src/tools/expfit.c

   1 /*
   2  * $Id: expfit.c,v 1.33 2005/08/29 19:39:11 lindahl Exp $
   3  *
   4  *                This source code is part of
   5  *
   6  *                 G   R   O   M   A   C   S
   7  *
   8  *          GROningen MAchine for Chemical Simulations
   9  *
  10  *                        VERSION 3.2.0
  11  * Written by David van der Spoel, Erik Lindahl, Berk Hess, and others.
  12  * Copyright (c) 1991-2000, University of Groningen, The Netherlands.
  13  * Copyright (c) 2001-2004, The GROMACS development team,
  14  * check out http://www.gromacs.org for more information.
  15
  16  * This program is free software; you can redistribute it and/or
  17  * modify it under the terms of the GNU General Public License
  18  * as published by the Free Software Foundation; either version 2
  19  * of the License, or (at your option) any later version.
  20  *
  21  * If you want to redistribute modifications, please consider that
  22  * scientific software is very special. Version control is crucial -
  23  * bugs must be traceable. We will be happy to consider code for
  24  * inclusion in the official distribution, but derived work must not
  25  * be called official GROMACS. Details are found in the README & COPYING
  26  * files - if they are missing, get the official version at www.gromacs.org.
  27  *
  28  * To help us fund GROMACS development, we humbly ask that you cite
  29  * the papers on the package - you can find them in the top README file.
  30  *
  31  * For more info, check our website at http://www.gromacs.org
  32  *
  33  * And Hey:
  34  * Green Red Orange Magenta Azure Cyan Skyblue
  35  */
  36 #ifdef HAVE_CONFIG_H
  37 #include <config.h>
  38 #endif
  39
  40
  41 #include <sysstuff.h>
  42 #include <string.h>
  43 #include <math.h>
  44 #include "typedefs.h"
  45 #include "smalloc.h"
  46 #include "xvgr.h"
  47 #include "futil.h"
  48 #include "gstat.h"
  49 #include "vec.h"
  50 #include "statutil.h"
  51 #include "index.h"
  52
  53 const int  nfp_ffn[effnNR] = { 0, 1, 2, 3, 2, 5, 7, 9, 4, 3};
  54
  55 const char *s_ffn[effnNR+2] = { NULL, "none", "exp", "aexp", "exp_exp", "vac",
  56                           "exp5", "exp7", "exp9","erffit", NULL, NULL };
  57 /* We don't allow errest as a choice on the command line */
  58
  59 const char *longs_ffn[effnNR] = {
  60   "no fit",
  61   "y = exp(-x/a1)",
  62   "y = a2 exp(-x/a1)",
  63   "y = a2 exp(-x/a1) + (1-a2) exp(-x/a3)",
  64   "y = exp(-v) (cosh(wv) + 1/w sinh(wv)), v = x/(2 a1), w = sqrt(1 - a2)",
  65   "y = a1 exp(-x/a2) +  a3 exp(-x/a4) + a5",
  66   "y = a1 exp(-x/a2) +  a3 exp(-x/a4) + a5 exp(-x/a6) + a7",
  67   "y = a1 exp(-x/a2) +  a3 exp(-x/a4) + a5 exp(-x/a6) + a7 exp(-x/a8) + a9",
  68   "y = 1/2*(a1+a2) - 1/2*(a1-a2)*erf( (x-a3) /a4^2)",
  69   "y = a2*ee(a1,x) + (1-a2)*ee(a2,x)"
  70 };
  71
  72 extern gmx_bool mrqmin_new(real x[],real y[],real sig[],int ndata,real a[],
  73                        int ia[],int ma,real **covar,real **alpha,real *chisq,
  74                        void (*funcs)(real, real [], real *, real []),
  75                        real *alamda);
