Actual source code: morethuente.c

      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);

     12: static PetscErrorCode TaoLineSearchDestroy_MT(TaoLineSearch ls)
     13: {
     14:   PetscErrorCode   ierr;
     15:   TaoLineSearch_MT *mt;

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

     28: static PetscErrorCode TaoLineSearchSetFromOptions_MT(PetscOptionItems *PetscOptionsObject,TaoLineSearch ls)
     29: {
     32:   return(0);
     33: }

     35: static PetscErrorCode TaoLineSearchMonitor_MT(TaoLineSearch ls)
     36: {
     37:   TaoLineSearch_MT *mt = (TaoLineSearch_MT*)ls->data;
     38:   PetscErrorCode   ierr;

     41:   PetscViewerASCIIPrintf(ls->viewer, "stx: %g, fx: %g, dgx: %g\n", (double)mt->stx, (double)mt->fx, (double)mt->dgx);
     42:   PetscViewerASCIIPrintf(ls->viewer, "sty: %g, fy: %g, dgy: %g\n", (double)mt->sty, (double)mt->fy, (double)mt->dgy);
     43:   return(0);
     44: }

     46: static PetscErrorCode TaoLineSearchApply_MT(TaoLineSearch ls, Vec x, PetscReal *f, Vec g, Vec s)
     47: {
     48:   PetscErrorCode   ierr;
     49:   TaoLineSearch_MT *mt;

     51:   PetscReal        xtrapf = 4.0;
     52:   PetscReal        finit, width, width1, dginit, fm, fxm, fym, dgm, dgxm, dgym;
     53:   PetscReal        dgx, dgy, dg, dg2, fx, fy, stx, sty, dgtest;
     54:   PetscReal        ftest1=0.0, ftest2=0.0;
     55:   PetscInt         i, stage1,n1,n2,nn1,nn2;
     56:   PetscReal        bstepmin1, bstepmin2, bstepmax;
     57:   PetscBool        g_computed=PETSC_FALSE; /* to prevent extra gradient computation */

     66:   TaoLineSearchMonitor(ls, 0, *f, 0.0);

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

     72:   /* Check work vector */
     73:   if (!mt->work) {
     74:     VecDuplicate(x,&mt->work);
     75:     mt->x = x;
     76:     PetscObjectReference((PetscObject)mt->x);
     77:   } else if (x != mt->x) {
     78:     VecDestroy(&mt->work);
     79:     VecDuplicate(x,&mt->work);
     80:     PetscObjectDereference((PetscObject)mt->x);
     81:     mt->x = x;
     82:     PetscObjectReference((PetscObject)mt->x);
     83:   }

     85:   if (ls->bounded) {
     86:     /* Compute step length needed to make all variables equal a bound */
     87:     /* Compute the smallest steplength that will make one nonbinding variable
     88:      equal the bound */
     89:     VecGetLocalSize(ls->upper,&n1);
     90:     VecGetLocalSize(mt->x, &n2);
     91:     VecGetSize(ls->upper,&nn1);
     92:     VecGetSize(mt->x,&nn2);
     93:     if (n1 != n2 || nn1 != nn2) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Variable vector not compatible with bounds vector");
     94:     VecScale(s,-1.0);
     95:     VecBoundGradientProjection(s,x,ls->lower,ls->upper,s);
     96:     VecScale(s,-1.0);
     97:     VecStepBoundInfo(x,s,ls->lower,ls->upper,&bstepmin1,&bstepmin2,&bstepmax);
     98:     ls->stepmax = PetscMin(bstepmax,1.0e15);
     99:   }

    101:   VecDot(g,s,&dginit);
    102:   if (PetscIsInfOrNanReal(dginit)) {
    103:     PetscInfo1(ls,"Initial Line Search step * g is Inf or Nan (%g)\n",(double)dginit);
    104:     ls->reason=TAOLINESEARCH_FAILED_INFORNAN;
    105:     return(0);
    106:   }
    107:   if (dginit >= 0.0) {
    108:     PetscInfo1(ls,"Initial Line Search step * g is not descent direction (%g)\n",(double)dginit);
    109:     ls->reason = TAOLINESEARCH_FAILED_ASCENT;
    110:     return(0);
    111:   }

