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
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
|
*> \brief \b SGET22
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition
* ==========
*
* SUBROUTINE SGET22( TRANSA, TRANSE, TRANSW, N, A, LDA, E, LDE, WR,
* WI, WORK, RESULT )
*
* .. Scalar Arguments ..
* CHARACTER TRANSA, TRANSE, TRANSW
* INTEGER LDA, LDE, N
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), E( LDE, * ), RESULT( 2 ), WI( * ),
* $ WORK( * ), WR( * )
* ..
*
* Purpose
* =======
*
*>\details \b Purpose:
*>\verbatim
*>
*> SGET22 does an eigenvector check.
*>
*> The basic test is:
*>
*> RESULT(1) = | A E - E W | / ( |A| |E| ulp )
*>
*> using the 1-norm. It also tests the normalization of E:
*>
*> RESULT(2) = max | m-norm(E(j)) - 1 | / ( n ulp )
*> j
*>
*> where E(j) is the j-th eigenvector, and m-norm is the max-norm of a
*> vector. If an eigenvector is complex, as determined from WI(j)
*> nonzero, then the max-norm of the vector ( er + i*ei ) is the maximum
*> of
*> |er(1)| + |ei(1)|, ... , |er(n)| + |ei(n)|
*>
*> W is a block diagonal matrix, with a 1 by 1 block for each real
*> eigenvalue and a 2 by 2 block for each complex conjugate pair.
*> If eigenvalues j and j+1 are a complex conjugate pair, so that
*> WR(j) = WR(j+1) = wr and WI(j) = - WI(j+1) = wi, then the 2 by 2
*> block corresponding to the pair will be:
*>
*> ( wr wi )
*> ( -wi wr )
*>
*> Such a block multiplying an n by 2 matrix ( ur ui ) on the right
*> will be the same as multiplying ur + i*ui by wr + i*wi.
*>
*> To handle various schemes for storage of left eigenvectors, there are
*> options to use A-transpose instead of A, E-transpose instead of E,
*> and/or W-transpose instead of W.
*>
*>\endverbatim
*
* Arguments
* =========
*
*> \param[in] TRANSA
*> \verbatim
*> TRANSA is CHARACTER*1
*> Specifies whether or not A is transposed.
*> = 'N': No transpose
*> = 'T': Transpose
*> = 'C': Conjugate transpose (= Transpose)
*> \endverbatim
*>
*> \param[in] TRANSE
*> \verbatim
*> TRANSE is CHARACTER*1
*> Specifies whether or not E is transposed.
*> = 'N': No transpose, eigenvectors are in columns of E
*> = 'T': Transpose, eigenvectors are in rows of E
*> = 'C': Conjugate transpose (= Transpose)
*> \endverbatim
*>
*> \param[in] TRANSW
*> \verbatim
*> TRANSW is CHARACTER*1
*> Specifies whether or not W is transposed.
*> = 'N': No transpose
*> = 'T': Transpose, use -WI(j) instead of WI(j)
*> = 'C': Conjugate transpose, use -WI(j) instead of WI(j)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is REAL array, dimension (LDA,N)
*> The matrix whose eigenvectors are in E.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is REAL array, dimension (LDE,N)
*> The matrix of eigenvectors. If TRANSE = 'N', the eigenvectors
*> are stored in the columns of E, if TRANSE = 'T' or 'C', the
*> eigenvectors are stored in the rows of E.
*> \endverbatim
*>
*> \param[in] LDE
*> \verbatim
*> LDE is INTEGER
*> The leading dimension of the array E. LDE >= max(1,N).
*> \endverbatim
*>
*> \param[in] WR
*> \verbatim
*> WR is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[in] WI
*> \verbatim
*> WI is REAL array, dimension (N)
*> \endverbatim
*> \verbatim
*> The real and imaginary parts of the eigenvalues of A.
*> Purely real eigenvalues are indicated by WI(j) = 0.
*> Complex conjugate pairs are indicated by WR(j)=WR(j+1) and
*> WI(j) = - WI(j+1) non-zero; the real part is assumed to be
*> stored in the j-th row/column and the imaginary part in
*> the (j+1)-th row/column.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (N*(N+1))
*> \endverbatim
*>
*> \param[out] RESULT
*> \verbatim
*> RESULT is REAL array, dimension (2)
*> RESULT(1) = | A E - E W | / ( |A| |E| ulp )
*> RESULT(2) = max | m-norm(E(j)) - 1 | / ( n ulp )
*> \endverbatim
*>
*
* Authors
* =======
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup single_eig
*
* =====================================================================
SUBROUTINE SGET22( TRANSA, TRANSE, TRANSW, N, A, LDA, E, LDE, WR,
$ WI, WORK, RESULT )
*
* -- LAPACK test routine (version 3.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER TRANSA, TRANSE, TRANSW
INTEGER LDA, LDE, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), E( LDE, * ), RESULT( 2 ), WI( * ),
$ WORK( * ), WR( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0, ONE = 1.0 )
* ..
* .. Local Scalars ..
CHARACTER NORMA, NORME
INTEGER IECOL, IEROW, INCE, IPAIR, ITRNSE, J, JCOL,
$ JVEC
REAL ANORM, ENORM, ENRMAX, ENRMIN, ERRNRM, TEMP1,
$ ULP, UNFL
* ..
* .. Local Arrays ..
REAL WMAT( 2, 2 )
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SLAMCH, SLANGE
EXTERNAL LSAME, SLAMCH, SLANGE
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SGEMM, SLASET
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, REAL
* ..
* .. Executable Statements ..
