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
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
|
<HTML>
<HEAD><TITLE>SB10FD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="SB10FD">SB10FD</A></H2>
<H3>
H-infinity (sub)optimal state controller for a continuous-time system
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To compute the matrices of an H-infinity (sub)optimal n-state
controller
| AK | BK |
K = |----|----|,
| CK | DK |
using modified Glover's and Doyle's 1988 formulas, for the system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---|
| C1 | D11 D12 | | C | D |
| C2 | D21 D22 |
and for a given value of gamma, where B2 has as column size the
number of control inputs (NCON) and C2 has as row size the number
of measurements (NMEAS) being provided to the controller.
It is assumed that
(A1) (A,B2) is stabilizable and (C2,A) is detectable,
(A2) D12 is full column rank and D21 is full row rank,
(A3) | A-j*omega*I B2 | has full column rank for all omega,
| C1 D12 |
(A4) | A-j*omega*I B1 | has full row rank for all omega.
| C2 D21 |
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE SB10FD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
$ C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
$ DK, LDDK, RCOND, TOL, IWORK, DWORK, LDWORK,
$ BWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDWORK, M, N, NCON, NMEAS, NP
DOUBLE PRECISION GAMMA, TOL
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ RCOND( 4 )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0,
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0,
M-NCON >= NMEAS.
GAMMA (input) DOUBLE PRECISION
The value of gamma. It is assumed that gamma is
sufficiently large so that the controller is admissible.
GAMMA >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N part of this array contains the
controller state matrix AK.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
The leading N-by-NMEAS part of this array contains the
controller input matrix BK.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading NCON-by-N part of this array contains the
controller output matrix CK.
LDCK INTEGER
The leading dimension of the array CK.
LDCK >= max(1,NCON).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
The leading NCON-by-NMEAS part of this array contains the
controller input/output matrix DK.
LDDK INTEGER
The leading dimension of the array DK.
LDDK >= max(1,NCON).
RCOND (output) DOUBLE PRECISION array, dimension (4)
RCOND(1) contains the reciprocal condition number of the
control transformation matrix;
RCOND(2) contains the reciprocal condition number of the
measurement transformation matrix;
RCOND(3) contains an estimate of the reciprocal condition
number of the X-Riccati equation;
RCOND(4) contains an estimate of the reciprocal condition
number of the Y-Riccati equation.
</PRE>
<B>Tolerances</B>
<PRE>
TOL DOUBLE PRECISION
Tolerance used for controlling the accuracy of the applied
transformations for computing the normalized form in
SLICOT Library routine SB10PD. Transformation matrices
whose reciprocal condition numbers are less than TOL are
not allowed. If TOL <= 0, then a default value equal to
sqrt(EPS) is used, where EPS is the relative machine
precision.
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (LIWORK), where
LIWORK = max(2*max(N,M-NCON,NP-NMEAS,NCON),N*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= N*M + NP*(N+M) + M2*M2 + NP2*NP2 +
max(1,LW1,LW2,LW3,LW4,LW5,LW6), where
LW1 = (N+NP1+1)*(N+M2) + max(3*(N+M2)+N+NP1,5*(N+M2)),
LW2 = (N+NP2)*(N+M1+1) + max(3*(N+NP2)+N+M1,5*(N+NP2)),
LW3 = M2 + NP1*NP1 + max(NP1*max(N,M1),3*M2+NP1,5*M2),
LW4 = NP2 + M1*M1 + max(max(N,NP1)*M1,3*NP2+M1,5*NP2),
LW5 = 2*N*N + N*(M+NP) +
max(1,M*M + max(2*M1,3*N*N+max(N*M,10*N*N+12*N+5)),
NP*NP + max(2*NP1,3*N*N +
max(N*NP,10*N*N+12*N+5))),
LW6 = 2*N*N + N*(M+NP) +
max(1, M2*NP2 + NP2*NP2 + M2*M2 +
max(D1*D1 + max(2*D1, (D1+D2)*NP2),
D2*D2 + max(2*D2, D2*M2), 3*N,
N*(2*NP2 + M2) +
max(2*N*M2, M2*NP2 +
max(M2*M2+3*M2, NP2*(2*NP2+
M2+max(NP2,N)))))),
with D1 = NP1 - M2, D2 = M1 - NP2,
NP1 = NP - NP2, M1 = M - M2.
