Actual source code: morethuente.c

petsc-3.6.1 2015-08-06
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  1: #include <petsc/private/taolinesearchimpl.h>
  2: #include <../src/tao/linesearch/impls/morethuente/morethuente.h>

  4: /*
  5:    This algorithm is taken from More' and Thuente, "Line search algorithms
  6:    with guaranteed sufficient decrease", Argonne National Laboratory,
  7:    Technical Report MCS-P330-1092.
  8: */

 10: static PetscErrorCode Tao_mcstep(TaoLineSearch ls,PetscReal *stx,PetscReal *fx,PetscReal *dx,PetscReal *sty,PetscReal *fy,PetscReal *dy,PetscReal *stp,PetscReal *fp,PetscReal *dp);

 14: static PetscErrorCode TaoLineSearchDestroy_MT(TaoLineSearch ls)
 15: {
 16:   PetscErrorCode   ierr;
 17:   TaoLineSearch_MT *mt;

 21:   mt = (TaoLineSearch_MT*)(ls->data);
 22:   if (mt->x) {
 23:     PetscObjectDereference((PetscObject)mt->x);
 24:   }
 25:   VecDestroy(&mt->work);
 26:   PetscFree(ls->data);
 27:   return(0);
 28: }

 32: static PetscErrorCode TaoLineSearchSetFromOptions_MT(PetscOptions *PetscOptionsObject,TaoLineSearch ls)
 33: {
 36:   return(0);
 37: }

 41: static PetscErrorCode TaoLineSearchView_MT(TaoLineSearch ls, PetscViewer pv)
 42: {
 44:   PetscBool      isascii;

 47:   PetscObjectTypeCompare((PetscObject)pv, PETSCVIEWERASCII, &isascii);
 48:   if (isascii) {
 49:     PetscViewerASCIIPrintf(pv,"  maxf=%D, ftol=%g, gtol=%g\n",ls->max_funcs,(double)ls->rtol,(double)ls->ftol);
 50:   }
 51:   return(0);
 52: }

 56: /* @ TaoApply_LineSearch - This routine takes step length of 1.0.

 58:    Input Parameters:
 59: +  tao - Tao context
 60: .  X - current iterate (on output X contains new iterate, X + step*S)
 61: .  f - objective function evaluated at X
 62: .  G - gradient evaluated at X
 63: -  D - search direction


 66:    Info is set to 0.

 68: @ */

 70: static PetscErrorCode TaoLineSearchApply_MT(TaoLineSearch ls, Vec x, PetscReal *f, Vec g, Vec s)
 71: {
 72:   PetscErrorCode   ierr;
 73:   TaoLineSearch_MT *mt;

 75:   PetscReal        xtrapf = 4.0;
 76:   PetscReal        finit, width, width1, dginit, fm, fxm, fym, dgm, dgxm, dgym;
 77:   PetscReal        dgx, dgy, dg, dg2, fx, fy, stx, sty, dgtest;
 78:   PetscReal        ftest1=0.0, ftest2=0.0;
 79:   PetscInt         i, stage1,n1,n2,nn1,nn2;
 80:   PetscReal        bstepmin1, bstepmin2, bstepmax;
 81:   PetscBool        g_computed=PETSC_FALSE; /* to prevent extra gradient computation */


 90:   /* comm,type,size checks are done in interface TaoLineSearchApply */
 91:   mt = (TaoLineSearch_MT*)(ls->data);
 92:   ls->reason = TAOLINESEARCH_CONTINUE_ITERATING;

 94:   /* Check work vector */
 95:   if (!mt->work) {
 96:     VecDuplicate(x,&mt->work);
 97:     mt->x = x;
 98:     PetscObjectReference((PetscObject)mt->x);
 99:   } else if (x != mt->x) {
100:     VecDestroy(&mt->work);
101:     VecDuplicate(x,&mt->work);
102:     PetscObjectDereference((PetscObject)mt->x);
103:     mt->x = x;
104:     PetscObjectReference((PetscObject)mt->x);
105:   }