    113:   /* Initialization */
    114:   mt->bracket = 0;
    115:   stage1 = 1;
    116:   finit = *f;
    117:   dgtest = ls->ftol * dginit;
    118:   width = ls->stepmax - ls->stepmin;
    119:   width1 = width * 2.0;
    120:   VecCopy(x,mt->work);
    121:   /* Variable dictionary:
    122:    stx, fx, dgx - the step, function, and derivative at the best step
    123:    sty, fy, dgy - the step, function, and derivative at the other endpoint
    124:    of the interval of uncertainty
    125:    step, f, dg - the step, function, and derivative at the current step */

    127:   stx = 0.0;
    128:   fx  = finit;
    129:   dgx = dginit;
    130:   sty = 0.0;
    131:   fy  = finit;
    132:   dgy = dginit;

    134:   ls->step=ls->initstep;
    135:   for (i=0; i< ls->max_funcs; i++) {
    136:     /* Set min and max steps to correspond to the interval of uncertainty */
    137:     if (mt->bracket) {
    138:       ls->stepmin = PetscMin(stx,sty);
    139:       ls->stepmax = PetscMax(stx,sty);
    140:     } else {
    141:       ls->stepmin = stx;
    142:       ls->stepmax = ls->step + xtrapf * (ls->step - stx);
    143:     }

    145:     /* Force the step to be within the bounds */
    146:     ls->step = PetscMax(ls->step,ls->stepmin);
    147:     ls->step = PetscMin(ls->step,ls->stepmax);

    149:     /* If an unusual termination is to occur, then let step be the lowest
    150:      point obtained thus far */
    151:     if ((stx!=0) && (((mt->bracket) && (ls->step <= ls->stepmin || ls->step >= ls->stepmax)) || ((mt->bracket) && (ls->stepmax - ls->stepmin <= ls->rtol * ls->stepmax)) ||
    152:                      ((ls->nfeval+ls->nfgeval) >= ls->max_funcs - 1) || (mt->infoc == 0))) {
    153:       ls->step = stx;
    154:     }

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

    159:     if (ls->bounded) {
    160:       VecMedian(ls->lower, mt->work, ls->upper, mt->work);
    161:     }
    162:     if (ls->usegts) {
    163:       TaoLineSearchComputeObjectiveAndGTS(ls,mt->work,f,&dg);
    164:       g_computed=PETSC_FALSE;
    165:     } else {
    166:       TaoLineSearchComputeObjectiveAndGradient(ls,mt->work,f,g);
    167:       g_computed=PETSC_TRUE;
    168:       if (ls->bounded) {
    169:         VecDot(g,x,&dg);
    170:         VecDot(g,mt->work,&dg2);
    171:         dg = (dg2 - dg)/ls->step;
    172:       } else {
    173:         VecDot(g,s,&dg);
    174:       }
    175:     }

    177:     /* update bracketing parameters in the MT context for printouts in monitor */
    178:     mt->stx = stx;
    179:     mt->fx = fx;
    180:     mt->dgx = dgx;
    181:     mt->sty = sty;
    182:     mt->fy = fy;
    183:     mt->dgy = dgy;
    184:     TaoLineSearchMonitor(ls, i+1, *f, ls->step);

    186:     if (0 == i) {
    187:       ls->f_fullstep=*f;
    188:     }

    190:     if (PetscIsInfOrNanReal(*f) || PetscIsInfOrNanReal(dg)) {
    191:       /* User provided compute function generated Not-a-Number, assume
    192:        domain violation and set function value and directional
    193:        derivative to infinity. */
    194:       *f = PETSC_INFINITY;
    195:       dg = PETSC_INFINITY;
    196:     }

