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Contents of /trunk/solvers/dopri5/dopri5.c

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Revision 1733 - (show annotations) (download) (as text)
Wed Feb 6 00:18:31 2008 UTC (12 years, 7 months ago) by jpye
File MIME type: text/x-csrc
File size: 20374 byte(s)
In process of moving DOPRI5 solver to 'solvers' dir.
1 /*
2 This is a modified form of the DOPRI5 code, originally written in Fortran 77
3 by E Hairer and G Wanner, then translated to C by J Colinge.
4 Changes have been made to suite ASCEND requirements. Minor only.
5 -- John Pye, May 2007.
6 *//*
7 DOPRI5
8 Copyright (c) 2004, Ernst Hairer
9
10 Redistribution and use in source and binary forms, with or without
11 modification, are permitted provided that the following conditions are
12 met:
13
14 - Redistributions of source code must retain the above copyright
15 notice, this list of conditions and the following disclaimer.
16
17 - Redistributions in binary form must reproduce the above copyright
18 notice, this list of conditions and the following disclaimer in the
19 documentation and/or other materials provided with the distribution.
20
21 THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS “AS
22 IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
23 TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
24 PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR
25 CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
26 EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
27 PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
28 PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
29 LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
30 NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
31 SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
32 */
33
34 #include <math.h>
35 #include <stdio.h>
36 #include <stdlib.h>
37 #include <limits.h>
38 #include <memory.h>
39 #include "dopri5.h"
40
41
42 static long nfcn, nstep, naccpt, nrejct;
43 static double hout, xold, xout;
44 static unsigned nrds, *indir;
45 static double *yy1, *k1, *k2, *k3, *k4, *k5, *k6, *ysti;
46 static double *rcont1, *rcont2, *rcont3, *rcont4, *rcont5;
47
48
49 long nfcnRead (void){
50 return nfcn;
51
52 } /* nfcnRead */
53
54
55 long nstepRead (void){
56 return nstep;
57
58 } /* stepRead */
59
60
61 long naccptRead (void){
62 return naccpt;
63
64 } /* naccptRead */
65
66
67 long nrejctRead (void){
68 return nrejct;
69
70 } /* nrejct */
71
72
73 double hRead (void){
74 return hout;
75
76 } /* hRead */
77
78
79 double xRead (void){
80 return xout;
81
82 } /* xRead */
83
84
85 static double sign (double a, double b){
86 return (b > 0.0) ? fabs(a) : -fabs(a);
87
88 } /* sign */
89
90
91 static double min_d (double a, double b){
92 return (a < b)?a:b;
93
94 } /* min_d */
95
96
97 static double max_d (double a, double b){
98 return (a > b)?a:b;
99
100 } /* max_d */
101
102
103 static double hinit (unsigned n, FcnEqDiff *fcn, double x, double* y,