107:   if (ls->bounded) {
108:     /* Compute step length needed to make all variables equal a bound */
109:     /* Compute the smallest steplength that will make one nonbinding variable
110:      equal the bound */
111:     VecGetLocalSize(ls->upper,&n1);
112:     VecGetLocalSize(mt->x, &n2);
113:     VecGetSize(ls->upper,&nn1);
114:     VecGetSize(mt->x,&nn2);
115:     if (n1 != n2 || nn1 != nn2) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Variable vector not compatible with bounds vector");
116:     VecScale(s,-1.0);
117:     VecBoundGradientProjection(s,x,ls->lower,ls->upper,s);
118:     VecScale(s,-1.0);
119:     VecStepBoundInfo(x,s,ls->lower,ls->upper,&bstepmin1,&bstepmin2,&bstepmax);
120:     ls->stepmax = PetscMin(bstepmax,1.0e15);
121:   }

123:   VecDot(g,s,&dginit);
124:   if (PetscIsInfOrNanReal(dginit)) {
125:     PetscInfo1(ls,"Initial Line Search step * g is Inf or Nan (%g)\n",(double)dginit);
126:     ls->reason=TAOLINESEARCH_FAILED_INFORNAN;
127:     return(0);
128:   }
129:   if (dginit >= 0.0) {
130:     PetscInfo1(ls,"Initial Line Search step * g is not descent direction (%g)\n",(double)dginit);
131:     ls->reason = TAOLINESEARCH_FAILED_ASCENT;
132:     return(0);
133:   }


136:   /* Initialization */
137:   mt->bracket = 0;
138:   stage1 = 1;
139:   finit = *f;
140:   dgtest = ls->ftol * dginit;
141:   width = ls->stepmax - ls->stepmin;
142:   width1 = width * 2.0;
143:   VecCopy(x,mt->work);
144:   /* Variable dictionary:
145:    stx, fx, dgx - the step, function, and derivative at the best step
146:    sty, fy, dgy - the step, function, and derivative at the other endpoint
147:    of the interval of uncertainty
148:    step, f, dg - the step, function, and derivative at the current step */

150:   stx = 0.0;
151:   fx  = finit;
152:   dgx = dginit;
153:   sty = 0.0;
154:   fy  = finit;
155:   dgy = dginit;

157:   ls->step=ls->initstep;
158:   for (i=0; i< ls->max_funcs; i++) {
159:     /* Set min and max steps to correspond to the interval of uncertainty */
160:     if (mt->bracket) {
161:       ls->stepmin = PetscMin(stx,sty);
162:       ls->stepmax = PetscMax(stx,sty);
163:     } else {
164:       ls->stepmin = stx;
165:       ls->stepmax = ls->step + xtrapf * (ls->step - stx);
166:     }

168:     /* Force the step to be within the bounds */
169:     ls->step = PetscMax(ls->step,ls->stepmin);
170:     ls->step = PetscMin(ls->step,ls->stepmax);

172:     /* If an unusual termination is to occur, then let step be the lowest
173:      point obtained thus far */
174:     if ((stx!=0) && (((mt->bracket) && (ls->step <= ls->stepmin || ls->step >= ls->stepmax)) || ((mt->bracket) && (ls->stepmax - ls->stepmin <= ls->rtol * ls->stepmax)) ||
175:                      ((ls->nfeval+ls->nfgeval) >= ls->max_funcs - 1) || (mt->infoc == 0))) {
176:       ls->step = stx;
177:     }

179:     VecCopy(x,mt->work);
180:     VecAXPY(mt->work,ls->step,s);   /* W = X + step*S */

182:     if (ls->bounded) {
183:       VecMedian(ls->lower, mt->work, ls->upper, mt->work);
184:     }
185:     if (ls->usegts) {
186:       TaoLineSearchComputeObjectiveAndGTS(ls,mt->work,f,&dg);
187:       g_computed=PETSC_FALSE;
188:     } else {
189:       TaoLineSearchComputeObjectiveAndGradient(ls,mt->work,f,g);
190:       g_computed=PETSC_TRUE;
191:       if (ls->bounded) {
192:         VecDot(g,x,&dg);
193:         VecDot(g,mt->work,&dg2);
194:         dg = (dg2 - dg)/ls->step;
195:       } else {
196:         VecDot(g,s,&dg);
197:       }
198:     }