  76
  77 static real myexp(real x,real A,real tau)
  78 {
  79   if ((A == 0) || (tau == 0))
  80     return 0;
  81   return A*exp(-x/tau);
  82 }
  83
  84 void erffit (real x, real a[], real *y, real dyda[])
  85 {
  86 /* Fuction
  87  *      y=(a1+a2)/2-(a1-a2)/2*erf((x-a3)/a4^2)
  88  */
  89
  90 real erfarg;
  91 real erfval;
  92 real erfarg2;
  93 real derf;
  94
  95  erfarg=(x-a[3])/(a[4]*a[4]);
  96         erfarg2=erfarg*erfarg;
  97         erfval=gmx_erf(erfarg)/2;
  98         derf=(2.0/sqrt(M_PI))*(a[1]-a[2])/2*exp(-erfarg2)/(a[4]*a[4]);
  99         *y=(a[1]+a[2])/2-(a[1]-a[2])*erfval;
 100         dyda[1]=1/2-erfval;
 101         dyda[2]=1/2+erfval;
 102         dyda[3]=derf;
 103         dyda[4]=2*derf*erfarg;
 104 }
 105
 106
 107
 108 static void exp_one_parm(real x,real a[],real *y,real dyda[])
 109 {
 110   /* Fit to function
 111    *
 112    * y = exp(-x/a1)
 113    *
 114    */
 115
 116   real e1;
 117
 118   e1      = exp(-x/a[1]);
 119   *y      = e1;
 120   dyda[1] = x*e1/sqr(a[1]);
 121 }
 122
 123 static void exp_two_parm(real x,real a[],real *y,real dyda[])
 124 {
 125   /* Fit to function
 126    *
 127    * y = a2 exp(-x/a1)
 128    *
 129    */
 130
 131   real e1;
 132
 133   e1      = exp(-x/a[1]);
 134   *y      = a[2]*e1;
 135   dyda[1] = x*a[2]*e1/sqr(a[1]);
 136   dyda[2] = e1;
 137 }
 138
 139 static void exp_3_parm(real x,real a[],real *y,real dyda[])
 140 {
 141   /* Fit to function
 142    *
 143    * y = a2 exp(-x/a1) + (1-a2) exp(-x/a3)
 144    *
 145    */
 146
 147   real e1,e2;
 148
 149   e1      = exp(-x/a[1]);
 150   e2      = exp(-x/a[3]);
 151   *y      = a[2]*e1 + (1-a[2])*e2;
 152   dyda[1] = x*a[2]*e1/sqr(a[1]);
 153   dyda[2] = e1-e2;
 154   dyda[3] = x*(1-a[2])*e2/sqr(a[3]);
 155 }
 156
 157 static void exp_5_parm(real x,real a[],real *y,real dyda[])
 158 {
 159   /* Fit to function
 160    *
 161    * y = a1 exp(-x/a2) + a3 exp(-x/a4) + a5
 162    *
 163    */
 164
 165   real e1,e2;
 166
 167   e1      = exp(-x/a[2]);
 168   e2      = exp(-x/a[4]);
 169   *y      = a[1]*e1 + a[3]*e2 + a[5];
 170
 171   if (debug)
 172     fprintf(debug,"exp_5_parm called: x = %10.3f  y = %10.3f\n"
 173             "a = ( %8.3f  %8.3f  %8.3f  %8.3f  %8.3f)\n",
 174             x,*y,a[1],a[2],a[3],a[4],a[5]);
 175   dyda[1] = e1;
 176   dyda[2] = x*e1/sqr(a[2]);
 177   dyda[3] = e2;
 178   dyda[4] = x*e2/sqr(a[4]);
 179   dyda[5] = 0;
 180 }
 181
 182 static void exp_7_parm(real x,real a[],real *y,real dyda[])
 183 {
 184   /* Fit to function
 185    *
 186    * y = a1 exp(-x/a2) + a3 exp(-x/a4) + a5 exp(-x/a6) + a7
 187    *
 188    */
 189
 190   real e1,e2,e3;
 191
 192   e1      = exp(-x/a[2]);
 193   e2      = exp(-x/a[4]);
 194   e3      = exp(-x/a[6]);
 195   *y      = a[1]*e1 + a[3]*e2 + a[5]*e3 + a[7];
 196
 197   dyda[1] = e1;
 198   dyda[2] = x*e1/sqr(a[2]);
 199   dyda[3] = e2;
 200   dyda[4] = x*e2/sqr(a[4]);
 201   dyda[5] = e3;
 202   dyda[6] = x*e3/sqr(a[6]);
 203   dyda[7] = 0;
 204 }
 205
 206 static void exp_9_parm(real x,real a[],real *y,real dyda[])
 207 {
 208   /* Fit to function
 209    *