104 double posneg, double* f0, double* f1, double* yy1, int iord,
105 double hmax, double* atoler, double* rtoler, int itoler
106 ,void *user_data
107 ){
108 double dnf, dny, atoli, rtoli, sk, h, h1, der2, der12, sqr;
109 unsigned i;
110
111 dnf = 0.0;
112 dny = 0.0;
113 atoli = atoler[0];
114 rtoli = rtoler[0];
115
116 if (!itoler)
117 for (i = 0; i < n; i++)
118 {
119 sk = atoli + rtoli * fabs(y[i]);
120 sqr = f0[i] / sk;
121 dnf += sqr*sqr;
122 sqr = y[i] / sk;
123 dny += sqr*sqr;
124 }
125 else
126 for (i = 0; i < n; i++)
127 {
128 sk = atoler[i] + rtoler[i] * fabs(y[i]);
129 sqr = f0[i] / sk;
130 dnf += sqr*sqr;
131 sqr = y[i] / sk;
132 dny += sqr*sqr;
133 }
134
135 if ((dnf <= 1.0E-10) || (dny <= 1.0E-10))
136 h = 1.0E-6;
137 else
138 h = sqrt (dny/dnf) * 0.01;
139
140 h = min_d (h, hmax);
141 h = sign (h, posneg);
142
143 /* perform an explicit Euler step */
144 for (i = 0; i < n; i++)
145 yy1[i] = y[i] + h * f0[i];
146 fcn (n, x+h, yy1, f1, user_data);
147
148 /* estimate the second derivative of the solution */
149 der2 = 0.0;
150 if (!itoler)
151 for (i = 0; i < n; i++)
152 {
153 sk = atoli + rtoli * fabs(y[i]);
154 sqr = (f1[i] - f0[i]) / sk;
155 der2 += sqr*sqr;
156 }
157 else
158 for (i = 0; i < n; i++)
159 {
160 sk = atoler[i] + rtoler[i] * fabs(y[i]);
161 sqr = (f1[i] - f0[i]) / sk;
162 der2 += sqr*sqr;
163 }
164 der2 = sqrt (der2) / h;
165
166 /* step size is computed such that h**iord * max_d(norm(f0),norm(der2)) = 0.01 */
167 der12 = max_d (fabs(der2), sqrt(dnf));
168 if (der12 <= 1.0E-15)
169 h1 = max_d (1.0E-6, fabs(h)*1.0E-3);
170 else
171 h1 = pow (0.01/der12, 1.0/(double)iord);
172 h = min_d (100.0 * h, min_d (h1, hmax));
173
174 return sign (h, posneg);
175
176 } /* hinit */
177
178
179 /* core integrator */
180 static int dopcor (unsigned n, FcnEqDiff *fcn, double x, double* y, double xend,
181 double hmax, double h, double* rtoler, double* atoler,
182 int itoler, FILE* fileout, SolTrait *solout, int iout,
183 long nmax, double uround, int meth, long nstiff, double safe,
184 double beta, double fac1, double fac2, unsigned* icont
185 , void *user_data
186 ){
187 double facold, expo1, fac, facc1, facc2, fac11, posneg, xph;
188 double atoli, rtoli, hlamb, err, sk, hnew, yd0, ydiff, bspl;
189 double stnum, stden, sqr;
190 int iasti, iord, irtrn, reject, last, nonsti;
191 unsigned i, j;
192 double c2, c3, c4, c5, e1, e3, e4, e5, e6, e7, d1, d3, d4, d5, d6, d7;
193 double a21, a31, a32, a41, a42, a43, a51, a52, a53, a54;
194 double a61, a62, a63, a64, a65, a71, a73, a74, a75, a76;
195
196 /* initialisations */
197 switch (meth)
198 {
199 case 1:
200
201 c2=0.2, c3=0.3, c4=0.8, c5=8.0/9.0;
202 a21=0.2, a31=3.0/40.0, a32=9.0/40.0;
203 a41=44.0/45.0, a42=-56.0/15.0; a43=32.0/9.0;
204 a51=19372.0/6561.0, a52=-25360.0/2187.0;
205 a53=64448.0/6561.0, a54=-212.0/729.0;
206 a61=9017.0/3168.0, a62=-355.0/33.0, a63=46732.0/5247.0;
207 a64=49.0/176.0, a65=-5103.0/18656.0;
208 a71=35.0/384.0, a73=500.0/1113.0, a74=125.0/192.0;
209 a75=-2187.0/6784.0, a76=11.0/84.0;
