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#include "volterra2.h"
#include "delay_handle.h"
#include "gear.h"
#include "ggets.h"
#include "markov.h"
#include <stdlib.h>
#include "xpplim.h"
#include "getvar.h"
#include "volterra.h"
#include <math.h>
#include <stdio.h>
#include "parserslow.h"
#define MAX(a,b) ((a)>(b)?(a):(b))
#define MIN(a,b) ((a)<(b)?(a):(b))
/* #define Set_ivar(a,b) variables[(a)]=(b) */
/* This is an implicit solver for volterra integral and integro-differential
equations. It is based on code found in Peter Linz's book
ont Volterra equations.
One tries to evaluate:
int_0^t ( (t-t')^-mu K(t,t',u) dt')
where 0 <= mu < 1 and K(t,t',u) is cts and Lipschitz.
The product method is used combined with the trapezoidal rule for integration.
The method is A-stable since it is an implicit scheme.
The kernel structure contains the constant mu and the expression for
evaluating K(t,t',u)
*/
#define CONV 2
extern KERNEL kernel[MAXKER];
extern int NODE,NMarkov,FIX_VAR,PrimeStart;
extern int NKernel;
extern double *Memory[MAXODE];
extern double T0,DELTA_T;
extern int MaxPoints;
extern int EqType[MAXODE];
int CurrentPoint;
int KnFlag;
int AutoEvaluate=0;
extern double variables[];
extern int NVAR;
extern int MaxEulIter;
extern double EulTol,NEWT_ERR;
double evaluate();
double ker_val();
double alpha1n();
double alpbetjn();
double betnn();
extern int *my_ode[];
double get_ivar();
double ker_val(in)
int in;
{
if(KnFlag)return(kernel[in].k_n);
return(kernel[in].k_n1);
}
void alloc_v_memory() /* allocate stuff for volterra equations */
{
int i,len,formula[256],j;
/* First parse the kernels since these were deferred */
for(i=0;i<NKernel;i++){
kernel[i].k_n=0.0;
if(add_expr(kernel[i].expr,formula,&len)){
plintf("Illegal kernel %s=%s\n",kernel[i].name,kernel[i].expr);
exit(0); /* fatal error ... */
}
kernel[i].formula=(int *)malloc((len+2)*sizeof(int));
for(j=0;j<len;j++){
kernel[i].formula[j]=formula[j];
}
if(kernel[i].flag==CONV){
if(add_expr(kernel[i].kerexpr,formula,&len)){
plintf("Illegal convolution %s=%s\n",
kernel[i].name,kernel[i].kerexpr);
exit(0); /* fatal error ... */
}
kernel[i].kerform=(int *)malloc((len+2)*sizeof(int));
for(j=0;j<len;j++){
kernel[i].kerform[j]=formula[j];
}
}
}
allocate_volterra(MaxPoints,0);
}
void allocate_volterra(npts,flag)
int npts,flag;
{
int i,oldmem=MaxPoints,j;
int ntot=NODE+FIX_VAR+NMarkov;
npts=abs(npts);
MaxPoints=npts;
/* now allocate the memory */
if(NKernel==0)return;
if(flag==1)for(i=0;i<ntot;i++)free(Memory[i]);
for(i=0;i<ntot;i++){
Memory[i]=(double *)malloc(sizeof(double)*MaxPoints);
if(Memory[i]==NULL)break;
}
if(i<ntot&&flag==0){
plintf("Not enough memory... make Maxpts smaller \n");
exit(0);
}
if(i<ntot){
MaxPoints=oldmem;
for(j=0;j<i;j++)free(Memory[j]);
for(i=0;i<ntot;i++)
Memory[i]=(double *)malloc(sizeof(double)*MaxPoints);
err_msg("Not enough memory...resetting");
}
CurrentPoint=0;
KnFlag=1;
alloc_kernels(flag);
}
void re_evaluate_kernels()
{
int i,j,n=MaxPoints;
if(AutoEvaluate==0)return;
for(i=0;i<NKernel;i++){
if(kernel[i].flag==CONV){
for(j=0;j<=n;j++){
SETVAR(0,T0+DELTA_T*j);
kernel[i].cnv[j]=evaluate(kernel[i].kerform);
}
}
}
}
void alloc_kernels(flag)
int flag;
{
int i,n=MaxPoints;
int j;
double mu;
for(i=0;i<NKernel;i++){
if(kernel[i].flag==CONV){
if(flag==1)free(kernel[i].cnv);
kernel[i].cnv=(double *)malloc((n+1)*sizeof(double));
for(j=0;j<=n;j++){
SETVAR(0,T0+DELTA_T*j);
kernel[i].cnv[j]=evaluate(kernel[i].kerform);
}
}
/* Do the alpha functions here later */
if(kernel[i].mu>0.0){
mu=kernel[i].mu;
if(flag==1)free(kernel[i].al);
kernel[i].al=(double *)malloc((n+1)*sizeof(double));
for(j=0;j<=n;j++)kernel[i].al[j]=alpbetjn(mu,DELTA_T,j);
}
}
}
/* the following is the main driver for evaluating the sums in the
kernel the results here are used in the implicit solver. The integral
up to t_n-1 is evaluated and placed in sum. Kn-> Kn-1
the weights al and bet are computed in general, but specifically
for mu=0,.5 since these involve no transcendental functions
*/
/*** FIX THIS TO DO MORE GENERAL STUFF
K(t,t',u,u') someday...