For good performance, LDWORK must generally be larger.
Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
2*Q*(3*Q+2*N)+max(1,(N+Q)*(N+Q+6),Q*(Q+max(N,Q,5)+1),
2*N*(N+2*Q)+max(1,4*Q*Q+
max(2*Q,3*N*N+max(2*N*Q,10*N*N+12*N+5)),
Q*(3*N+3*Q+max(2*N,4*Q+max(N,Q))))).
BWORK LOGICAL array, dimension (2*N)
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix | A-j*omega*I B2 | had not full
| C1 D12 |
column rank in respect to the tolerance EPS;
= 2: if the matrix | A-j*omega*I B1 | had not full row
| C2 D21 |
rank in respect to the tolerance EPS;
= 3: if the matrix D12 had not full column rank in
respect to the tolerance TOL;
= 4: if the matrix D21 had not full row rank in respect
to the tolerance TOL;
= 5: if the singular value decomposition (SVD) algorithm
did not converge (when computing the SVD of one of
the matrices |A B2 |, |A B1 |, D12 or D21).
|C1 D12| |C2 D21|
= 6: if the controller is not admissible (too small value
of gamma);
= 7: if the X-Riccati equation was not solved
successfully (the controller is not admissible or
there are numerical difficulties);
= 8: if the Y-Riccati equation was not solved
successfully (the controller is not admissible or
there are numerical difficulties);
= 9: if the determinant of Im2 + Tu*D11HAT*Ty*D22 is
zero [3].
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The routine implements the Glover's and Doyle's 1988 formulas [1],
[2] modified to improve the efficiency as described in [3].
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Glover, K. and Doyle, J.C.
State-space formulae for all stabilizing controllers that
satisfy an Hinf norm bound and relations to risk sensitivity.
Systems and Control Letters, vol. 11, pp. 167-172, 1988.
[2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
Smith, R.
mu-Analysis and Synthesis Toolbox.
The MathWorks Inc., Natick, Mass., 1995.
[3] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
Fortran 77 routines for Hinf and H2 design of continuous-time
linear control systems.
Rep. 98-14, Department of Engineering, Leicester University,
Leicester, U.K., 1998.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The accuracy of the result depends on the condition numbers of the
input and output transformations and on the condition numbers of
the two Riccati equations, as given by the values of RCOND(1),
RCOND(2), RCOND(3) and RCOND(4), respectively.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
* SB10FD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX, N2MAX
PARAMETER ( NMAX = 10, MMAX = 10, PMAX = 10, N2MAX = 20 )
INTEGER LDA, LDB, LDC, LDD, LDAK, LDBK, LDCK, LDDK,
$ LDAC, LDBC, LDCC, LDDC
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
$ LDAK = NMAX, LDBK = NMAX, LDCK = MMAX,
$ LDDK = MMAX, LDAC = 2*NMAX, LDBC = 2*NMAX,
$ LDCC = PMAX, LDDC = PMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( 2*MAX( NMAX, MMAX, PMAX ),
$ NMAX*NMAX ) )
INTEGER MPMX
PARAMETER ( MPMX = MAX( MMAX, PMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = 2*MPMX*( 3*MPMX + 2*NMAX ) +
$ MAX( ( NMAX + MPMX )*( NMAX + MPMX + 6 ),
$ MPMX*( MPMX + MAX( NMAX, MPMX, 5 ) + 1 ),
$ 2*NMAX*( NMAX + 2*MPMX ) +
$ MAX( 4*MPMX*MPMX + MAX( 2*MPMX, 3*NMAX*NMAX +
$ MAX( 2*NMAX*MPMX, 10*NMAX*NMAX+12*NMAX+5 ) ),
$ MPMX*( 3*NMAX + 3*MPMX +
$ MAX( 2*NMAX, 4*MPMX +
$ MAX( NMAX, MPMX ) ) ) ) ) )
* .. Local Scalars ..