210 e1=71.0/57600.0, e3=-71.0/16695.0, e4=71.0/1920.0;
211 e5=-17253.0/339200.0, e6=22.0/525.0, e7=-1.0/40.0;
212 d1=-12715105075.0/11282082432.0, d3=87487479700.0/32700410799.0;
213 d4=-10690763975.0/1880347072.0, d5=701980252875.0/199316789632.0;
214 d6=-1453857185.0/822651844.0, d7=69997945.0/29380423.0;
215
216 break;
217 }
218
219 facold = 1.0E-4;
220 expo1 = 0.2 - beta * 0.75;
221 facc1 = 1.0 / fac1;
222 facc2 = 1.0 / fac2;
223 posneg = sign (1.0, xend-x);
224
225 /* initial preparations */
226 atoli = atoler[0];
227 rtoli = rtoler[0];
228 last = 0;
229 hlamb = 0.0;
230 iasti = 0;
231 fcn(n, x, y, k1, user_data);
232 hmax = fabs (hmax);
233 iord = 5;
234 if (h == 0.0)
235 h = hinit (n, fcn, x, y, posneg, k1, k2, k3, iord, hmax, atoler, rtoler, itoler, user_data);
236 nfcn += 2;
237 reject = 0;
238 xold = x;
239 if (iout)
240 {
241 irtrn = 1;
242 hout = h;
243 xout = x;
244 solout (naccpt+1, xold, x, y, n, &irtrn, user_data);
245 if (irtrn < 0)
246 {
247 if (fileout)
248 fprintf (fileout, "Exit of dopri5 at x = %.16e\r\n", x);
249 return 2;
250 }
251 }
252
253 /* basic integration step */
254 while (1)
255 {
256 if (nstep > nmax)
257 {
258 if (fileout)
259 fprintf (fileout, "Exit of dopri5 at x = %.16e, more than nmax = %li are needed\r\n", x, nmax);
260 xout = x;
261 hout = h;
262 return -2;
263 }
264
265 if (0.1 * fabs(h) <= fabs(x) * uround)
266 {
267 if (fileout)
268 fprintf (fileout, "Exit of dopri5 at x = %.16e, step size too small h = %.16e\r\n", x, h);
269 xout = x;
270 hout = h;
271 return -3;
272 }
273
274 if ((x + 1.01*h - xend) * posneg > 0.0)
275 {
276 h = xend - x;
277 last = 1;
278 }
279
280 nstep++;
281
282 /* the first 6 stages */
283 for (i = 0; i < n; i++)
284 yy1[i] = y[i] + h * a21 * k1[i];
285 fcn (n, x+c2*h, yy1, k2, user_data);
286 for (i = 0; i < n; i++)
287 yy1[i] = y[i] + h * (a31*k1[i] + a32*k2[i]);
288 fcn (n, x+c3*h, yy1, k3, user_data);
289 for (i = 0; i < n; i++)
290 yy1[i] = y[i] + h * (a41*k1[i] + a42*k2[i] + a43*k3[i]);
291 fcn (n, x+c4*h, yy1, k4, user_data);
292 for (i = 0; i <n; i++)
293 yy1[i] = y[i] + h * (a51*k1[i] + a52*k2[i] + a53*k3[i] + a54*k4[i]);
294 fcn (n, x+c5*h, yy1, k5, user_data);
295 for (i = 0; i < n; i++)
296 ysti[i] = y[i] + h * (a61*k1[i] + a62*k2[i] + a63*k3[i] + a64*k4[i] + a65*k5[i]);
297 xph = x + h;
298 fcn (n, xph, ysti, k6, user_data);
299 for (i = 0; i < n; i++)
300 yy1[i] = y[i] + h * (a71*k1[i] + a73*k3[i] + a74*k4[i] + a75*k5[i] + a76*k6[i]);
301 fcn (n, xph, yy1, k2, user_data);
302 if (iout == 2){
303 if (nrds == n){
304 for (i = 0; i < n; i++)
305 {
306 rcont5[i] = h * (d1*k1[i] + d3*k3[i] + d4*k4[i] + d5*k5[i] + d6*k6[i] + d7*k2[i]);
307 }
308 }else{
309 for (j = 0; j < nrds; j++)
310 {
311 i = icont[j];
312 rcont5[j] = h * (d1*k1[i] + d3*k3[i] + d4*k4[i] + d5*k5[i] + d6*k6[i] + d7*k2[i]);
313 }
314 }
315 }
316 for (i = 0; i < n; i++)
317 k4[i] = h * (e1*k1[i] + e3*k3[i] + e4*k4[i] + e5*k5[i] + e6*k6[i] + e7*k2[i]);
318 nfcn += 6;
319
320 /* error estimation */
321 err = 0.0;