 210    * y = a1 exp(-x/a2) + a3 exp(-x/a4) + a5 exp(-x/a6) + a7
 211    *
 212    */
 213
 214   real e1,e2,e3,e4;
 215
 216   e1      = exp(-x/a[2]);
 217   e2      = exp(-x/a[4]);
 218   e3      = exp(-x/a[6]);
 219   e4      = exp(-x/a[8]);
 220   *y      = a[1]*e1 + a[3]*e2 + a[5]*e3 + a[7]*e4 + a[9];
 221
 222   dyda[1] = e1;
 223   dyda[2] = x*e1/sqr(a[2]);
 224   dyda[3] = e2;
 225   dyda[4] = x*e2/sqr(a[4]);
 226   dyda[5] = e3;
 227   dyda[6] = x*e3/sqr(a[6]);
 228   dyda[7] = e4;
 229   dyda[8] = x*e4/sqr(a[8]);
 230   dyda[9] = 0;
 231 }
 232
 233 static void vac_2_parm(real x,real a[],real *y,real dyda[])
 234 {
 235   /* Fit to function
 236    *
 237    * y = 1/2 (1 - 1/w) exp(-(1+w)v) + 1/2 (1 + 1/w) exp(-(1-w)v)
 238    *
 239    *   = exp(-v) (cosh(wv) + 1/w sinh(wv))
 240    *
 241    *    v = x/(2 a1)
 242    *    w = sqrt(1 - a2)
 243    *
 244    *    For tranverse current autocorrelation functions:
 245    *       a1 = tau
 246    *       a2 = 4 tau (eta/rho) k^2
 247    *
 248    */
 249
 250   double v,det,omega,omega2,em,ec,es;
 251
 252   v   = x/(2*a[1]);
 253   det = 1 - a[2];
 254   em  = exp(-v);
 255   if (det != 0) {
 256     omega2  = fabs(det);
 257     omega   = sqrt(omega2);
 258     if (det > 0) {
 259       ec = em*0.5*(exp(omega*v)+exp(-omega*v));
 260       es = em*0.5*(exp(omega*v)-exp(-omega*v))/omega;
 261     } else {
 262       ec = em*cos(omega*v);
 263       es = em*sin(omega*v)/omega;
 264     }
 265     *y      = ec + es;
 266     dyda[2] = (v/det*ec+(v-1/det)*es)/(-2.0);
 267     dyda[1] = (1-det)*v/a[1]*es;
 268   } else {
 269     *y      = (1+v)*em;
 270     dyda[2] = -v*v*em*(0.5+v/6);
 271     dyda[1] = v*v/a[1]*em;
 272   }
 273 }
 274
 275 static void errest_3_parm(real x,real a[],real *y,real dyda[])
 276 {
 277   real e1,e2,v1,v2;
 278
 279   if (a[1])
 280     e1 = exp(-x/a[1]) - 1;
 281   else
 282     e1 = 0;
 283   if (a[3])
 284     e2 = exp(-x/a[3]) - 1;
 285   else
 286     e2 = 0;
 287
 288   if (x > 0) {
 289     v1 = 2*a[1]*(e1*a[1]/x + 1);
 290     v2 = 2*a[3]*(e2*a[3]/x + 1);
 291     *y      = a[2]*v1 + (1-a[2])*v2;
 292     dyda[1] = 2*     a[2] *(v1/a[1] + e1);
 293     dyda[3] = 2*(1 - a[2])*(v2/a[3] + e2);
 294     dyda[2] = (v1 - v2);
 295   } else {
 296     *y      = 0;
 297     dyda[1] = 0;
 298     dyda[3] = 0;
 299     dyda[2] = 0;
 300   }
 301 }
 302
 303 typedef void (*myfitfn)(real x,real a[],real *y,real dyda[]);
 304 myfitfn mfitfn[effnNR] = {
 305   exp_one_parm, exp_one_parm, exp_two_parm, exp_3_parm, vac_2_parm,
 306   exp_5_parm,   exp_7_parm,   exp_9_parm, erffit,  errest_3_parm
 307 };
 308
 309 real fit_function(int eFitFn,real *parm,real x)
 310 {
 311   static real y,dum[8];
 312
 313   mfitfn[eFitFn](x,parm-1,&y,dum);
 314
 315   return y;
 316 }
 317
 318 /* lmfit_exp supports up to 3 parameter fitting of exponential functions */
 319 static gmx_bool lmfit_exp(int nfit,real x[],real y[],real dy[],real ftol,
 320                       real parm[],real dparm[],gmx_bool bVerbose,
 321                       int eFitFn,int fix)
 322 {
 323   real chisq,ochisq,alamda;
 324   real *a,**covar,**alpha,*dum;
