1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
|
SUBROUTINE CLAEIN( RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK,
$ EPS3, SMLNUM, INFO )
*
* -- LAPACK auxiliary routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*
* .. Scalar Arguments ..
LOGICAL NOINIT, RIGHTV
INTEGER INFO, LDB, LDH, N
REAL EPS3, SMLNUM
COMPLEX W
* ..
* .. Array Arguments ..
REAL RWORK( * )
COMPLEX B( LDB, * ), H( LDH, * ), V( * )
* ..
*
* Purpose
* =======
*
* CLAEIN uses inverse iteration to find a right or left eigenvector
* corresponding to the eigenvalue W of a complex upper Hessenberg
* matrix H.
*
* Arguments
* =========
*
* RIGHTV (input) LOGICAL
* = .TRUE. : compute right eigenvector;
* = .FALSE.: compute left eigenvector.
*
* NOINIT (input) LOGICAL
* = .TRUE. : no initial vector supplied in V
* = .FALSE.: initial vector supplied in V.
*
* N (input) INTEGER
* The order of the matrix H. N >= 0.
*
* H (input) COMPLEX array, dimension (LDH,N)
* The upper Hessenberg matrix H.
*
* LDH (input) INTEGER
* The leading dimension of the array H. LDH >= max(1,N).
*
* W (input) COMPLEX
* The eigenvalue of H whose corresponding right or left
* eigenvector is to be computed.
*
* V (input/output) COMPLEX array, dimension (N)
* On entry, if NOINIT = .FALSE., V must contain a starting
* vector for inverse iteration; otherwise V need not be set.
* On exit, V contains the computed eigenvector, normalized so
* that the component of largest magnitude has magnitude 1; here
* the magnitude of a complex number (x,y) is taken to be
* |x| + |y|.
*
* B (workspace) COMPLEX array, dimension (LDB,N)
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* RWORK (workspace) REAL array, dimension (N)
*
* EPS3 (input) REAL
* A small machine-dependent value which is used to perturb
* close eigenvalues, and to replace zero pivots.
*
* SMLNUM (input) REAL
* A machine-dependent value close to the underflow threshold.
*
* INFO (output) INTEGER
* = 0: successful exit
* = 1: inverse iteration did not converge; V is set to the
* last iterate.
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, TENTH
PARAMETER ( ONE = 1.0E+0, TENTH = 1.0E-1 )
COMPLEX ZERO
PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
CHARACTER NORMIN, TRANS
INTEGER I, IERR, ITS, J
REAL GROWTO, NRMSML, ROOTN, RTEMP, SCALE, VNORM
COMPLEX CDUM, EI, EJ, TEMP, X
* ..
* .. External Functions ..
INTEGER ICAMAX
REAL SCASUM, SCNRM2
COMPLEX CLADIV
EXTERNAL ICAMAX, SCASUM, SCNRM2, CLADIV
* ..
* .. External Subroutines ..
EXTERNAL CLATRS, CSSCAL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, MAX, REAL, SQRT
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* .. Statement Function definitions ..
CABS1( CDUM ) = ABS( REAL( CDUM ) ) + ABS( AIMAG( CDUM ) )
* ..
* .. Executable Statements ..
*
INFO = 0
*
* GROWTO is the threshold used in the acceptance test for an
* eigenvector.
*
ROOTN = SQRT( REAL( N ) )
GROWTO = TENTH / ROOTN
NRMSML = MAX( ONE, EPS3*ROOTN )*SMLNUM
*
* Form B = H - W*I (except that the subdiagonal elements are not
* stored).
*
DO 20 J = 1, N
DO 10 I = 1, J - 1
B( I, J ) = H( I, J )
10 CONTINUE
B( J, J ) = H( J, J ) - W
20 CONTINUE
*
IF( NOINIT ) THEN
*
* Initialize V.
*
DO 30 I = 1, N
V( I ) = EPS3
30 CONTINUE
ELSE
*
* Scale supplied initial vector.
*
VNORM = SCNRM2( N, V, 1 )
CALL CSSCAL( N, ( EPS3*ROOTN ) / MAX( VNORM, NRMSML ), V, 1 )
END IF
*
IF( RIGHTV ) THEN
*
* LU decomposition with partial pivoting of B, replacing zero
* pivots by EPS3.
*
DO 60 I = 1, N - 1
EI = H( I+1, I )
IF( CABS1( B( I, I ) ).LT.CABS1( EI ) ) THEN
*
* Interchange rows and eliminate.
*
X = CLADIV( B( I, I ), EI )
B( I, I ) = EI
DO 40 J = I + 1, N
TEMP = B( I+1, J )
B( I+1, J ) = B( I, J ) - X*TEMP
B( I, J ) = TEMP
40 CONTINUE
ELSE
*
* Eliminate without interchange.
*
IF( B( I, I ).EQ.ZERO )
$ B( I, I ) = EPS3
X = CLADIV( EI, B( I, I ) )
IF( X.NE.ZERO ) THEN
DO 50 J = I + 1, N
B( I+1, J ) = B( I+1, J ) - X*B( I, J )
50 CONTINUE
END IF
END IF
60 CONTINUE
IF( B( N, N ).EQ.ZERO )
$ B( N, N ) = EPS3
*
TRANS = 'N'
*
ELSE
*
* UL decomposition with partial pivoting of B, replacing zero
* pivots by EPS3.
*
DO 90 J = N, 2, -1
EJ = H( J, J-1 )
IF( CABS1( B( J, J ) ).LT.CABS1( EJ ) ) THEN
*
* Interchange columns and eliminate.
*
X = CLADIV( B( J, J ), EJ )
B( J, J ) = EJ
DO 70 I = 1, J - 1
TEMP = B( I, J-1 )
B( I, J-1 ) = B( I, J ) - X*TEMP
B( I, J ) = TEMP
70 CONTINUE
ELSE
*
* Eliminate without interchange.
*
IF( B( J, J ).EQ.ZERO )
$ B( J, J ) = EPS3
X = CLADIV( EJ, B( J, J ) )
IF( X.NE.ZERO ) THEN
DO 80 I = 1, J - 1
B( I, J-1 ) = B( I, J-1 ) - X*B( I, J )
80 CONTINUE
END IF
END IF
90 CONTINUE
IF( B( 1, 1 ).EQ.ZERO )
$ B( 1, 1 ) = EPS3
*
TRANS = 'C'
*
END IF
*
NORMIN = 'N'
DO 110 ITS = 1, N
*
* Solve U*x = scale*v for a right eigenvector
* or U**H *x = scale*v for a left eigenvector,
* overwriting x on v.
*
CALL CLATRS( 'Upper', TRANS, 'Nonunit', NORMIN, N, B, LDB, V,
$ SCALE, RWORK, IERR )
NORMIN = 'Y'
*
* Test for sufficient growth in the norm of v.
*
VNORM = SCASUM( N, V, 1 )
IF( VNORM.GE.GROWTO*SCALE )
$ GO TO 120
*
* Choose new orthogonal starting vector and try again.
*
RTEMP = EPS3 / ( ROOTN+ONE )
V( 1 ) = EPS3
DO 100 I = 2, N
V( I ) = RTEMP
100 CONTINUE
V( N-ITS+1 ) = V( N-ITS+1 ) - EPS3*ROOTN
110 CONTINUE
*
* Failure to find eigenvector in N iterations.
*
INFO = 1
*
120 CONTINUE
*
* Normalize eigenvector.
*
I = ICAMAX( N, V, 1 )
CALL CSSCAL( N, ONE / CABS1( V( I ) ), V, 1 )
*
RETURN
*
* End of CLAEIN
*
END
|