322 if (!itoler)
323 for (i = 0; i < n; i++)
324 {
325 sk = atoli + rtoli * max_d (fabs(y[i]), fabs(yy1[i]));
326 sqr = k4[i] / sk;
327 err += sqr*sqr;
328 }
329 else
330 for (i = 0; i < n; i++)
331 {
332 sk = atoler[i] + rtoler[i] * max_d (fabs(y[i]), fabs(yy1[i]));
333 sqr = k4[i] / sk;
334 err += sqr*sqr;
335 }
336 err = sqrt (err / (double)n);
337
338 /* computation of hnew */
339 fac11 = pow (err, expo1);
340 /* Lund-stabilization */
341 fac = fac11 / pow(facold,beta);
342 /* we require fac1 <= hnew/h <= fac2 */
343 fac = max_d (facc2, min_d (facc1, fac/safe));
344 hnew = h / fac;
345
346 if (err <= 1.0)
347 {
348 /* step accepted */
349
350 facold = max_d (err, 1.0E-4);
351 naccpt++;
352
353 /* stiffness detection */
354 if (!(naccpt % nstiff) || (iasti > 0))
355 {
356 stnum = 0.0;
357 stden = 0.0;
358 for (i = 0; i < n; i++)
359 {
360 sqr = k2[i] - k6[i];
361 stnum += sqr*sqr;
362 sqr = yy1[i] - ysti[i];
363 stden += sqr*sqr;
364 }
365 if (stden > 0.0){
366 hlamb = h * sqrt (stnum / stden);
367 }
368
369 if (hlamb > 3.25){
370 nonsti = 0;
371 iasti++;
372 if (iasti == 15){
373 if (fileout){
374 fprintf (fileout, "The problem seems to become stiff at x = %.16e\r\n", x);
375 }else{
376 xout = x;
377 hout = h;
378 return -4;
379 }
380 }
381 }
382 else
383 {
384 nonsti++;
385 if (nonsti == 6)
386 iasti = 0;
387 }
388 }
389
390 if (iout == 2){
391 if (nrds == n){
392 for (i = 0; i < n; i++)
393 {
394 yd0 = y[i];
395 ydiff = yy1[i] - yd0;
396 bspl = h * k1[i] - ydiff;
397 rcont1[i] = y[i];
398 rcont2[i] = ydiff;
399 rcont3[i] = bspl;
400 rcont4[i] = -h * k2[i] + ydiff - bspl;
401 }
402 }else{
403 for (j = 0; j < nrds; j++)
404 {
405 i = icont[j];
406 yd0 = y[i];
407 ydiff = yy1[i] - yd0;
408 bspl = h * k1[i] - ydiff;
409 rcont1[j] = y[i];
410 rcont2[j] = ydiff;
411 rcont3[j] = bspl;
412 rcont4[j] = -h * k2[i] + ydiff - bspl;
413 }
414 }
415 }
416 memcpy (k1, k2, n * sizeof(double));
417 memcpy (y, yy1, n * sizeof(double));
418 xold = x;
419 x = xph;
420
421 if (iout)
422 {
423 hout = h;
424 xout = x;
425 solout (naccpt+1, xold, x, y, n, &irtrn, user_data);
426 if (irtrn < 0)
427 {
428 if (fileout)
429 fprintf (fileout, "Exit of dopri5 at x = %.16e\r\n", x);
430 return 2;
431 }
432 }
433
434 /* normal exit */
435 if (last)
436 {
437 hout=hnew;
438 xout = x;
439 return 1;
440 }
441
442 if (fabs(hnew) > hmax)
443 hnew = posneg * hmax;
444 if (reject)
445 hnew = posneg * min_d (fabs(hnew), fabs(h));
446
447 reject = 0;
448 }
449 else
450 {
451 /* step rejected */
452 hnew = h / min_d (facc1, fac11/safe);
453 reject = 1;
454 if (naccpt >= 1)
455 nrejct=nrejct + 1;
456 last = 0;
457 }
458
459 h = hnew;
460 }
461
462 } /* dopcor */
463
464
465 /* front-end */
466 int dopri5(
467 unsigned n, FcnEqDiff *fcn, double x, double* y, double xend, double* rtoler,
468 double* atoler, int itoler, SolTrait *solout, int iout, FILE* fileout, double uround,
469 double safe, double fac1, double fac2, double beta, double hmax, double h,
470 long nmax, int meth, long nstiff, unsigned nrdens, unsigned* icont, unsigned licont
