/**
* 2D trapezoidal integration
*/
float trapzd2d(Func2D *func, float x1, float x2,
float y1, float y2, int n)
{
TrapezFunc2D trapezFunc2D( func, x1, x2 );
return trapzd( &trapezFunc2D, n, y1, y2, n );
}
double qromb(double (*funcd)(), double es, double Wind, double AirDens, double ZO,
double EactAir, double F, double hsalt, double phi_r, double ushear, double Zrh,
double a, double b)
// Returns the integral of the function func from a to b. Integration is performed
// by Romberg's method: Numerical Recipes in C Section 4.3
{
void polint(double xa[], double ya[], int n, double x, double *y, double *dy);
double trapzd(double (*funcd)(), double es, double Wind, double AirDens,
double ZO, double EactAir, double F, double hsalt, double phi_r,
double ushear, double Zrh, double a, double b, int n);
double ss, dss;
double s[MAX_ITER+1], h[MAX_ITER+2];
int j;
h[1] = 1.0;
for(j=1; j<=MAX_ITER; j++) {
s[j]=trapzd(funcd,es, Wind, AirDens, ZO, EactAir, F, hsalt, phi_r,
ushear, Zrh, a,b,j);
if(j >= K) {
polint(&h[j-K],&s[j-K],K,0.0,&ss,&dss);
if (fabs(dss) <= MACHEPS*fabs(ss)) return ss;
}
h[j+1]=0.25*h[j];
}
nrerror("Too many steps in routine qromb");
return 0.0;
}
double CMesh::qromb(meshMemFunc mf, double a, double b) {
const int JMAX=20, JMAXP=JMAX+1, K=5;
const double EPS=1E-10;
double ss, dss;
std::vector<double> s, h, s_t, h_t;
int i,j;
h.push_back(1.0);
for (j=1;j<=JMAX;j++){
s.push_back(trapzd(mf,a,b,j));
if (j>=K){
for (i=0;i<K;i++){
h_t.push_back(h[j-K+i]);
s_t.push_back(s[j-K+i]);
}
polint(h_t,s_t,0.0,ss,dss);
if (fabs(dss) <= EPS*fabs(ss)) {
return ss;
}
h_t.clear();
s_t.clear();
}
h.push_back(0.25*h[j-1]);
}
fprintf(stderr,"Too many steps in routine qromb\n");
exit(1);
return 0.0;
}
template < class T > T qtrap(T (* func) (T), T a, T b )
{
/*
Integrira funkciju 'func' koristeæi trapezoidalno pravilo
NINETGR_EPS - željena relativna toènost
JMAX - max. broj koraka
*/
int j;
T s, olds;
olds = T(-1.e30);
for( j=1; j<=JMAX; j++ )
{
s = trapzd(func,a,b,j);
if( fabs(s-olds) < NINTEGR_EPS*fabs(olds) )
return s;
if( s == 0.0 && olds == 0.0 && j > 6 )
return s;
olds = s;
}
printf("To many steps in NIntegrate !!!");
return T();
}
main(){
int count,n;
double y1, y2, y3;
printf("Numerical integration using Trapzoidal rule\n");
printf("1: 1/(1+0.5*cos(x)) from 0 to pi\n");
printf("2: x*log(abs(x)) from 0 to 1 \n");
printf("3: sin(x) from 0 to pi \n\n");
printf("%10s %15s %15s %19s\n","n","y1","y2","sin(x)");
n=1;
for (count=0; count<10; count++){
n *=2;
y1=trapzd(fun1, 0.0, M_PI, n);
y2=trapzd(fun2, 0.0, 1.0, n);
y3=trapzd(sin, 0.0, M_PI, n);
printf("%10d \t %12.4f\t %12.4f\t %12.4f\n",n, y1, y2, y3);
}
system("pause");
}
void initialize_global()
{
real yo = 0;
real dy = YMAX/NG;
g[0] = 0;
for (int i = 1; i <= NG; i++) {
real y = yo + dy;
g[i] = g[i-1] + trapzd(gaus2, yo, y, EPS);
yo = y;
}
}
int main(void)
{
int i;
float a=0.0,b=PIO2,s;
printf("\nIntegral of func with 2^(n-1) points\n");
printf("Actual value of integral is %10.6f\n",fint(b)-fint(a));
printf("%6s %24s\n","n","approx. integral");
for (i=1;i<=NMAX;i++) {
s=trapzd(func,a,b,i);
printf("%6d %20.6f\n",i,s);
}
return 0;
}
double
qromb(double (*func)(double), double a, double b, double eps, int imax)
{
double ss = 0;
double dss;
int i;
/* allocate enough storage */
if (imax+1 > nmax) {
#ifdef DEBUG
(void)fprintf(stdout, "qromb: malloc %d cells %d bytes\n",
imax+1, (imax+1) * sizeof(double));
#endif