 325   gmx_bool bCont;
 326   int  i,j,ma,mfit,*lista,*ia;
 327
 328   if ((eFitFn < 0) || (eFitFn >= effnNR))
 329     gmx_fatal(FARGS,"fitfn = %d, should be in 0..%d (%s,%d)",
 330                 effnNR-1,eFitFn,__FILE__,__LINE__);
 331
 332   ma=mfit=nfp_ffn[eFitFn];         /* number of parameters to fit */
 333   snew(a,ma+1);
 334   snew(covar,ma+1);
 335   snew(alpha,ma+1);
 336   snew(lista,ma+1);
 337   snew(ia,ma+1);
 338   snew(dum,ma+1);
 339   for(i=1; (i<ma+1); i++) {
 340     lista[i] = i;
 341     ia[i] = 1; /* fixed bug B.S.S 19/11  */
 342     snew(covar[i],ma+1);
 343     snew(alpha[i],ma+1);
 344   }
 345   if (fix) {
 346     if (bVerbose)
 347       fprintf(stderr,"Will keep parameters fixed during fit procedure: %d\n",
 348               fix);
 349     for(i=0; i<ma; i++)
 350       if (fix & 1<<i)
 351         ia[i+1] = 0;
 352   }
 353   if (debug)
 354     fprintf(debug,"%d parameter fit\n",mfit);
 355
 356   /* Initial params */
 357   alamda = -1;    /* Starting value   */
 358   chisq  = 1e12;
 359   for(i=0; (i<mfit); i++)
 360     a[i+1] = parm[i];
 361
 362   j = 0;
 363   if (bVerbose)
 364     fprintf(stderr,"%4s  %10s  %10s  %10s  %10s  %10s %10s\n",
 365             "Step","chi^2","Lambda","A1","A2","A3","A4");
 366   do {
 367     ochisq = chisq;
 368     /* mrqmin(x-1,y-1,dy-1,nfit,a,ma,lista,mfit,covar,alpha,
 369      *   &chisq,expfn[mfit-1],&alamda)
 370      */
 371     if (!mrqmin_new(x-1,y-1,dy-1,nfit,a,ia,ma,covar,alpha,&chisq,
 372                     mfitfn[eFitFn],&alamda))
 373       return FALSE;
 374
 375     if (bVerbose) {
 376       fprintf(stderr,"%4d  %10.5e  %10.5e  %10.5e",
 377               j,chisq,alamda,a[1]);
 378       if (mfit > 1)
 379         fprintf(stderr,"  %10.5e",a[2]);
 380       if (mfit > 2)
 381         fprintf(stderr,"  %10.5e",a[3]);
 382       if (mfit > 3)
 383          fprintf(stderr," %10.5e",a[4]);
 384       fprintf(stderr,"\n");
 385     }
 386     j++;
 387     bCont = ((fabs(ochisq - chisq) > fabs(ftol*chisq)) ||
 388              ((ochisq == chisq)));
 389   } while (bCont && (alamda != 0.0) && (j < 50));
 390   if (bVerbose)
 391     fprintf(stderr,"\n");
 392
 393   /* Now get the covariance matrix out */
 394   alamda = 0;
 395
 396   /*  mrqmin(x-1,y-1,dy-1,nfit,a,ma,lista,mfit,covar,alpha,
 397    * &chisq,expfn[mfit-1],&alamda)
 398    */
 399   if ( !mrqmin_new(x-1,y-1,dy-1,nfit,a,ia,ma,covar,alpha,&chisq,
 400                    mfitfn[eFitFn],&alamda))
 401     return FALSE;
 402
 403   for(j=0; (j<mfit); j++) {
 404     parm[j]  = a[j+1];
 405     dparm[j] = covar[j+1][j+1];
 406   }
 407
 408   for(i=0; (i<ma+1); i++) {
 409     sfree(covar[i]);
 410     sfree(alpha[i]);
 411   }
 412   sfree(a);
 413   sfree(covar);
 414   sfree(alpha);
 415   sfree(lista);
 416   sfree(dum);
 417
 418   return TRUE;
 419 }
 420
 421 real do_lmfit(int ndata,real c1[],real sig[],real dt,real x0[],
 422               real begintimefit,real endtimefit,const output_env_t oenv,
 423               gmx_bool bVerbose, int eFitFn,real fitparms[],int fix)
 424 {
 425   FILE *fp;
 426   char buf[32];
 427