*
* Initialize RESULT (in case N=0)
*
RESULT( 1 ) = ZERO
RESULT( 2 ) = ZERO
IF( N.LE.0 )
$ RETURN
*
UNFL = SLAMCH( 'Safe minimum' )
ULP = SLAMCH( 'Precision' )
*
ITRNSE = 0
INCE = 1
NORMA = 'O'
NORME = 'O'
*
IF( LSAME( TRANSA, 'T' ) .OR. LSAME( TRANSA, 'C' ) ) THEN
NORMA = 'I'
END IF
IF( LSAME( TRANSE, 'T' ) .OR. LSAME( TRANSE, 'C' ) ) THEN
NORME = 'I'
ITRNSE = 1
INCE = LDE
END IF
*
* Check normalization of E
*
ENRMIN = ONE / ULP
ENRMAX = ZERO
IF( ITRNSE.EQ.0 ) THEN
*
* Eigenvectors are column vectors.
*
IPAIR = 0
DO 30 JVEC = 1, N
TEMP1 = ZERO
IF( IPAIR.EQ.0 .AND. JVEC.LT.N .AND. WI( JVEC ).NE.ZERO )
$ IPAIR = 1
IF( IPAIR.EQ.1 ) THEN
*
* Complex eigenvector
*
DO 10 J = 1, N
TEMP1 = MAX( TEMP1, ABS( E( J, JVEC ) )+
$ ABS( E( J, JVEC+1 ) ) )
10 CONTINUE
ENRMIN = MIN( ENRMIN, TEMP1 )
ENRMAX = MAX( ENRMAX, TEMP1 )
IPAIR = 2
ELSE IF( IPAIR.EQ.2 ) THEN
IPAIR = 0
ELSE
*
* Real eigenvector
*
DO 20 J = 1, N
TEMP1 = MAX( TEMP1, ABS( E( J, JVEC ) ) )
20 CONTINUE
ENRMIN = MIN( ENRMIN, TEMP1 )
ENRMAX = MAX( ENRMAX, TEMP1 )
IPAIR = 0
END IF
30 CONTINUE
*
ELSE
*
* Eigenvectors are row vectors.
*
DO 40 JVEC = 1, N
WORK( JVEC ) = ZERO
40 CONTINUE
*
DO 60 J = 1, N
IPAIR = 0
DO 50 JVEC = 1, N
IF( IPAIR.EQ.0 .AND. JVEC.LT.N .AND. WI( JVEC ).NE.ZERO )
$ IPAIR = 1
IF( IPAIR.EQ.1 ) THEN
WORK( JVEC ) = MAX( WORK( JVEC ),
$ ABS( E( J, JVEC ) )+ABS( E( J,
$ JVEC+1 ) ) )
WORK( JVEC+1 ) = WORK( JVEC )
ELSE IF( IPAIR.EQ.2 ) THEN
IPAIR = 0
ELSE
WORK( JVEC ) = MAX( WORK( JVEC ),
$ ABS( E( J, JVEC ) ) )
IPAIR = 0
END IF
50 CONTINUE
60 CONTINUE
*
DO 70 JVEC = 1, N
ENRMIN = MIN( ENRMIN, WORK( JVEC ) )
ENRMAX = MAX( ENRMAX, WORK( JVEC ) )
70 CONTINUE
END IF
*
* Norm of A:
*
ANORM = MAX( SLANGE( NORMA, N, N, A, LDA, WORK ), UNFL )
*
* Norm of E:
*
ENORM = MAX( SLANGE( NORME, N, N, E, LDE, WORK ), ULP )
*
* Norm of error:
*
* Error = AE - EW
*
CALL SLASET( 'Full', N, N, ZERO, ZERO, WORK, N )
*
IPAIR = 0
IEROW = 1
IECOL = 1
*
DO 80 JCOL = 1, N
IF( ITRNSE.EQ.1 ) THEN
IEROW = JCOL
ELSE
IECOL = JCOL
END IF
*
IF( IPAIR.EQ.0 .AND. WI( JCOL ).NE.ZERO )
$ IPAIR = 1
*
IF( IPAIR.EQ.1 ) THEN
WMAT( 1, 1 ) = WR( JCOL )
WMAT( 2, 1 ) = -WI( JCOL )
WMAT( 1, 2 ) = WI( JCOL )
WMAT( 2, 2 ) = WR( JCOL )
CALL SGEMM( TRANSE, TRANSW, N, 2, 2, ONE, E( IEROW, IECOL ),
$ LDE, WMAT, 2, ZERO, WORK( N*( JCOL-1 )+1 ), N )
IPAIR = 2
ELSE IF( IPAIR.EQ.2 ) THEN
IPAIR = 0
*
ELSE
*
CALL SAXPY( N, WR( JCOL ), E( IEROW, IECOL ), INCE,
$ WORK( N*( JCOL-1 )+1 ), 1 )
IPAIR = 0
END IF
*
80 CONTINUE
*
CALL SGEMM( TRANSA, TRANSE, N, N, N, ONE, A, LDA, E, LDE, -ONE,
$ WORK, N )
*
ERRNRM = SLANGE( 'One', N, N, WORK, N, WORK( N*N+1 ) ) / ENORM
*
* Compute RESULT(1) (avoiding under/overflow)
*
IF( ANORM.GT.ERRNRM ) THEN
RESULT( 1 ) = ( ERRNRM / ANORM ) / ULP
ELSE
IF( ANORM.LT.ONE ) THEN
RESULT( 1 ) = ( MIN( ERRNRM, ANORM ) / ANORM ) / ULP
ELSE
RESULT( 1 ) = MIN( ERRNRM / ANORM, ONE ) / ULP
END IF
END IF
*
* Compute RESULT(2) : the normalization error in E.
*
RESULT( 2 ) = MAX( ABS( ENRMAX-ONE ), ABS( ENRMIN-ONE ) ) /
$ ( REAL( N )*ULP )
*
RETURN
*
* End of SGET22
*
END
|