471 , void *user_data
472 ){
473 int arret, idid;
474 unsigned i;
475
476 /* initialisations */
477 nfcn = nstep = naccpt = nrejct = arret = 0;
478 rcont1 = rcont2 = rcont3 = rcont4 = rcont5 = NULL;
479 indir = NULL;
480
481 /* n, the dimension of the system */
482 if (n == UINT_MAX)
483 {
484 if (fileout)
485 fprintf (fileout, "System too big, max. n = %u\r\n", UINT_MAX-1);
486 arret = 1;
487 }
488
489 /* nmax, the maximal number of steps */
490 if (!nmax)
491 nmax = 100000;
492 else if (nmax <= 0)
493 {
494 if (fileout)
495 fprintf (fileout, "Wrong input, nmax = %li\r\n", nmax);
496 arret = 1;
497 }
498
499 /* meth, coefficients of the method */
500 if (!meth)
501 meth = 1;
502 else if ((meth <= 0) || (meth >= 2))
503 {
504 if (fileout)
505 fprintf (fileout, "Curious input, meth = %i\r\n", meth);
506 arret = 1;
507 }
508
509 /* nstiff, parameter for stiffness detection */
510 if (!nstiff)
511 nstiff = 1000;
512 else if (nstiff < 0)
513 nstiff = nmax + 10;
514
515 /* iout, switch for calling solout */
516 if ((iout < 0) || (iout > 2))
517 {
518 if (fileout)
519 fprintf (fileout, "Wrong input, iout = %i\r\n", iout);
520 arret = 1;
521 }
522
523 /* nrdens, number of dense output components */
524 if (nrdens > n)
525 {
526 if (fileout)
527 fprintf (fileout, "Curious input, nrdens = %u\r\n", nrdens);
528 arret = 1;
529 }
530 else if (nrdens)
531 {
532 /* is there enough memory to allocate rcont12345&indir ? */
533 rcont1 = (double*) malloc (nrdens*sizeof(double));
534 rcont2 = (double*) malloc (nrdens*sizeof(double));
535 rcont3 = (double*) malloc (nrdens*sizeof(double));
536 rcont4 = (double*) malloc (nrdens*sizeof(double));
537 rcont5 = (double*) malloc (nrdens*sizeof(double));
538 if (nrdens < n)
539 indir = (unsigned*) malloc (n*sizeof(unsigned));
540
541 if (!rcont1 || !rcont2 || !rcont3 || !rcont4 || !rcont5 || (!indir && (nrdens < n)))
542 {
543 if (fileout)
544 fprintf (fileout, "Not enough free memory for rcont12345&indir\r\n");
545 arret = 1;
546 }
547
548 /* control of length of icont */
549 if (nrdens == n)
550 {
551 if (icont && fileout)
552 fprintf (fileout, "Warning : when nrdens = n there is no need allocating memory for icont\r\n");
553 nrds = n;
554 }
555 else if (licont < nrdens)
556 {
557 if (fileout)
558 fprintf (fileout, "Insufficient storage for icont, min. licont = %u\r\n", nrdens);
559 arret = 1;
560 }
561 else
562 {
563 if ((iout < 2) && fileout)
564 fprintf (fileout, "Warning : put iout = 2 for dense output\r\n");
565 nrds = nrdens;
566 for (i = 0; i < n; i++)
567 indir[i] = UINT_MAX;
568 for (i = 0; i < nrdens; i++)
569 indir[icont[i]] = i;
570 }
571 }
572
573 /* uround, smallest number satisfying 1.0+uround > 1.0 */
574 if (uround == 0.0)
575 uround = 2.3E-16;
576 else if ((uround <= 1.0E-35) || (uround >= 1.0))
577 {
578 if (fileout)
579 fprintf (fileout, "Which machine do you have ? Your uround was : %.16e\r\n", uround);
580 arret = 1;
581 }
582
583 /* safety factor */
584 if (safe == 0.0)
585 safe = 0.9;
