ. of Tennessee, --
154 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
155 * December 2016
156 *
157 * .. Scalar Arguments ..
158  LOGICAL NOINIT, RIGHTV
159  INTEGER INFO, LDB, LDH, N
160  DOUBLE PRECISION EPS3, SMLNUM
161  COMPLEX*16 W
162 * ..
163 * .. Array Arguments ..
164  DOUBLE PRECISION RWORK( * )
165  COMPLEX*16 B( LDB, * ), H( LDH, * ), V( * )
166 * ..
167 *
168 * =====================================================================
169 *
170 * .. Parameters ..
171  DOUBLE PRECISION ONE, TENTH
172  parameter( one = 1.0d+0, tenth = 1.0d-1 )
173  COMPLEX*16 ZERO
174  parameter( zero = ( 0.0d+0, 0.0d+0 ) )
175 * ..
176 * .. Local Scalars ..
177  CHARACTER NORMIN, TRANS
178  INTEGER I, IERR, ITS, J
179  DOUBLE PRECISION GROWTO, NRMSML, ROOTN, RTEMP, SCALE, VNORM
180  COMPLEX*16 CDUM, EI, EJ, TEMP, X
181 * ..
182 * .. External Functions ..
183  INTEGER IZAMAX
184  DOUBLE PRECISION DZASUM, DZNRM2
185  COMPLEX*16 ZLADIV
186  EXTERNAL izamax, dzasum, dznrm2, zladiv
187 * ..
188 * .. External Subroutines ..
189  EXTERNAL zdscal, zlatrs
190 * ..
191 * .. Intrinsic Functions ..
192  INTRINSIC abs, dble, dimag, max, sqrt
193 * ..
194 * .. Statement Functions ..
195  DOUBLE PRECISION CABS1
196 * ..
197 * .. Statement Function definitions ..
198  cabs1( cdum ) = abs( dble( cdum ) ) + abs( dimag( cdum ) )
199 * ..
200 * .. Executable Statements ..
201 *
202  info = 0
203 *
204 * GROWTO is the threshold used in the acceptance test for an
205 * eigenvector.
206 *
207  rootn = sqrt( dble( n ) )
208  growto = tenth / rootn
209  nrmsml = max( one, eps3*rootn )*smlnum
210 *
211 * Form B = H - W*I (except that the subdiagonal elements are not
212 * stored).
213 *
214  DO 20 j = 1, n
215  DO 10 i = 1, j - 1
216  b( i, j ) = h( i, j )
217  10 CONTINUE
218  b( j, j ) = h( j, j ) - w
219  20 CONTINUE
220 *
221  IF( noinit ) THEN
222 *
223 * Initialize V.
224 *
225  DO 30 i = 1, n
226  v( i ) = eps3
227  30 CONTINUE
228  ELSE
229 *
230 * Scale supplied initial vector.
231 *
232  vnorm = dznrm2( n, v, 1 )
233  CALL zdscal( n, ( eps3*rootn ) / max( vnorm, nrmsml ), v, 1 )
234  END IF
235 *
236  IF( rightv ) THEN
237 *
238 * LU decomposition with partial pivoting of B, replacing zero
239 * pivots by EPS3.
240 *
241  DO 60 i = 1, n - 1
242  ei = h( i+1, i )
243  IF( cabs1( b( i, i ) ).LT.cabs1( ei ) ) THEN
244 *
245 * Interchange rows and eliminate.
246 *
247  x = zladiv( b( i, i ), ei )
248  b( i, i ) = ei
249  DO 40 j = i + 1, n
250  temp = b( i+1, j )
251  b( i+1, j ) = b( i, j ) - x*temp
252  b( i, j ) = temp
253  40 CONTINUE
254  ELSE
255 *
256 * Eliminate without interchange.
257 *
258  IF( b( i, i ).EQ.zero )
259  $ b( i, i ) = eps3
260  x = zladiv( ei, b( i, i ) )
261  IF( x.NE.zero ) THEN
262  DO 50 j = i + 1, n
263  b( i+1, j ) = b( i+1, j ) - x*b( i, j )
264  50 CONTINUE
265  END IF
266  END IF
267  60 CONTINUE
268  IF( b( n, n ).EQ.zero )
269  $ b( n, n ) = eps3
270 *
271  trans = 'N'
272 *
273  ELSE
274 *
275 * UL decomposition with partial pivoting of B, replacing zero
276 * pivots by EPS3.
277 *
278  DO 90 j = n, 2, -1
279  ej = h( j, j-1 )
280  IF( cabs1( b( j, j ) ).LT.cabs1( ej ) ) THEN
281 *
282 * Interchange columns and eliminate.
283 *
284  x = zladiv( b( j, j ), ej )
285  b( j, j ) = ej
286  DO 70 i = 1, j - 1
287  temp = b( i, j-1 )
288  b( i, j-1 ) = b( i, j ) - x*temp
289  b( i, j ) = temp
290  70 CONTINUE
291  ELSE
292 *
293 * Eliminate without interchange.
294 *
295  IF( b( j, j ).EQ.zero )
296  $ b( j, j ) = eps3
297  x = zladiv( ej, b( j, j ) )
298  IF( x.NE.zero ) THEN
299  DO 80 i = 1, j - 1
300  b( i, j-1 ) = b( i, j-1 ) - x*b( i, j )
301  80 CONTINUE
302  END IF
303  END IF
304  90 CONTINUE
305  IF( b( 1, 1 ).EQ.zero )
306  $ b( 1, 1 ) = eps3
307 *
308  trans = 'C'
309 *
310  END IF
311 *
312  normin = 'N'
313  DO 110 its = 1, n
314 *
315 * Solve U*x = scale*v for a right eigenvector
316 * or U**H *x = scale*v for a left eigenvector,
317 * overwriting x on v.
318 *
319  CALL zlatrs( 'Upper', trans, 'Nonunit', normin, n, b, ldb, v,
320  $ scale, rwork, ierr )
321  normin = 'Y'
322 *
323 * Test for sufficient growth in the norm of v.
324 *
325  vnorm = dzasum( n, v, 1 )
326  IF( vnorm.GE.growto*scale )
327  $ GO TO 120
328 *
329 * Choose new orthogonal starting vector and try again.
330 *
331  rtemp = eps3 / ( rootn+one )
332  v( 1 ) = eps3
333  DO 100 i = 2, n
334  v( i ) = rtemp
335  100 CONTINUE
336  v( n-its+1 ) = v( n-its+1 ) - eps3*rootn
337  110 CONTINUE
338 *
339 * Failure to find eigenvector in N iterations.
340 *
341  info = 1
342 *
343  120 CONTINUE
344 *
345 * Normalize eigenvector.
346 *
347  i = izamax( n, v, 1 )
348  CALL zdscal( n, one / cabs1( v( i ) ), v, 1 )
349 *
350  RETURN
351 *
352 * End of ZLAEIN
353 *
354  END
zdscal
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:80
zlatrs
subroutine zlatrs(UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO)
ZLATRS solves a triangular system of equations with the scale factor set to prevent overflow.
Definition: zlatrs.f:241
zlaein
subroutine zlaein(RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK, EPS3, SMLNUM, INFO)
ZLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iterat...
Definition: zlaein.f:151