if (h != NULL) {
free(h);
}
h = (double *)malloc((imax+1) * sizeof(double));
if (s != NULL) {
free(s);
}
s = (double *)malloc((imax+1) * sizeof(double));
nmax = imax+1;
}
h[0] = 1;
for (i = 0; i < imax; i++) {
s[i] = trapzd(func, a, b, i);
if (i >= K-1) {
ss = polint(&h[i-(K-1)], &s[i-(K-1)], K, 0.0, &dss);
#ifdef DEBUG
(void)fprintf(stdout, "qromb: a %.15e b %.15e i %d ss %.15e dss %.15e eps %.15e\n",
a, b, i, ss, dss, eps);
#endif
if (fabs(dss) < eps * fabs(ss)) {
return(ss);
}
}
h[i+1] = 0.25 * h[i];
s[i+1] = s[i];
}
#ifdef DEBUG
(void)fprintf(stdout, "qromb: a %.15e b %.15e i %d ss %.15e\n", a, b, i, ss);
#endif
return(ss);
}
float qsimp(float (*func)(float), float a, float b) {
int j;
float s,st,ost=0.0,os=0.0;
for (j=1;j<=JMAX;j++){
st=trapzd(func,a,b,j);
s=(4.0*st-ost)/3.0;
if (j > 5)
if (fabs(s-os) < EPS*fabs(os) ||
(s == 0.0 && os == 0.0)) return s;
os=s;
ost=st;
}
error("Too many steps in routine qsimp", 10);
}
float qtrap(float (*func)(float), float a, float b)
{
float trapzd(float (*func)(float), float a, float b, int n);
void nrerror(char error_text[]);
int j;
float s,olds;
olds = -1.0e30;
for (j=1;j<=JMAX;j++) {
s=trapzd(func,a,b,j);
if (fabs(s-olds) < EPS*fabs(olds)) return s;
olds=s;
}
nrerror("Too many steps in routine qtrap");
return 0.0;
}
Real qsimp(Real (*func)(Real), const Real a, const Real b)
{
int j;
Real s,st,ost,os;
ost = os = -1.0e30;
for (j=1; j<=JMAX; j++) {
st = trapzd(func,a,b,j,ost);
s = (4.0*st-ost)/3.0; /* EQUIVALENT TO SIMPSON'S RULE */
if (j > 5) /* AVOID SPURIOUS EARLY CONVERGENCE. */
if (fabs(s-os) < EPS*fabs(os) || (s == 0.0 && os == 0.0)) return s;
os=s;
ost=st;
}
ath_error("[qsimp]: Too many steps!\n");
return 0.0;
}
double qsimp(double (*func)(double, double *, double *, int, double *,int),
double a,
double b,
double *ss,
double p[],
double *c,
int npts,
double xx[],
int idx)
/*Returns the integral of the function func from a to b. The parameters
EPS can be set to the desired fractional accuracy and JMAX so that 2 to
the power JMAX-1 is the maximum allowed number of steps. Integration is
performed by Simpson's rule. */
{
void trapzd(double (*func)(double, double *, double *, int, double *,int),
double a,
double b,
int n,
double *ss,
double p[],
double *c,
int npts,
double xx[],
int idx);
/*void nrerror(char error_text[]);*/
int j;
double s,st,ost=0.0,os=0.0;
for (j=1;j<=JMAX;j++) {
trapzd(func,a,b,j,ss,p,c,npts,xx,idx);
st = *ss;
s=(4.0*st-ost)/3.0;
if (j > 5) /*Avoid spurious early convergence.*/
if (fabs(s-os) < EPS*fabs(os) ||
(s == 0.0 && os == 0.0))
return s;
os=s;
ost=st;
}
/*nrerror("Too many steps in routine qsimp");*/
return s; /*Never get here.*/
}
float qsimp(float (*func)(float), float a, float b)
{
float trapzd(float (*func)(float), float a, float b, int n);
void nrerror(char error_text[]);
int j;
float s,st,ost=0.0,os=0.0;
for (j=1;j<=JMAX;j++) {
st=trapzd(func,a,b,j);
s=(4.0*st-ost)/3.0;
if (j > 5)
if (fabs(s-os) < EPS*fabs(os) ||
(s == 0.0 && os == 0.0)) return s;
os=s;
ost=st;
}
nrerror("Too many steps in routine qsimp");
return 0.0;
}
double romb(float (*func)(float), float a, float b) {
double s[MAX];
int m,n;
double var;
for (m = 0; m < MAX; m++) s[m] = 1;
for (n = 0; n < MAX; n++) {
for (m = 0; m <= n; m++) {
if (m==0) {
var = s[m];
s[m] = trapzd(func, a, b, n);
} else {
s[n]= ( pow(4 , m-1) * s[m-1] - var )
/ (pow(4, m-1) - 1);
var = s[m]; s[m]= s[n];
}
}
}
return s[MAX-1];
}