***/
void init_sums(t0,n,dt,i0,iend,ishift)
double t0,dt;
int n,i0,iend,ishift;
{
double t=t0+n*dt,tp=t0+i0*dt;
double sum[MAXODE],al,alpbet,mu;
int nvar=FIX_VAR+NODE+NMarkov;
int l,ioff,ker,i;
SETVAR(0,t);
SETVAR(PrimeStart,tp);
for(l=0;l<nvar;l++)SETVAR(l+1,Memory[l][ishift]);
for(ker=0;ker<NKernel;ker++){
kernel[ker].k_n1=kernel[ker].k_n;
mu=kernel[ker].mu;
if(mu==0.0)al=.5*dt;
else al=alpha1n(mu,dt,t,tp);
sum[ker]=al*evaluate(kernel[ker].formula);
if(kernel[ker].flag==CONV)
sum[ker]=sum[ker]*kernel[ker].cnv[n-i0];
}
for(i=1;i<=iend;i++){
ioff=(ishift+i)%MaxPoints;
tp+=dt;
SETVAR(PrimeStart,tp);
for(l=0;l<nvar;l++)SETVAR(l+1,Memory[l][ioff]);
for(ker=0;ker<NKernel;ker++){
mu=kernel[ker].mu;
if(mu==0.0)alpbet=dt;
else alpbet=kernel[ker].al[n-i0-i]; /* alpbetjn(mu,dt,t,tp); */
if(kernel[ker].flag==CONV)
sum[ker]+=(alpbet*evaluate(kernel[ker].formula)
*kernel[ker].cnv[n-i0-i]);
else sum[ker]+=(alpbet*evaluate(kernel[ker].formula));
}
}
for(ker=0;ker<NKernel;ker++){
kernel[ker].sum=sum[ker];
}
}
/* the following functions compute integrals for the piecewise
-- constant -- product integration rule. Thus they agree with
the trapezoid rule for mu=0 and there is a special case for mu=.5
since that involves no transcendentals. Later I will put in the
piecewise --linear-- method
*/
double alpha1n(mu,dt,t,t0)
double mu,dt,t,t0;
{
double m1;
if(mu==.5)return(sqrt(fabs(t-t0))-sqrt(fabs(t-t0-dt)));
m1=1-mu;
return(.5*(pow(fabs(t-t0),m1)-pow(fabs(t-t0-dt),m1))/m1);
}
double alpbetjn(mu,dt,l)
double dt,mu;
int l;
{
double m1;
double dif=l*dt;
if(mu==.5)return(sqrt(dif+dt)-sqrt(fabs(dif-dt)));
m1=1-mu;
return(.5*(pow(dif+dt,m1)-pow(fabs(dif-dt),m1))/m1);
}
double betnn(mu,dt,t0,t)
double mu,dt,t0,t;
{
double m1;
if(mu==.5)return(sqrt(dt));
m1=1-mu;
return(.5*pow(dt,m1)/m1);
}
void get_kn(y,t) /* uses the guessed value y to update Kn */
double t,*y;
{
int i;
SETVAR(0,t);
SETVAR(PrimeStart,t);
for(i=0;i<NODE;i++)
SETVAR(i+1,y[i]);
for(i=NODE;i<NODE+FIX_VAR;i++)
SETVAR(i+1,evaluate(my_ode[i]));
for(i=0;i<NKernel;i++){
if(kernel[i].flag==CONV)
kernel[i].k_n=kernel[i].sum+
kernel[i].betnn*evaluate(kernel[i].formula)*kernel[i].cnv[0];
else
kernel[i].k_n=kernel[i].sum+kernel[i].betnn*evaluate(kernel[i].formula);
/* plintf(" Value t=%g %d =%g %g\n",t,i,kernel[i].k_n,y[i]); */
}
}
int volterra(y,t,dt,nt,neq,istart,work)
double *y,*t,dt,*work;
int nt,neq,*istart;
{
double *jac,*yg,*yp,*yp2,*ytemp,*errvec;
double z,mu,bet;
int i,j;
yp=work;
yg=yp+neq;
ytemp=yg+neq;
errvec=ytemp+neq;
yp2=errvec+neq;
jac=yp2+neq;
/* Initialization of everything */
if(*istart==1){
CurrentPoint=0;
KnFlag=1;
for(i=0;i<NKernel;i++){ /* zero the integrals */
kernel[i].k_n=0.0;
kernel[i].k_n1=0.0;
mu=kernel[i].mu; /* compute bet_nn */