    198:     ftest1 = finit + ls->step * dgtest;
    199:     if (ls->bounded) {
    200:       ftest2 = finit + ls->step * dgtest * ls->ftol;
    201:     }
    202:     /* Convergence testing */
    203:     if (((*f - ftest1 <= 1.0e-10 * PetscAbsReal(finit)) &&  (PetscAbsReal(dg) + ls->gtol*dginit <= 0.0))) {
    204:       PetscInfo(ls, "Line search success: Sufficient decrease and directional deriv conditions hold\n");
    205:       ls->reason = TAOLINESEARCH_SUCCESS;
    206:       break;
    207:     }

    209:     /* Check Armijo if beyond the first breakpoint */
    210:     if (ls->bounded && (*f <= ftest2) && (ls->step >= bstepmin2)) {
    211:       PetscInfo(ls,"Line search success: Sufficient decrease.\n");
    212:       ls->reason = TAOLINESEARCH_SUCCESS;
    213:       break;
    214:     }

    216:     /* Checks for bad cases */
    217:     if (((mt->bracket) && (ls->step <= ls->stepmin||ls->step >= ls->stepmax)) || (!mt->infoc)) {
    218:       PetscInfo(ls,"Rounding errors may prevent further progress.  May not be a step satisfying\n");
    219:       PetscInfo(ls,"sufficient decrease and curvature conditions. Tolerances may be too small.\n");
    220:       ls->reason = TAOLINESEARCH_HALTED_OTHER;
    221:       break;
    222:     }
    223:     if ((ls->step == ls->stepmax) && (*f <= ftest1) && (dg <= dgtest)) {
    224:       PetscInfo1(ls,"Step is at the upper bound, stepmax (%g)\n",(double)ls->stepmax);
    225:       ls->reason = TAOLINESEARCH_HALTED_UPPERBOUND;
    226:       break;
    227:     }
    228:     if ((ls->step == ls->stepmin) && (*f >= ftest1) && (dg >= dgtest)) {
    229:       PetscInfo1(ls,"Step is at the lower bound, stepmin (%g)\n",(double)ls->stepmin);
    230:       ls->reason = TAOLINESEARCH_HALTED_LOWERBOUND;
    231:       break;
    232:     }
    233:     if ((mt->bracket) && (ls->stepmax - ls->stepmin <= ls->rtol*ls->stepmax)){
    234:       PetscInfo1(ls,"Relative width of interval of uncertainty is at most rtol (%g)\n",(double)ls->rtol);
    235:       ls->reason = TAOLINESEARCH_HALTED_RTOL;
    236:       break;
    237:     }

    239:     /* In the first stage, we seek a step for which the modified function
    240:      has a nonpositive value and nonnegative derivative */
    241:     if ((stage1) && (*f <= ftest1) && (dg >= dginit * PetscMin(ls->ftol, ls->gtol))) {
    242:       stage1 = 0;
    243:     }

    245:     /* A modified function is used to predict the step only if we
    246:      have not obtained a step for which the modified function has a
    247:      nonpositive function value and nonnegative derivative, and if a
    248:      lower function value has been obtained but the decrease is not
    249:      sufficient */

    251:     if ((stage1) && (*f <= fx) && (*f > ftest1)) {
    252:       fm   = *f - ls->step * dgtest;    /* Define modified function */
    253:       fxm  = fx - stx * dgtest;         /* and derivatives */
    254:       fym  = fy - sty * dgtest;
    255:       dgm  = dg - dgtest;
    256:       dgxm = dgx - dgtest;
    257:       dgym = dgy - dgtest;

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

    263:       fx  = fxm + stx * dgtest; /* Reset the function and */
    264:       fy  = fym + sty * dgtest; /* gradient values */
    265:       dgx = dgxm + dgtest;
    266:       dgy = dgym + dgtest;
    267:     } else {
    268:       /* Update the interval of uncertainty and compute the new step */
    269:       Tao_mcstep(ls,&stx,&fx,&dgx,&sty,&fy,&dgy,&ls->step,f,&dg);
    270:     }