float qromb(float (*func)(float), float a, float b)
{
void polint(float xa[], float ya[], int n, float x, float *y, float *dy);
float trapzd(float (*func)(float), float a, float b, int n);
float ss,dss;
float s[JMAXP+1],h[JMAXP+1];
int j;
h[1]=1.0;
for (j=1;j<=JMAX;j++) {
s[j]=trapzd(func,a,b,j);
if (j >= K) {
polint(&h[j-K],&s[j-K],K,0.0,&ss,&dss);
if (fabs(dss) < EPS*fabs(ss)) return ss;
}
s[j+1]=s[j];
h[j+1]=0.25*h[j];
}
error("Too many steps in routine qromb");
return 0.0;
}
double
qsimp (double (*func)(double),
double a, /* lower limit */
double b) /* upper limit */
{
int j;
long double s, st, ost, os;
ost = os = -1.0e30;
for (j = 1; j <= JMAX; j++){
st = trapzd (func, a, b, j);
s = (4.0 * st - ost)/3.0;
if (fabs (s - os) < EPS * fabs(os))
return s;
os = s;
ost = st;
}
fprintf (stderr, "Too many steps in routine QSIMP\n");
// exit (1);
return s;
}
template < class T > T qsimp( T (*func)(T), T a, T b )
{
int j;
T s, st, ost, os;
ost = os = T(-1.0e30);
for( j=1; j<=JMAX; j++ )
{
st = trapzd(func, a, b, j);
s = ( 4.0 * st - ost) / 3.0;
if( fabs(s-os) < EPS * fabs(os) )
return s;
if( s == 0.0 && os == 0.0 && j > 6 )
return s;
os = s;
ost = st;
}
printf("Previse koraka u qsimp");
return T(0.0);
}
/******************************************************************************
* @brief Integration is performed by Romberg's method: Numerical Recipes
* in C Section 4.3
*****************************************************************************/
double
qromb(double (*funcd)(),
double es,
double Wind,
double AirDens,
double ZO,
double EactAir,
double F,
double hsalt,
double phi_r,
double ushear,
double Zrh,
double a,
double b)
{
extern parameters_struct param;
double ss, dss;
double s[param.BLOWING_MAX_ITER + 1];
double h[param.BLOWING_MAX_ITER + 2];
int j;
h[1] = 1.0;
for (j = 1; j <= param.BLOWING_MAX_ITER; j++) {
s[j] = trapzd(funcd, es, Wind, AirDens, ZO, EactAir, F, hsalt, phi_r,
ushear, Zrh, a, b, j);
if (j >= param.BLOWING_K) {
polint(&h[j - param.BLOWING_K], &s[j - param.BLOWING_K],
param.BLOWING_K, 0.0, &ss, &dss);
if (fabs(dss) <= DBL_EPSILON * fabs(ss)) {
return ss;
}
}
h[j + 1] = 0.25 * h[j];
}
log_err("Too many steps");
return 0.; // To avoid warnings.
}
/** Romberg integration.
\param func Pointer to a member function of class model_parameters.
\param a Lower limit of integration.
\param b Upper limit of inegration.
\param ns
\return The integral of the function from a to b using Romberg's method
*/
dvariable function_minimizer::adromb(
dvariable (model_parameters::*func)(const dvariable&),
double a, const dvariable& b, int ns)
{
const double base = 4;
int MAXN = min(JMAX, ns);
dvar_vector s(1,MAXN+1);
for(int j=1; j<=MAXN+1; j++)
{
s[j] = trapzd(func,a,b,j);
}
for(int iter=1; iter<=MAXN+1; iter++)
{
for(int j=1; j<=MAXN+1-iter; j++)
{
s[j] = (pow(base,iter)*s[j+1]-s[j])/(pow(base,iter)-1);
}
}
return s[1];
}
template < class T > T qromb( T (*func)(T), T a, T b )
{
T ss, dss;
T s [JMAXP+1], h[JMAXP+1];
int j;
h[1-1] = T(1.0);
for( j=1; j<=JMAX; j++ )
{
s[j-1] = trapzd(func, a, b, j);
if( j >= K )
{
polint(&h[j-K-1], &s[j-K-1], K, T(0.0), &ss, &dss );
if( fabs(dss) <= EPS*fabs(ss) )
return ss;
}
s[j+1-1] = s[j-1];
h[j+1-1] = 0.25 * h[j-1];
}
puts("To many steps in qromb");
return T(0.0);
}
/***********************************************************************//**
* @brief Perform Romberg integration
*
* @param[in] a Left integration boundary.