200:     if (0 == i) {
201:       ls->f_fullstep=*f;
202:     }

204:     if (PetscIsInfOrNanReal(*f) || PetscIsInfOrNanReal(dg)) {
205:       /* User provided compute function generated Not-a-Number, assume
206:        domain violation and set function value and directional
207:        derivative to infinity. */
208:       *f = PETSC_INFINITY;
209:       dg = PETSC_INFINITY;
210:     }

212:     ftest1 = finit + ls->step * dgtest;
213:     if (ls->bounded) {
214:       ftest2 = finit + ls->step * dgtest * ls->ftol;
215:     }
216:     /* Convergence testing */
217:     if (((*f - ftest1 <= 1.0e-10 * PetscAbsReal(finit)) &&  (PetscAbsReal(dg) + ls->gtol*dginit <= 0.0))) {
218:       PetscInfo(ls, "Line search success: Sufficient decrease and directional deriv conditions hold\n");
219:       ls->reason = TAOLINESEARCH_SUCCESS;
220:       break;
221:     }

223:     /* Check Armijo if beyond the first breakpoint */
224:     if (ls->bounded && (*f <= ftest2) && (ls->step >= bstepmin2)) {
225:       PetscInfo(ls,"Line search success: Sufficient decrease.\n");
226:       ls->reason = TAOLINESEARCH_SUCCESS;
227:       break;
228:     }

230:     /* Checks for bad cases */
231:     if (((mt->bracket) && (ls->step <= ls->stepmin||ls->step >= ls->stepmax)) || (!mt->infoc)) {
232:       PetscInfo(ls,"Rounding errors may prevent further progress.  May not be a step satisfying\n");
233:       PetscInfo(ls,"sufficient decrease and curvature conditions. Tolerances may be too small.\n");
234:       ls->reason = TAOLINESEARCH_HALTED_OTHER;
235:       break;
236:     }
237:     if ((ls->step == ls->stepmax) && (*f <= ftest1) && (dg <= dgtest)) {
238:       PetscInfo1(ls,"Step is at the upper bound, stepmax (%g)\n",(double)ls->stepmax);
239:       ls->reason = TAOLINESEARCH_HALTED_UPPERBOUND;
240:       break;
241:     }
242:     if ((ls->step == ls->stepmin) && (*f >= ftest1) && (dg >= dgtest)) {
243:       PetscInfo1(ls,"Step is at the lower bound, stepmin (%g)\n",(double)ls->stepmin);
244:       ls->reason = TAOLINESEARCH_HALTED_LOWERBOUND;
245:       break;
246:     }
247:     if ((mt->bracket) && (ls->stepmax - ls->stepmin <= ls->rtol*ls->stepmax)){
248:       PetscInfo1(ls,"Relative width of interval of uncertainty is at most rtol (%g)\n",(double)ls->rtol);
249:       ls->reason = TAOLINESEARCH_HALTED_RTOL;
250:       break;
251:     }

253:     /* In the first stage, we seek a step for which the modified function
254:      has a nonpositive value and nonnegative derivative */
255:     if ((stage1) && (*f <= ftest1) && (dg >= dginit * PetscMin(ls->ftol, ls->gtol))) {
256:       stage1 = 0;
257:     }

259:     /* A modified function is used to predict the step only if we
260:      have not obtained a step for which the modified function has a
261:      nonpositive function value and nonnegative derivative, and if a
262:      lower function value has been obtained but the decrease is not
263:      sufficient */

265:     if ((stage1) && (*f <= fx) && (*f > ftest1)) {
266:       fm   = *f - ls->step * dgtest;    /* Define modified function */
267:       fxm  = fx - stx * dgtest;         /* and derivatives */
268:       fym  = fy - sty * dgtest;
269:       dgm  = dg - dgtest;
270:       dgxm = dgx - dgtest;
271:       dgym = dgy - dgtest;

273:       /* if (dgxm * (ls->step - stx) >= 0.0) */
274:       /* Update the interval of uncertainty and compute the new step */
275:       Tao_mcstep(ls,&stx,&fxm,&dgxm,&sty,&fym,&dgym,&ls->step,&fm,&dgm);

277:       fx  = fxm + stx * dgtest; /* Reset the function and */
278:       fy  = fym + sty * dgtest; /* gradient values */
279:       dgx = dgxm + dgtest;
280:       dgy = dgym + dgtest;
281:     } else {
282:       /* Update the interval of uncertainty and compute the new step */
283:       Tao_mcstep(ls,&stx,&fx,&dgx,&sty,&fy,&dgy,&ls->step,f,&dg);
284:     }

