/* density of a stable distribution */
void stable(int *n, double *y, double *beta, double *alpha, int *npt,
double *up, double *eps, int *type, int *err, double *ffy)
{
int i, j;
double h, s, *eta, *seta, *ceta, *sa;
*err=0;
eta=(double*)R_alloc((size_t)(*n),sizeof(double));
seta=(double*)R_alloc((size_t)(*n),sizeof(double));
ceta=(double*)R_alloc((size_t)(*n),sizeof(double));
sa=(double*)R_alloc((size_t)(*n),sizeof(double));
nn=*n;
if(!eta||!seta||!ceta||!sa){
*err=1;
return;}
for(i=0;i<*n;i++){
ffy[i]=0.0;
eta[i]=beta[i]*(1.0-fabs(1.0-alpha[i]))*M_PI/2.0;
seta[i]=sin(eta[i]);
ceta[i]=cos(eta[i]);}
if(*type==1){
*npt=*npt-*npt%2;
h=*up/ *npt;
for(j=0;j<=*npt;j++){
s=(*npt-j)*h;
for(i=0;i<*n;i++){
sa[i]=pow(s,alpha[i]);
ffy[i]=ffy[i]+(4-2*(j%2==0)-(j==1||j==*npt))*cos(-y[i]*s+sa[i]*seta[i])*exp(-sa[i]*ceta[i]);}}
for(i=0;i<*n;i++)ffy[i]=ffy[i]*h/3.0/M_PI;}
else {
for(i=0;i<*n;i++){
alphai=alpha[i];
yi=y[i];
setai=seta[i];
cetai=ceta[i];
ffy[i]=(romberg(fcn1, *eps)+romberg(fcn2, *eps))/M_PI;}}}
/* cdf of a stable distribution */
void pstable(int *n, double *y, double *beta, double *alpha,
double *eps, int *err, double *ffy)
{
int i;
*err=0;
nn=*n;
for(i=0;i<*n;i++){
ffy[i]=0.0;
etai=beta[i]*(1.0-fabs(1.0-alpha[i]))*M_PI/2.0;
setai=sin(etai);
cetai=cos(etai);
alphai=alpha[i];
yyi=y[i];
if(etai==0.&&yyi==0)
ffy[i]=0.5;
else ffy[i]=0.5+(romberg(fcn3, *eps)+romberg(fcn4, *eps))/M_PI;}}
/* P(Z_j\in (a_j,b_j)): deriv wrt rho */
void r_emvndrh(int *m, double *w, double *x, double *rh, double *eps,
double *deriv)
{ double r_grh(double),der,tem,sum;
double romberg(double (*)(double), double, double, double);
double pnorms(double),phi(double);
int i,k;
double *t;
extern int mm;
extern double *ww,*xx,rs,r1,r32;
mm=*m; rs=sqrt(*rh); r1=sqrt(1.-(*rh)); r32=r1*(1.-(*rh));
xx=(double *) malloc(mm * sizeof(double));
ww=(double *) malloc(mm * sizeof(double));
t=(double *) malloc(mm * sizeof(double));
for(i=0;i<mm;i++) { ww[i]=w[i]; xx[i]=x[i]; }
if((*rh)>=0.) der=romberg(r_grh,-UB,UB,*eps);
else /* rho=0 */
{ for(i=0;i<mm;i++) t[i]=pnorms(x[i])-pnorms(w[i]);
for(k=0,sum=0.;k<mm;k++)
{ for(i=0,tem=1.;i<mm;i++)
{ if(i!=k) { tem*=t[i]; }
else tem*=(x[i]*phi(x[i])-w[i]*phi(w[i]));
// maybe check if x>10 or w<-10?