INTEGER SDIM
LOGICAL SELECT
DOUBLE PRECISION GAMMA, TOL
INTEGER I, INFO1, INFO2, INFO3, J, M, N, NCON, NMEAS, NP
* .. Local Arrays ..
LOGICAL BWORK(N2MAX)
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), AK(LDAK,NMAX), AC(LDAC,N2MAX),
$ B(LDB,MMAX), BK(LDBK,PMAX), BC(LDBC,MMAX),
$ C(LDC,NMAX), CK(LDCK,NMAX), CC(LDCC,N2MAX),
$ D(LDD,MMAX), DK(LDDK,PMAX), DC(LDDC,MMAX),
$ DWORK(LDWORK), RCOND( 4 )
* .. External Subroutines ..
EXTERNAL SB10FD, SB10LD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, NP, NCON, NMEAS
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) N
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) M
ELSE IF ( NP.LT.0 .OR. NP.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99985 ) NP
ELSE IF ( NCON.LT.0 .OR. NCON.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99984 ) NCON
ELSE IF ( NMEAS.LT.0 .OR. NMEAS.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99983 ) NMEAS
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,NP )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,NP )
READ ( NIN, FMT = * ) GAMMA, TOL
* Compute the suboptimal controller
CALL SB10FD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
$ C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
$ DK, LDDK, RCOND, TOL, IWORK, DWORK, LDWORK,
$ BWORK, INFO1 )
*
IF ( INFO1.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99989 ) ( AK(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99989 ) ( BK(I,J), J = 1,NMEAS )
20 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 30 I = 1, NCON
WRITE ( NOUT, FMT = 99989 ) ( CK(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NCON
WRITE ( NOUT, FMT = 99989 ) ( DK(I,J), J = 1,NMEAS )
40 CONTINUE
WRITE( NOUT, FMT = 99992 )
WRITE( NOUT, FMT = 99988 ) ( RCOND(I), I = 1, 4 )
* Compute the closed-loop matrices
CALL SB10LD(N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
$ D, LDD, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
$ AC, LDAC, BC, LDBC, CC, LDCC, DC, LDDC, IWORK,
$ DWORK, LDWORK, INFO2 )
*
IF ( INFO2.EQ.0 ) THEN
* Compute the closed-loop poles
CALL DGEES( 'N','N', SELECT, 2*N, AC, LDAC, SDIM,
$ DWORK(1), DWORK(2*N+1), DWORK, 2*N,
$ DWORK(4*N+1), LDWORK-4*N, BWORK, INFO3)
*
IF( INFO3.EQ.0 ) THEN
WRITE( NOUT, FMT = 99991 )
WRITE( NOUT, FMT = 99988 ) (DWORK(I), I =1, 2*N)
WRITE( NOUT, FMT = 99990 )
WRITE( NOUT, FMT = 99988 ) (DWORK(2*N+I), I =1, 2*N)
ELSE
WRITE( NOUT, FMT = 99996 ) INFO3
END IF
ELSE
WRITE( NOUT, FMT = 99997 ) INFO2
END IF
ELSE
WRITE( NOUT, FMT = 99998 ) INFO1
END IF
END IF
STOP
*
99999 FORMAT (' SB10FD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10FD =',I2)
99997 FORMAT (/' INFO on exit from SB10LD =',I2)