286:     /* Force a sufficient decrease in the interval of uncertainty */
287:     if (mt->bracket) {
288:       if (PetscAbsReal(sty - stx) >= 0.66 * width1) ls->step = stx + 0.5*(sty - stx);
289:       width1 = width;
290:       width = PetscAbsReal(sty - stx);
291:     }
292:   }
293:   if ((ls->nfeval+ls->nfgeval) > ls->max_funcs) {
294:     PetscInfo2(ls,"Number of line search function evals (%D) > maximum (%D)\n",(ls->nfeval+ls->nfgeval),ls->max_funcs);
295:     ls->reason = TAOLINESEARCH_HALTED_MAXFCN;
296:   }

298:   /* Finish computations */
299:   PetscInfo2(ls,"%D function evals in line search, step = %g\n",(ls->nfeval+ls->nfgeval),(double)ls->step);

301:   /* Set new solution vector and compute gradient if needed */
302:   VecCopy(mt->work,x);
303:   if (!g_computed) {
304:     TaoLineSearchComputeGradient(ls,mt->work,g);
305:   }
306:   return(0);
307: }

311: PETSC_EXTERN PetscErrorCode TaoLineSearchCreate_MT(TaoLineSearch ls)
312: {
313:   PetscErrorCode   ierr;
314:   TaoLineSearch_MT *ctx;

318:   PetscNewLog(ls,&ctx);
319:   ctx->bracket=0;
320:   ctx->infoc=1;
321:   ls->data = (void*)ctx;
322:   ls->initstep = 1.0;
323:   ls->ops->setup=0;
324:   ls->ops->reset=0;
325:   ls->ops->apply=TaoLineSearchApply_MT;
326:   ls->ops->view =TaoLineSearchView_MT;
327:   ls->ops->destroy=TaoLineSearchDestroy_MT;
328:   ls->ops->setfromoptions=TaoLineSearchSetFromOptions_MT;
329:   return(0);
330: }

332: /*
333:      The subroutine mcstep is taken from the work of Jorge Nocedal.
334:      this is a variant of More' and Thuente's routine.

336:      subroutine mcstep

338:      the purpose of mcstep is to compute a safeguarded step for
339:      a linesearch and to update an interval of uncertainty for
340:      a minimizer of the function.

342:      the parameter stx contains the step with the least function
343:      value. the parameter stp contains the current step. it is
344:      assumed that the derivative at stx is negative in the
345:      direction of the step. if bracket is set true then a
346:      minimizer has been bracketed in an interval of uncertainty
347:      with endpoints stx and sty.

349:      the subroutine statement is

351:      subroutine mcstep(stx,fx,dx,sty,fy,dy,stp,fp,dp,bracket,
352:                        stpmin,stpmax,info)

354:      where

356:        stx, fx, and dx are variables which specify the step,
357:          the function, and the derivative at the best step obtained
358:          so far. The derivative must be negative in the direction
359:          of the step, that is, dx and stp-stx must have opposite
360:          signs. On output these parameters are updated appropriately.

362:        sty, fy, and dy are variables which specify the step,
363:          the function, and the derivative at the other endpoint of
364:          the interval of uncertainty. On output these parameters are
365:          updated appropriately.

367:        stp, fp, and dp are variables which specify the step,
368:          the function, and the derivative at the current step.
369:          If bracket is set true then on input stp must be
370:          between stx and sty. On output stp is set to the new step.

372:        bracket is a logical variable which specifies if a minimizer
373:          has been bracketed.  If the minimizer has not been bracketed
374:          then on input bracket must be set false.  If the minimizer
375:          is bracketed then on output bracket is set true.

377:        stpmin and stpmax are input variables which specify lower
378:          and upper bounds for the step.

380:        info is an integer output variable set as follows:
381:          if info = 1,2,3,4,5, then the step has been computed
382:          according to one of the five cases below. otherwise
383:          info = 0, and this indicates improper input parameters.