}
sum+=tem;
}
der=.5*sum;
}
free(xx); free(ww); free(t);
*deriv=der;
}
double romberg(double f(double), double a, double b, int n, int m) {
if((n == 0) && (m == 0)) {
return (double)(1/2) * (b - a) * (f(a) + f(b));
}
else if(m == 0) {
double coefficient = (b - a)/pow(2, n);
double sum = 0;
int k;
for(k = 0; k < (pow(2, n-1)+1); k++) {
sum += f(a + (2*k - 1)*coefficient);
}
return 0.5 * romberg(f, a, b, n-1, 0) + coefficient*sum;
}
double r1 = romberg(f, a, b, n, m-1);
double r2 = romberg(f, a, b, n-1, m-1);
double coefficient = 1/(pow(4,(double)m) - 1);
return coefficient * (pow(4,m)*r1 - r2);
}
//this method is here for 1-d integration (a convenience method)
//the integration limits are placed in arrays of size 1, and other methods are called
double RecursiveIntegration::romberg(MultiVarFunction* f, double a, double b, int n)
{
double* aa = new double[1];
aa[0] = a;
double* bb = new double[1];
bb[0] = b;
double result = romberg(f, aa, bb, n, 1);
delete[] aa;
delete[] bb;
return result;
}
/* P(Z_j\in (a_j,b_j)): deriv wrt a_k or b_k,
ks=-1 for a_k, ks=1 for b_k
*/
void r_emvnd(int *m, double *w, double *x, double *rh, int *k, int *ks,
double *eps, double *deriv)
{ double r_gd(double),der;
double romberg(double (*)(double), double, double, double);
int i;
extern int mm,kk,ksign;
extern double *ww,*xx,rs,r1;
mm=*m; kk=(*k-1); rs=sqrt(*rh); r1=sqrt(1.-(*rh)); ksign=*ks;
xx=(double *) malloc(mm * sizeof(double));
ww=(double *) malloc(mm * sizeof(double));
for(i=0;i<mm;i++) { ww[i]=w[i]; xx[i]=x[i]; }
der=romberg(r_gd,-UB,UB,*eps);
free(xx); free(ww);
*deriv= ksign*der;
}
QString Integration::logInfo()
{
ApplicationWindow *app = (ApplicationWindow *)parent();
QLocale locale = app->locale();
int prec = app->d_decimal_digits;
QString logInfo = "[" + QDateTime::currentDateTime().toString(Qt::LocalDate);
if (d_integrand == AnalyticalFunction){
logInfo += "\n" + tr("Numerical integration of") + " f(" + d_variable + ") = " + d_formula + " ";
logInfo += tr("using a %1 order method").arg(d_method) + "\n";
logInfo += tr("From") + " x = " + locale.toString(d_from, 'g', prec) + " ";
logInfo += tr("to") + " x = " + locale.toString(d_to, 'g', prec) + "\n";
logInfo += tr("Tolerance") + " = " + locale.toString(d_tolerance, 'g', prec) + "\n";
logInfo += tr("Iterations") + ": " + QString::number(romberg()) + "\n";
} else if (d_integrand == DataSet){
if (d_graph)
logInfo += tr("\tPlot")+ ": ''" + d_graph->multiLayer()->objectName() + "'']\n";
else
logInfo += "\n";
QString dataSet;
if (d_curve)
dataSet = d_curve->title().text();
else
dataSet = d_y_col_name;
logInfo += "\n" + tr("Numerical integration of") + ": " + dataSet + " ";
logInfo += tr("using the Trapezoidal Rule") + "\n";
logInfo += tr("Points") + ": " + QString::number(d_n) + " " + tr("from") + " x = " + locale.toString(d_from, 'g', prec) + " ";
logInfo += tr("to") + " x = " + locale.toString(d_to, 'g', prec) + "\n";
// use GSL to find maximum value of data set
gsl_vector *aux = gsl_vector_alloc(d_n);
for(int i=0; i < d_n; i++)
gsl_vector_set (aux, i, fabs(d_y[i]));
int maxID = gsl_vector_max_index (aux);
gsl_vector_free(aux);
logInfo += tr("Peak at") + " x = " + locale.toString(d_x[maxID], 'g', prec)+"\t";
logInfo += "y = " + locale.toString(d_y[maxID], 'g', prec)+"\n";
d_area = trapez();
}
logInfo += tr("Area") + "=" + locale.toString(d_area, 'g', prec);
logInfo += "\n-------------------------------------------------------------\n";
return logInfo;
}
main() {
double EPS = 1.e-12;
double ans = 10000000;
double prevans = 1;
int n = 1;
int m = 2;
double a = 0.0;
double b = 1.0;
printf(" m n ans (anser - prev answer) \n");
while(fabs(ans - prevans) > EPS) {
prevans = ans;
ans = romberg(f, a, b, n, m);
printf(" %d %d %12.8f %12.8f \n", m, n, ans, ans - prevans);
n = n*2;
}
}
int main(int narg, char* argc[])
{
std::cout.setf( std::ios::fixed, std:: ios::floatfield );
std::cout.precision(8);
timestamp_t inicio, fim;
inicio = get_timestamp();
std::vector<Function*> functions;
functions.push_back( new Function1() );
functions.push_back( new Function2() );
RombergIntegration romberg(argc[1], functions, atoi(argc[2]));
std::cout << "Integral: " << romberg.calculateIntegral() << "\n";
fim = get_timestamp();
std::cout << "tempo: " << (fim - inicio)/1000.0L << " milissegundos.\n";
return 0;
}
int main(){
printf("%lf\n",romberg(test,0,1));
printf("%lf\n",simpson(test,0,1,(int)1e6));
}