 428   int  i,j,nparm,nfitpnts;
 429   real integral,ttt;
 430   real *parm,*dparm;
 431   real *x,*y,*dy;
 432   real ftol = 1e-4;
 433
 434   nparm = nfp_ffn[eFitFn];
 435   if (debug) {
 436     fprintf(debug,"There are %d points to fit %d vars!\n",ndata,nparm);
 437     fprintf(debug,"Fit to function %d from %g through %g, dt=%g\n",
 438             eFitFn,begintimefit,endtimefit,dt);
 439   }
 440
 441   snew(x,ndata);
 442   snew(y,ndata);
 443   snew(dy,ndata);
 444
 445   j=0;
 446   for(i=0; i<ndata; i++) {
 447     ttt = x0 ? x0[i] : dt*i;
 448     if (ttt>=begintimefit && ttt<=endtimefit) {
 449       x[j] = ttt;
 450       y[j] = c1[i];
 451
 452       /* mrqmin does not like sig to be zero */
 453       if (sig[i]<1.0e-7)
 454         dy[j]=1.0e-7;
 455       else
 456         dy[j]=sig[i];
 457       if (debug)
 458         fprintf(debug,"j= %d, i= %d, x= %g, y= %g, dy= %g\n",
 459                 j,i,x[j],y[j],dy[j]);
 460       j++;
 461     }
 462   }
 463   nfitpnts = j;
 464   integral = 0;
 465   if (nfitpnts < nparm)
 466     fprintf(stderr,"Not enough data points for fitting!\n");
 467   else {
 468     snew(parm,nparm);
 469     snew(dparm,nparm);
 470     if (fitparms)
 471       for(i=0; (i < nparm); i++)
 472         parm[i]=fitparms[i];
 473
 474     if (!lmfit_exp(nfitpnts,x,y,dy,ftol,parm,dparm,bVerbose,eFitFn,fix))
 475       fprintf(stderr,"Fit failed!\n");
 476     else if (nparm <= 3) {
 477       /* Compute the integral from begintimefit */
 478       if (nparm == 3)
 479         integral=(parm[0]*myexp(begintimefit,parm[1],  parm[0]) +
 480                   parm[2]*myexp(begintimefit,1-parm[1],parm[2]));
 481       else if (nparm == 2)
 482         integral=parm[0]*myexp(begintimefit,parm[1],  parm[0]);
 483       else if (nparm == 1)
 484         integral=parm[0]*myexp(begintimefit,1,  parm[0]);
 485       else
 486         gmx_fatal(FARGS,"nparm = %d in file %s, line %d",
 487                     nparm,__FILE__,__LINE__);
 488
 489       /* Generate THE output */
 490       if (bVerbose) {
 491         fprintf(stderr,"FIT: # points used in fit is: %d\n",nfitpnts);
 492         fprintf(stderr,"FIT: %21s%21s%21s\n",
 493                 "parm0     ","parm1 (ps)   ","parm2 (ps)    ");
 494         fprintf(stderr,"FIT: ------------------------------------------------------------\n");
 495         fprintf(stderr,"FIT: %8.3g +/- %8.3g%9.4g +/- %8.3g%8.3g +/- %8.3g\n",
 496                 parm[0],dparm[0],parm[1],dparm[1],parm[2],dparm[2]);
 497         fprintf(stderr,"FIT: Integral (calc with fitted function) from %g ps to inf. is: %g\n",
 498                 begintimefit,integral);
 499
 500         sprintf(buf,"test%d.xvg",nfitpnts);
 501         fp = xvgropen(buf,"C(t) + Fit to C(t)","Time (ps)","C(t)",oenv);
 502         fprintf(fp,"# parm0 = %g, parm1 = %g, parm2 = %g\n",
 503                 parm[0],parm[1],parm[2]);
 504         for(j=0; (j<nfitpnts); j++) {
 505           ttt = x0 ? x0[j] : dt*j;
 506           fprintf(fp,"%10.5e  %10.5e  %10.5e\n",
 507                   ttt,c1[j],fit_function(eFitFn,parm,ttt));
 508         }
 509         xvgrclose(fp);