385:      subprograms called

387:        fortran-supplied ... abs,max,min,sqrt

389:      argonne national laboratory. minpack project. june 1983
390:      jorge j. more', david j. thuente

392: */

396: static PetscErrorCode Tao_mcstep(TaoLineSearch ls,PetscReal *stx,PetscReal *fx,PetscReal *dx,PetscReal *sty,PetscReal *fy,PetscReal *dy,PetscReal *stp,PetscReal *fp,PetscReal *dp)
397: {
398:   TaoLineSearch_MT *mtP = (TaoLineSearch_MT *) ls->data;
399:   PetscReal        gamma1, p, q, r, s, sgnd, stpc, stpf, stpq, theta;
400:   PetscInt         bound;

403:   /* Check the input parameters for errors */
404:   mtP->infoc = 0;
405:   if (mtP->bracket && (*stp <= PetscMin(*stx,*sty) || (*stp >= PetscMax(*stx,*sty)))) SETERRQ(PETSC_COMM_SELF,1,"bad stp in bracket");
406:   if (*dx * (*stp-*stx) >= 0.0) SETERRQ(PETSC_COMM_SELF,1,"dx * (stp-stx) >= 0.0");
407:   if (ls->stepmax < ls->stepmin) SETERRQ(PETSC_COMM_SELF,1,"stepmax > stepmin");

409:   /* Determine if the derivatives have opposite sign */
410:   sgnd = *dp * (*dx / PetscAbsReal(*dx));

412:   if (*fp > *fx) {
413:     /* Case 1: a higher function value.
414:      The minimum is bracketed. If the cubic step is closer
415:      to stx than the quadratic step, the cubic step is taken,
416:      else the average of the cubic and quadratic steps is taken. */

418:     mtP->infoc = 1;
419:     bound = 1;
420:     theta = 3 * (*fx - *fp) / (*stp - *stx) + *dx + *dp;
421:     s = PetscMax(PetscAbsReal(theta),PetscAbsReal(*dx));
422:     s = PetscMax(s,PetscAbsReal(*dp));
423:     gamma1 = s*PetscSqrtScalar(PetscPowScalar(theta/s,2.0) - (*dx/s)*(*dp/s));
424:     if (*stp < *stx) gamma1 = -gamma1;
425:     /* Can p be 0?  Check */
426:     p = (gamma1 - *dx) + theta;
427:     q = ((gamma1 - *dx) + gamma1) + *dp;
428:     r = p/q;
429:     stpc = *stx + r*(*stp - *stx);
430:     stpq = *stx + ((*dx/((*fx-*fp)/(*stp-*stx)+*dx))*0.5) * (*stp - *stx);

432:     if (PetscAbsReal(stpc-*stx) < PetscAbsReal(stpq-*stx)) {
433:       stpf = stpc;
434:     } else {
435:       stpf = stpc + 0.5*(stpq - stpc);
436:     }
437:     mtP->bracket = 1;
438:   } else if (sgnd < 0.0) {
439:     /* Case 2: A lower function value and derivatives of
440:      opposite sign. The minimum is bracketed. If the cubic
441:      step is closer to stx than the quadratic (secant) step,
442:      the cubic step is taken, else the quadratic step is taken. */

444:     mtP->infoc = 2;
445:     bound = 0;
446:     theta = 3*(*fx - *fp)/(*stp - *stx) + *dx + *dp;
447:     s = PetscMax(PetscAbsReal(theta),PetscAbsReal(*dx));
448:     s = PetscMax(s,PetscAbsReal(*dp));
449:     gamma1 = s*PetscSqrtScalar(PetscPowScalar(theta/s,2.0) - (*dx/s)*(*dp/s));
450:     if (*stp > *stx) gamma1 = -gamma1;
451:     p = (gamma1 - *dp) + theta;
452:     q = ((gamma1 - *dp) + gamma1) + *dx;
453:     r = p/q;
454:     stpc = *stp + r*(*stx - *stp);
455:     stpq = *stp + (*dp/(*dp-*dx))*(*stx - *stp);

457:     if (PetscAbsReal(stpc-*stp) > PetscAbsReal(stpq-*stp)) {
458:       stpf = stpc;
459:     } else {
460:       stpf = stpq;
461:     }
462:     mtP->bracket = 1;
463:   } else if (PetscAbsReal(*dp) < PetscAbsReal(*dx)) {
464:     /* Case 3: A lower function value, derivatives of the
465:      same sign, and the magnitude of the derivative decreases.
466:      The cubic step is only used if the cubic tends to infinity
467:      in the direction of the step or if the minimum of the cubic
468:      is beyond stp. Otherwise the cubic step is defined to be
469:      either stepmin or stepmax. The quadratic (secant) step is also
470:      computed and if the minimum is bracketed then the step
471:      closest to stx is taken, else the step farthest away is taken. */