if(mu==0.0)bet=.5*dt;
else bet=betnn(mu,dt,*t,*t);
kernel[i].betnn=bet;
}
SETVAR(0,*t);
SETVAR(PrimeStart,*t);
for(i=0;i<NODE;i++)
if(!EqType[i])SETVAR(i+1,y[i]); /* assign initial data */
for(i=NODE;i<NODE+FIX_VAR;i++)
SETVAR(i+1,evaluate(my_ode[i])); /* set fixed variables for pass 1 */
for(i=0;i<NODE;i++)
if(EqType[i]){
z=evaluate(my_ode[i]); /* reset IC for integral eqns */
SETVAR(i+1,z);
y[i]=z;
}
for(i=NODE;i<NODE+FIX_VAR;i++) /* pass 2 for fixed variables */
SETVAR(i+1,evaluate(my_ode[i]));
for(i=0;i<NODE+FIX_VAR+NMarkov;i++)
Memory[i][0]=get_ivar(i+1); /* save everything */
CurrentPoint=1;
*istart=0;
}
for(i=0;i<nt;i++) /* the real computation */
{
*t=*t+dt;
set_wieners(dt,y,*t);
if((j=volt_step(y,*t,dt,neq,yg,yp,yp2,ytemp,errvec,jac))!=0)
return(j);
stor_delay(y);
}
return(0);
}
int volt_step(y,t,dt,neq,yg,yp,yp2,ytemp,errvec,jac)
double *y,t,dt,*yg,*yp,*yp2,*ytemp,*errvec,*jac;
int neq;
{
int i0,iend,ishift,i,iter=0,info,ipivot[MAXODE1],j,ind;
int n1=NODE+1;
double dt2=.5*dt,err;
double del,yold,fac,delinv;
i0=MAX(0,CurrentPoint-MaxPoints);
iend=MIN(CurrentPoint-1,MaxPoints-1);
ishift=i0%MaxPoints;
init_sums(T0,CurrentPoint,dt,i0,iend,ishift); /* initialize all the sums */
KnFlag=0;
for(i=0;i<neq;i++){
SETVAR(i+1,y[i]);
yg[i]=y[i];
}
for(i=NODE;i<NODE+NMarkov;i++)
SETVAR(i+1+FIX_VAR,y[i]);
SETVAR(0,t-dt);
for(i=NODE;i<NODE+FIX_VAR;i++)
SETVAR(i+1,evaluate(my_ode[i]));
for(i=0;i<NODE;i++){
if(!EqType[i])yp2[i]=y[i]+dt2*evaluate(my_ode[i]);
else yp2[i]=0.0;
}
KnFlag=1;
while(1){
get_kn(yg,t);
for(i=NODE;i<NODE+FIX_VAR;i++)
SETVAR(i+1,evaluate(my_ode[i]));
for(i=0;i<NODE;i++){
yp[i]=evaluate(my_ode[i]);
/* plintf(" yp[%d]=%g\n",i,yp[i]); */
if(EqType[i])errvec[i]=-yg[i]+yp[i];
else errvec[i]=-yg[i]+dt2*yp[i]+yp2[i];
}
/* Compute Jacobian */
for(i=0;i<NODE;i++){
del=NEWT_ERR*MAX(NEWT_ERR,fabs(yg[i]));
yold=yg[i];
yg[i]+=del;
delinv=1./del;
get_kn(yg,t);
for(j=NODE;j<NODE+FIX_VAR;j++)
SETVAR(j+1,evaluate(my_ode[j]));
for(j=0;j<NODE;j++){
fac=delinv;
if(!EqType[j])fac*=dt2;
jac[j*NODE+i]=(evaluate(my_ode[j])-yp[j])*fac;
}
yg[i]=yold;
}
for(i=0;i<NODE;i++)
jac[n1*i]-=1.0;
sgefa(jac,NODE,NODE,ipivot,&info);
if(info!=-1)
{
return(-1); /* Jacobian is singular */
}
err=0.0;
sgesl(jac,NODE,NODE,ipivot,errvec,0);
for(i=0;i<NODE;i++){
err=MAX(fabs(errvec[i]),err);
yg[i]-=errvec[i];
}
if(err<EulTol) break;
iter++;
if(iter>MaxEulIter)return(-2); /* too many iterates */
}
/* We have a good point; lets save it */
get_kn(yg,t);
/* for(i=NODE;i<NODE+FIX_VAR;i++)
SETVAR(i+1,evaluate(my_ode[i])); */
for(i=0;i<NODE;i++)y[i]=yg[i];
ind=CurrentPoint%MaxPoints;
for(i=0;i<NODE+FIX_VAR+NMarkov;i++)
Memory[i][ind]=GETVAR(i+1);
CurrentPoint++;
return(0);
}
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