 510       }
 511     }
 512     if (fitparms)
 513     {
 514         for(i=0;(i<nparm);i++)
 515         {
 516             fitparms[i] = parm[i];
 517         }
 518     }
 519     sfree(parm);
 520     sfree(dparm);
 521   }
 522
 523   sfree(x);
 524   sfree(y);
 525   sfree(dy);
 526
 527   return integral;
 528 }
 529
 530 void do_expfit(int ndata,real c1[],real dt,real begintimefit,real endtimefit)
 531 {
 532   int i,n;
 533   real *x,*y,*Dy;
 534   real aa,bb,saa,sbb,A,tau,dA,dtau;
 535
 536   fprintf(stderr,"Will fit data from %g (ps) to %g (ps).\n",
 537           begintimefit,endtimefit);
 538
 539   snew(x,ndata);   /* allocate the maximum necessary space */
 540   snew(y,ndata);
 541   snew(Dy,ndata);
 542   n=0;
 543
 544   for(i=0; (i<ndata); i++) {
 545     if ( (dt*i >= begintimefit) && (dt*i <= endtimefit) ) {
 546       x[n]=dt*i;
 547       y[n]=c1[i];
 548       Dy[n]=0.5;
 549       fprintf(stderr,"n= %d, i= %d, x= %g, y= %g\n",n,i,x[n],y[n]);
 550       n++;
 551     }
 552   }
 553   fprintf(stderr,"# of data points used in the fit is : %d\n\n",n);
 554   expfit(n,x,y,Dy,&aa,&saa,&bb,&sbb);
 555
 556   A=exp(aa);
 557   dA=exp(aa)*saa;
 558   tau=-1.0/bb;
 559   dtau=sbb/sqr(bb);
 560   fprintf(stderr,"Fitted to y=exp(a+bx):\n");
 561   fprintf(stderr,"a = %10.5f\t b = %10.5f",aa,bb);
 562   fprintf(stderr,"\n");
 563   fprintf(stderr,"Fitted to y=Aexp(-x/tau):\n");
 564   fprintf(stderr,"A  = %10.5f\t tau  = %10.5f\n",A,tau);
 565   fprintf(stderr,"dA = %10.5f\t dtau = %10.5f\n",dA,dtau);
 566 }
 567
 568
 569 void expfit(int n, real *x, real *y, real *Dy, real *a, real *sa,
 570             real *b, real *sb)
 571 {
 572   real *w,*ly,A,SA,B,SB;
 573   int  i;
 574   real sum,xbar,ybar,Sxx,Sxy,wr2,chi2,gamma,Db;
 575
 576 #define ZERO 0.0
 577 #define ONE 1.0
 578 #define ONEP5 1.5
 579 #define TWO 2.0
 580
 581 #define sqr(x) ((x)*(x))
 582
 583   /*allocate memory */
 584   snew(w,n);
 585   snew(ly,n);
 586
 587   /* Calculate weights and values of ln(y). */
 588   for(i=0;(i<n); i++){
 589     w[i]=sqr(y[i]/Dy[i]);
 590     ly[i]=log(y[i]);
 591   }
 592
 593   /* The weighted averages of x and y: xbar and ybar. */
 594   sum=ZERO;
 595   xbar=ZERO;
 596   ybar=ZERO;
 597   for(i=0;(i<n);i++){
 598     sum+=w[i];
 599     xbar+=w[i]*x[i];
 600     ybar+=w[i]*ly[i];
 601   }
 602   xbar/=sum;
 603   ybar/=sum;
 604
 605   /* The centered product sums Sxx and Sxy, and hence A and B. */
 606   Sxx=ZERO;
 607   Sxy=ZERO;
 608   for(i=0;(i<n);i++){
 609     Sxx+=w[i]*sqr(x[i]-xbar);
 610     Sxy+=w[i]*(x[i]-xbar)*(ly[i]-ybar);
 611   }
 612   B=Sxy/Sxx;
 613   A=ybar-B*xbar;
 614
 615   /* Chi-squared (chi2) and gamma. */
 616   chi2=ZERO;
 617   gamma=ZERO;
 618   for(i=0;(i<n);i++){
 619     wr2=w[i]*sqr(ly[i]-A-B*x[i]);
 620     chi2+=wr2;
 621     gamma+=wr2*(x[i]-xbar);
 622   }
 623
 624   /* Refined values of A and B. Also SA and SB. */
 625   Db=-ONEP5*gamma/Sxx;
 626   B+=Db;
 627   A-=ONEP5*chi2/sum-xbar*Db;
 628   SB=sqrt((chi2/(n-2))/Sxx);
 629   SA=SB*sqrt(Sxx/sum+sqr(xbar));
 630   *a=A;
 631   *b=B;
 632   *sa=SA;
 633   *sb=SB;
 634 }
 635
 636