99996 FORMAT (' The controller state matrix AK is'/)
99995 FORMAT (/' The controller input matrix BK is'/)
99994 FORMAT (/' The controller output matrix CK is'/)
99993 FORMAT (/' The controller matrix DK is'/)
99992 FORMAT (/' The estimated condition numbers are'/)
99991 FORMAT (/' The real parts of the closed-loop system poles are'/)
99990 FORMAT (/' The imaginary parts of the closed-loop system',
$ ' poles are'/)
99989 FORMAT (10(1X,F8.4))
99988 FORMAT ( 5(1X,D12.5))
99987 FORMAT (/' N is out of range.',/' N = ',I5)
99986 FORMAT (/' M is out of range.',/' M = ',I5)
99985 FORMAT (/' N is out of range.',/' N = ',I5)
99984 FORMAT (/' NCON is out of range.',/' NCON = ',I5)
99983 FORMAT (/' NMEAS is out of range.',/' NMEAS = ',I5)
END
</PRE>
<B>Program Data</B>
<PRE>
SB10FD EXAMPLE PROGRAM DATA
6 5 5 2 2
-1.0 0.0 4.0 5.0 -3.0 -2.0
-2.0 4.0 -7.0 -2.0 0.0 3.0
-6.0 9.0 -5.0 0.0 2.0 -1.0
-8.0 4.0 7.0 -1.0 -3.0 0.0
2.0 5.0 8.0 -9.0 1.0 -4.0
3.0 -5.0 8.0 0.0 2.0 -6.0
-3.0 -4.0 -2.0 1.0 0.0
2.0 0.0 1.0 -5.0 2.0
-5.0 -7.0 0.0 7.0 -2.0
4.0 -6.0 1.0 1.0 -2.0
-3.0 9.0 -8.0 0.0 5.0
1.0 -2.0 3.0 -6.0 -2.0
1.0 -1.0 2.0 -4.0 0.0 -3.0
-3.0 0.0 5.0 -1.0 1.0 1.0
-7.0 5.0 0.0 -8.0 2.0 -2.0
9.0 -3.0 4.0 0.0 3.0 7.0
0.0 1.0 -2.0 1.0 -6.0 -2.0
1.0 -2.0 -3.0 0.0 0.0
0.0 4.0 0.0 1.0 0.0
5.0 -3.0 -4.0 0.0 1.0
0.0 1.0 0.0 1.0 -3.0
0.0 0.0 1.0 7.0 1.0
15.0 0.00000001
</PRE>
<B>Program Results</B>
<PRE>
SB10FD EXAMPLE PROGRAM RESULTS
The controller state matrix AK is
-2.8043 14.7367 4.6658 8.1596 0.0848 2.5290
4.6609 3.2756 -3.5754 -2.8941 0.2393 8.2920
-15.3127 23.5592 -7.1229 2.7599 5.9775 -2.0285
-22.0691 16.4758 12.5523 -16.3602 4.4300 -3.3168
30.6789 -3.9026 -1.3868 26.2357 -8.8267 10.4860
-5.7429 0.0577 10.8216 -11.2275 1.5074 -10.7244
The controller input matrix BK is
-0.1581 -0.0793
-0.9237 -0.5718
0.7984 0.6627
0.1145 0.1496
-0.6743 -0.2376
0.0196 -0.7598
The controller output matrix CK is
-0.2480 -0.1713 -0.0880 0.1534 0.5016 -0.0730
2.8810 -0.3658 1.3007 0.3945 1.2244 2.5690
The controller matrix DK is
0.0554 0.1334
-0.3195 0.0333
The estimated condition numbers are
0.10000D+01 0.10000D+01 0.11241D-01 0.80492D-03
The real parts of the closed-loop system poles are
-0.10731D+03 -0.66556D+02 -0.38269D+02 -0.38269D+02 -0.20089D+02
-0.62557D+01 -0.62557D+01 -0.32405D+01 -0.32405D+01 -0.17178D+01
-0.41466D+01 -0.76437D+01
The imaginary parts of the closed-loop system poles are
0.00000D+00 0.00000D+00 0.13114D+02 -0.13114D+02 0.00000D+00
0.12961D+02 -0.12961D+02 0.67998D+01 -0.67998D+01 0.00000D+00
0.00000D+00 0.00000D+00
</PRE>
<HR>
<p>
<A HREF=..\libindex.html><B>Return to index</B></A></BODY>
</HTML>
|