    272:     /* Force a sufficient decrease in the interval of uncertainty */
    273:     if (mt->bracket) {
    274:       if (PetscAbsReal(sty - stx) >= 0.66 * width1) ls->step = stx + 0.5*(sty - stx);
    275:       width1 = width;
    276:       width = PetscAbsReal(sty - stx);
    277:     }
    278:   }
    279:   if ((ls->nfeval+ls->nfgeval) > ls->max_funcs) {
    280:     PetscInfo2(ls,"Number of line search function evals (%D) > maximum (%D)\n",(ls->nfeval+ls->nfgeval),ls->max_funcs);
    281:     ls->reason = TAOLINESEARCH_HALTED_MAXFCN;
    282:   }

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

    287:   /* Set new solution vector and compute gradient if needed */
    288:   VecCopy(mt->work,x);
    289:   if (!g_computed) {
    290:     TaoLineSearchComputeGradient(ls,mt->work,g);
    291:   }
    292:   return(0);
    293: }

    295: /*MC
    296:    TAOLINESEARCHMT - Line-search type with cubic interpolation that satisfies both the sufficient decrease and
    297:    curvature conditions. This method can take step lengths greater than 1.

    299:    More-Thuente line-search can be selected with "-tao_ls_type more-thuente".

    301:    References:
    302: .     1. - JORGE J. MORE AND DAVID J. THUENTE, LINE SEARCH ALGORITHMS WITH GUARANTEED SUFFICIENT DECREASE.
    303:           ACM Trans. Math. Software 20, no. 3 (1994): 286-307.

    305:    Level: developer

    307: .seealso: TaoLineSearchCreate(), TaoLineSearchSetType(), TaoLineSearchApply()

    309: .keywords: Tao, linesearch
    310: M*/
    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 = NULL;
    324:   ls->ops->reset = NULL;
    325:   ls->ops->apply = TaoLineSearchApply_MT;
    326:   ls->ops->destroy = TaoLineSearchDestroy_MT;
    327:   ls->ops->setfromoptions = TaoLineSearchSetFromOptions_MT;
    328:   ls->ops->monitor = TaoLineSearchMonitor_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: */

    394: static PetscErrorCode Tao_mcstep(TaoLineSearch ls,PetscReal *stx,PetscReal *fx,PetscReal *dx,PetscReal *sty,PetscReal *fy,PetscReal *dy,PetscReal *stp,PetscReal *fp,PetscReal *dp)
    395: {
    396:   TaoLineSearch_MT *mtP = (TaoLineSearch_MT *) ls->data;
    397:   PetscReal        gamma1, p, q, r, s, sgnd, stpc, stpf, stpq, theta;
    398:   PetscInt         bound;

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

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

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

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

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

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

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

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

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

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

    508:     mtP->infoc = 4;
    509:     bound = 0;
    510:     if (mtP->bracket) {
    511:       theta = 3*(*fp - *fy)/(*sty - *stp) + *dy + *dp;
    512:       s = PetscMax(PetscAbsReal(theta),PetscAbsReal(*dy));
    513:       s = PetscMax(s,PetscAbsReal(*dp));
    514:       gamma1 = s*PetscSqrtScalar(PetscPowScalar(theta/s,2.0) - (*dy/s)*(*dp/s));
    515:       if (*stp > *sty) gamma1 = -gamma1;
    516:       p = (gamma1 - *dp) + theta;
    517:       q = ((gamma1 - *dp) + gamma1) + *dy;
    518:       r = p/q;
    519:       stpc = *stp + r*(*sty - *stp);
    520:       stpf = stpc;
    521:     } else if (*stp > *stx) {
    522:       stpf = ls->stepmax;
    523:     } else {
    524:       stpf = ls->stepmin;
    525:     }
    526:   }

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

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

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