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
|
*
*
SUBROUTINE SSTEIN2( N, D, E, M, W, IBLOCK, ISPLIT, ORFAC, Z, LDZ,
$ WORK, IWORK, IFAIL, INFO )
*
* -- ScaLAPACK routine (version 1.7) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* May 1, 1997
*
* .. Scalar Arguments ..
INTEGER INFO, LDZ, M, N
REAL ORFAC
* ..
* .. Array Arguments ..
INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
$ IWORK( * )
REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* SSTEIN2 computes the eigenvectors of a real symmetric tridiagonal
* matrix T corresponding to specified eigenvalues, using inverse
* iteration.
*
* The maximum number of iterations allowed for each eigenvector is
* specified by an internal parameter MAXITS (currently set to 5).
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix. N >= 0.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of the tridiagonal matrix T.
*
* E (input) REAL array, dimension (N)
* The (n-1) subdiagonal elements of the tridiagonal matrix
* T, in elements 1 to N-1. E(N) need not be set.
*
* M (input) INTEGER
* The number of eigenvectors to be found. 0 <= M <= N.
*
* W (input) REAL array, dimension (N)
* The first M elements of W contain the eigenvalues for
* which eigenvectors are to be computed. The eigenvalues
* should be grouped by split-off block and ordered from
* smallest to largest within the block. ( The output array
* W from SSTEBZ with ORDER = 'B' is expected here. )
*
* IBLOCK (input) INTEGER array, dimension (N)
* The submatrix indices associated with the corresponding
* eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
* the first submatrix from the top, =2 if W(i) belongs to
* the second submatrix, etc. ( The output array IBLOCK
* from SSTEBZ is expected here. )
*
* ISPLIT (input) INTEGER array, dimension (N)
* The splitting points, at which T breaks up into submatrices.
* The first submatrix consists of rows/columns 1 to
* ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
* through ISPLIT( 2 ), etc.
* ( The output array ISPLIT from SSTEBZ is expected here. )
*
* ORFAC (input) REAL
* ORFAC specifies which eigenvectors should be
* orthogonalized. Eigenvectors that correspond to eigenvalues
* which are within ORFAC*||T|| of each other are to be
* orthogonalized.
*
* Z (output) REAL array, dimension (LDZ, M)
* The computed eigenvectors. The eigenvector associated
* with the eigenvalue W(i) is stored in the i-th column of
* Z. Any vector which fails to converge is set to its current
* iterate after MAXITS iterations.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= max(1,N).
*
* WORK (workspace) REAL array, dimension (5*N)
*
* IWORK (workspace) INTEGER array, dimension (N)
*
* IFAIL (output) INTEGER array, dimension (M)
* On normal exit, all elements of IFAIL are zero.
* If one or more eigenvectors fail to converge after
* MAXITS iterations, then their indices are stored in
* array IFAIL.
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, then i eigenvectors failed to converge
* in MAXITS iterations. Their indices are stored in
* array IFAIL.
*
* Internal Parameters
* ===================
*
* MAXITS INTEGER, default = 5
* The maximum number of iterations performed.
*
* EXTRA INTEGER, default = 2
* The number of iterations performed after norm growth
* criterion is satisfied, should be at least 1.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TEN, ODM1
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TEN = 1.0E+1,
$ ODM1 = 1.0E-1 )
INTEGER MAXITS, EXTRA
PARAMETER ( MAXITS = 5, EXTRA = 2 )
* ..
* .. Local Scalars ..
INTEGER B1, BLKSIZ, BN, GPIND, I, IINFO, INDRV1,
$ INDRV2, INDRV3, INDRV4, INDRV5, ITS, J, J1,
$ JBLK, JMAX, NBLK, NRMCHK
REAL EPS, EPS1, NRM, ONENRM, ORTOL, PERTOL, SCL,
$ SEP, STPCRT, TOL, XJ, XJM, ZTR
* ..
* .. Local Arrays ..
INTEGER ISEED( 4 )
* ..
* .. External Functions ..
INTEGER ISAMAX
REAL SASUM, SDOT, SLAMCH, SNRM2
EXTERNAL ISAMAX, SASUM, SDOT, SLAMCH, SNRM2
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SCOPY, SLAGTF, SLAGTS, SLARNV, SSCAL,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
DO 10 I = 1, M
IFAIL( I ) = 0
10 CONTINUE
*
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
INFO = -4
ELSE IF( ORFAC.LT.ZERO ) THEN
INFO = -8
ELSE IF( LDZ.LT.MAX( 1, N ) ) THEN
INFO = -10
ELSE
DO 20 J = 2, M
IF( IBLOCK( J ).LT.IBLOCK( J-1 ) ) THEN
INFO = -6
GO TO 30
END IF
IF( IBLOCK( J ).EQ.IBLOCK( J-1 ) .AND. W( J ).LT.W( J-1 ) )
$ THEN
INFO = -5
GO TO 30
END IF
20 CONTINUE
30 CONTINUE
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SSTEIN2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. M.EQ.0 ) THEN
RETURN
ELSE IF( N.EQ.1 ) THEN
Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
EPS = SLAMCH( 'Precision' )
*
* Initialize seed for random number generator SLARNV.
*
DO 40 I = 1, 4
ISEED( I ) = 1
40 CONTINUE
*
* Initialize pointers.
*
INDRV1 = 0
INDRV2 = INDRV1 + N
INDRV3 = INDRV2 + N
INDRV4 = INDRV3 + N
INDRV5 = INDRV4 + N
*
* Compute eigenvectors of matrix blocks.