586 else if ((safe >= 1.0) || (safe <= 1.0E-4))
587 {
588 if (fileout)
589 fprintf (fileout, "Curious input for safety factor, safe = %.16e\r\n", safe);
590 arret = 1;
591 }
592
593 /* fac1, fac2, parameters for step size selection */
594 if (fac1 == 0.0)
595 fac1 = 0.2;
596 if (fac2 == 0.0)
597 fac2 = 10.0;
598
599 /* beta for step control stabilization */
600 if (beta == 0.0)
601 beta = 0.04;
602 else if (beta < 0.0)
603 beta = 0.0;
604 else if (beta > 0.2)
605 {
606 if (fileout)
607 fprintf (fileout, "Curious input for beta : beta = %.16e\r\n", beta);
608 arret = 1;
609 }
610
611 /* maximal step size */
612 if (hmax == 0.0)
613 hmax = xend - x;
614
615 /* is there enough free memory for the method ? */
616 yy1 = (double*) malloc (n*sizeof(double));
617 k1 = (double*) malloc (n*sizeof(double));
618 k2 = (double*) malloc (n*sizeof(double));
619 k3 = (double*) malloc (n*sizeof(double));
620 k4 = (double*) malloc (n*sizeof(double));
621 k5 = (double*) malloc (n*sizeof(double));
622 k6 = (double*) malloc (n*sizeof(double));
623 ysti = (double*) malloc (n*sizeof(double));
624
625 if (!yy1 || !k1 || !k2 || !k3 || !k4 || !k5 || !k6 || !ysti)
626 {
627 if (fileout)
628 fprintf (fileout, "Not enough free memory for the method\r\n");
629 arret = 1;
630 }
631
632 /* when a failure has occured, we return -1 */
633 if (arret)
634 {
635 if (ysti)
636 free (ysti);
637 if (k6)
638 free (k6);
639 if (k5)
640 free (k5);
641 if (k4)
642 free (k4);
643 if (k3)
644 free (k3);
645 if (k2)
646 free (k2);
647 if (k1)
648 free (k1);
649 if (yy1)
650 free (yy1);
651 if (indir)
652 free (indir);
653 if (rcont5)
654 free (rcont5);
655 if (rcont4)
656 free (rcont4);
657 if (rcont3)
658 free (rcont3);
659 if (rcont2)
660 free (rcont2);
661 if (rcont1)
662 free (rcont1);
663
664 return -1;
665 }
666 else
667 {
668 idid = dopcor (n, fcn, x, y, xend, hmax, h, rtoler, atoler, itoler, fileout
669 , solout, iout, nmax, uround, meth, nstiff, safe, beta, fac1, fac2, icont
670 , user_data
671 );
672 free (ysti);
673 free (k6);
674 free (k5); /* reverse order freeing too increase chances */
675 free (k4); /* of efficient dynamic memory managing */
676 free (k3);
677 free (k2);
678 free (k1);
679 free (yy1);
680 if (indir)
681 free (indir);
682 if (rcont5)
683 {
684 free (rcont5);
685 free (rcont4);
686 free (rcont3);
687 free (rcont2);
688 free (rcont1);
689 }
690
691 return idid;
692 }
693
694 } /* dopri5 */
695
696
697 /* dense output function */
698 double contd5 (unsigned ii, double x){
699 unsigned i;
700 double theta, theta1;
701
702 i = UINT_MAX;
703
704 if (!indir)
705 i = ii;
706 else
707 i = indir[ii];
708
709 if (i == UINT_MAX)
710 {
711 printf ("No dense output available for %uth component", ii);
712 return 0.0;
713 }
714
715 theta = (x - xold) / hout;
716 theta1 = 1.0 - theta;
717
718 return rcont1[i] + theta*(rcont2[i] + theta1*(rcont3[i] + theta*(rcont4[i] + theta1*rcont5[i])));
719
720 } /* contd5 */
721

john.pye@anu.edu.au
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