473:     mtP->infoc = 3;
474:     bound = 1;
475:     theta = 3*(*fx - *fp)/(*stp - *stx) + *dx + *dp;
476:     s = PetscMax(PetscAbsReal(theta),PetscAbsReal(*dx));
477:     s = PetscMax(s,PetscAbsReal(*dp));

479:     /* The case gamma1 = 0 only arises if the cubic does not tend
480:        to infinity in the direction of the step. */
481:     gamma1 = s*PetscSqrtScalar(PetscMax(0.0,PetscPowScalar(theta/s,2.0) - (*dx/s)*(*dp/s)));
482:     if (*stp > *stx) gamma1 = -gamma1;
483:     p = (gamma1 - *dp) + theta;
484:     q = (gamma1 + (*dx - *dp)) + gamma1;
485:     r = p/q;
486:     if (r < 0.0 && gamma1 != 0.0) stpc = *stp + r*(*stx - *stp);
487:     else if (*stp > *stx)        stpc = ls->stepmax;
488:     else                         stpc = ls->stepmin;
489:     stpq = *stp + (*dp/(*dp-*dx)) * (*stx - *stp);

491:     if (mtP->bracket) {
492:       if (PetscAbsReal(*stp-stpc) < PetscAbsReal(*stp-stpq)) {
493:         stpf = stpc;
494:       } else {
495:         stpf = stpq;
496:       }
497:     } else {
498:       if (PetscAbsReal(*stp-stpc) > PetscAbsReal(*stp-stpq)) {
499:         stpf = stpc;
500:       } else {
501:         stpf = stpq;
502:       }
503:     }
504:   } else {
505:     /* Case 4: A lower function value, derivatives of the
506:        same sign, and the magnitude of the derivative does
507:        not decrease. If the minimum is not bracketed, the step
508:        is either stpmin or stpmax, else the cubic step is taken. */

510:     mtP->infoc = 4;
511:     bound = 0;
512:     if (mtP->bracket) {
513:       theta = 3*(*fp - *fy)/(*sty - *stp) + *dy + *dp;
514:       s = PetscMax(PetscAbsReal(theta),PetscAbsReal(*dy));
515:       s = PetscMax(s,PetscAbsReal(*dp));
516:       gamma1 = s*PetscSqrtScalar(PetscPowScalar(theta/s,2.0) - (*dy/s)*(*dp/s));
517:       if (*stp > *sty) gamma1 = -gamma1;
518:       p = (gamma1 - *dp) + theta;
519:       q = ((gamma1 - *dp) + gamma1) + *dy;
520:       r = p/q;
521:       stpc = *stp + r*(*sty - *stp);
522:       stpq = *stp + (*dp/(*dp-*dx)) * (*stx - *stp);

524:       stpf = stpc;
525:     } else if (*stp > *stx) {
526:       stpf = ls->stepmax;
527:     } else {
528:       stpf = ls->stepmin;
529:     }
530:   }

532:   /* Update the interval of uncertainty.  This update does not
533:      depend on the new step or the case analysis above. */

535:   if (*fp > *fx) {
536:     *sty = *stp;
537:     *fy = *fp;
538:     *dy = *dp;
539:   } else {
540:     if (sgnd < 0.0) {
541:       *sty = *stx;
542:       *fy = *fx;
543:       *dy = *dx;
544:     }
545:     *stx = *stp;
546:     *fx = *fp;
547:     *dx = *dp;
548:   }

550:   /* Compute the new step and safeguard it. */
551:   stpf = PetscMin(ls->stepmax,stpf);
552:   stpf = PetscMax(ls->stepmin,stpf);
553:   *stp = stpf;
554:   if (mtP->bracket && bound) {
555:     if (*sty > *stx) {
556:       *stp = PetscMin(*stx+0.66*(*sty-*stx),*stp);
557:     } else {
558:       *stp = PetscMax(*stx+0.66*(*sty-*stx),*stp);
559:     }
560:   }
561:   return(0);
562: }