*
J1 = 1
DO 160 NBLK = 1, IBLOCK( M )
*
* Find starting and ending indices of block nblk.
*
IF( NBLK.EQ.1 ) THEN
B1 = 1
ELSE
B1 = ISPLIT( NBLK-1 ) + 1
END IF
BN = ISPLIT( NBLK )
BLKSIZ = BN - B1 + 1
IF( BLKSIZ.EQ.1 )
$ GO TO 60
GPIND = J1
*
* Compute reorthogonalization criterion and stopping criterion.
*
ONENRM = ABS( D( B1 ) ) + ABS( E( B1 ) )
ONENRM = MAX( ONENRM, ABS( D( BN ) )+ABS( E( BN-1 ) ) )
DO 50 I = B1 + 1, BN - 1
ONENRM = MAX( ONENRM, ABS( D( I ) )+ABS( E( I-1 ) )+
$ ABS( E( I ) ) )
50 CONTINUE
ORTOL = ORFAC*ONENRM
*
STPCRT = SQRT( ODM1 / BLKSIZ )
*
* Loop through eigenvalues of block nblk.
*
60 CONTINUE
JBLK = 0
DO 150 J = J1, M
IF( IBLOCK( J ).NE.NBLK ) THEN
J1 = J
GO TO 160
END IF
JBLK = JBLK + 1
XJ = W( J )
*
* Skip all the work if the block size is one.
*
IF( BLKSIZ.EQ.1 ) THEN
WORK( INDRV1+1 ) = ONE
GO TO 120
END IF
*
* If eigenvalues j and j-1 are too close, add a relatively
* small perturbation.
*
IF( JBLK.GT.1 ) THEN
EPS1 = ABS( EPS*XJ )
PERTOL = TEN*EPS1
SEP = XJ - XJM
IF( SEP.LT.PERTOL )
$ XJ = XJM + PERTOL
END IF
*
ITS = 0
NRMCHK = 0
*
* Get random starting vector.
*
CALL SLARNV( 2, ISEED, BLKSIZ, WORK( INDRV1+1 ) )
*
* Copy the matrix T so it won't be destroyed in factorization.
*
CALL SCOPY( BLKSIZ, D( B1 ), 1, WORK( INDRV4+1 ), 1 )
CALL SCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV2+2 ), 1 )
CALL SCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV3+1 ), 1 )
*
* Compute LU factors with partial pivoting ( PT = LU )
*
TOL = ZERO
CALL SLAGTF( BLKSIZ, WORK( INDRV4+1 ), XJ, WORK( INDRV2+2 ),
$ WORK( INDRV3+1 ), TOL, WORK( INDRV5+1 ), IWORK,
$ IINFO )
*
* Update iteration count.
*
70 CONTINUE
ITS = ITS + 1
IF( ITS.GT.MAXITS )
$ GO TO 100
*
* Normalize and scale the righthand side vector Pb.
*
SCL = BLKSIZ*ONENRM*MAX( EPS,
$ ABS( WORK( INDRV4+BLKSIZ ) ) ) /
$ SASUM( BLKSIZ, WORK( INDRV1+1 ), 1 )
CALL SSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
*
* Solve the system LU = Pb.
*
CALL SLAGTS( -1, BLKSIZ, WORK( INDRV4+1 ), WORK( INDRV2+2 ),
$ WORK( INDRV3+1 ), WORK( INDRV5+1 ), IWORK,
$ WORK( INDRV1+1 ), TOL, IINFO )
*
* Reorthogonalize by modified Gram-Schmidt if eigenvalues are
* close enough.
*
IF( JBLK.EQ.1 )
$ GO TO 90
IF( ABS( XJ-XJM ).GT.ORTOL )
$ GPIND = J
*
IF( GPIND.NE.J ) THEN
DO 80 I = GPIND, J - 1
ZTR = -SDOT( BLKSIZ, WORK( INDRV1+1 ), 1, Z( B1, I ),
$ 1 )
CALL SAXPY( BLKSIZ, ZTR, Z( B1, I ), 1,
$ WORK( INDRV1+1 ), 1 )
80 CONTINUE
END IF
*
* Check the infinity norm of the iterate.
*
90 CONTINUE
JMAX = ISAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
NRM = ABS( WORK( INDRV1+JMAX ) )
*
* Continue for additional iterations after norm reaches
* stopping criterion.
*
IF( NRM.LT.STPCRT )
$ GO TO 70
NRMCHK = NRMCHK + 1
IF( NRMCHK.LT.EXTRA+1 )
$ GO TO 70
*
GO TO 110
*
* If stopping criterion was not satisfied, update info and
* store eigenvector number in array ifail.
*
100 CONTINUE
INFO = INFO + 1
IFAIL( INFO ) = J
*
* Accept iterate as jth eigenvector.
*
110 CONTINUE
SCL = ONE / SNRM2( BLKSIZ, WORK( INDRV1+1 ), 1 )
JMAX = ISAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
IF( WORK( INDRV1+JMAX ).LT.ZERO )
$ SCL = -SCL
CALL SSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
120 CONTINUE
DO 130 I = 1, N
Z( I, J ) = ZERO
130 CONTINUE
DO 140 I = 1, BLKSIZ
Z( B1+I-1, J ) = WORK( INDRV1+I )
140 CONTINUE
*
* Save the shift to check eigenvalue spacing at next
* iteration.
*
XJM = XJ
*
150 CONTINUE
160 CONTINUE
*
RETURN
*
* End of SSTEIN2
*
END
|