* @param[in] b Right integration boundary.
* @param[in] k Integration order (default: k=5)
*
* Returns the integral of the integrand from a to b. Integration is
* performed by Romberg's method of order 2K, where e.g. K=2 in Simpson's
* rule.
* The number of iterations is limited by m_max_iter. m_eps specifies the
* requested fractional accuracy. By default it is set to 1e-6.
*
* @todo Check that k is smaller than m_max_iter
* @todo Make use of std::vector class
***************************************************************************/
double GIntegral::romb(const double& a, const double& b, const int& k)
{
// Initialise result
double result = 0.0;
// Continue only if integration range is valid
if (b > a) {
// Initialise variables
bool converged = false;
double ss = 0.0;
double dss = 0.0;
// Allocate temporal storage
double* s = new double[m_max_iter+2];
double* h = new double[m_max_iter+2];
// Initialise step size
h[1] = 1.0;
s[0] = 0.0;
// Iterative loop
for (m_iter = 1; m_iter <= m_max_iter; ++m_iter) {
// Integration using Trapezoid rule
s[m_iter] = trapzd(a, b, m_iter, s[m_iter-1]);
// Compile option: Check for NaN/Inf
#if defined(G_NAN_CHECK)
if (is_notanumber(s[m_iter]) || is_infinite(s[m_iter])) {
std::cout << "*** ERROR: GIntegral::romb";
std::cout << "(a=" << a << ", b=" << b << ", k=" << k << "):";
std::cout << " NaN/Inf encountered";
std::cout << " (s[" << m_iter << "]=" << s[m_iter] << ")";
std::cout << std::endl;
}
#endif
// Starting from iteration k on, use polynomial interpolation
if (m_iter >= k) {
ss = polint(&h[m_iter-k], &s[m_iter-k], k, 0.0, &dss);
if (std::abs(dss) <= m_eps * std::abs(ss)) {
converged = true;
result = ss;
break;
}
}
// Reduce step size
h[m_iter+1]= 0.25 * h[m_iter];
} // endfor: iterative loop
// Free temporal storage
delete [] s;
delete [] h;
// Dump warning
if (!m_silent) {
if (!converged) {
std::string msg = "Integration uncertainty "+
gammalib::str(std::abs(dss))+
" exceeds tolerance of "+
gammalib::str(m_eps * std::abs(ss))+
" after "+gammalib::str(m_iter)+
" iterations. Result "+
gammalib::str(result)+
" is inaccurate.";
gammalib::warning(G_ROMB, msg);
}
}
} // endif: integration range was valid
// Return result
return result;
}
virtual float func( float y, float n )
{
return trapzd( innerFunc, y, x1, x2, (int)n );
}
/***********************************************************************//**
* @brief Perform Romberg integration
*
* @param[in] a Left integration boundary.
* @param[in] b Right integration boundary.
* @param[in] k Integration order (default: k=5)
*
* Returns the integral of the integrand from a to b. Integration is
* performed by Romberg's method of order 2K, where e.g. K=2 in Simpson's
* rule.
* The number of iterations is limited by m_max_iter. m_eps specifies the
* requested fractional accuracy. By default it is set to 1e-6.
*
* @todo Check that k is smaller than m_max_iter
* @todo Make use of std::vector class
***************************************************************************/
double GIntegral::romb(double a, double b, int k)
{
// Initialise result
bool converged = false;
double result = 0.0;
double ss = 0.0;
double dss = 0.0;
// Continue only if integration range is valid
if (b > a) {
// Allocate temporal storage
double* s = new double[m_max_iter+2];
double* h = new double[m_max_iter+2];
// Initialise step size
h[1] = 1.0;
s[0] = 0.0;
// Iterative loop
for (m_iter = 1; m_iter <= m_max_iter; ++m_iter) {
// Integration using Trapezoid rule
s[m_iter] = trapzd(a, b, m_iter, s[m_iter-1]);
// Compile option: Check for NaN/Inf
#if defined(G_NAN_CHECK)
if (isnotanumber(s[m_iter]) || isinfinite(s[m_iter])) {
std::cout << "*** ERROR: GIntegral::romb";
std::cout << "(a=" << a << ", b=" << b << ", k=" << k << "):";
std::cout << " NaN/Inf encountered";
std::cout << " (s[" << m_iter << "]=" << s[m_iter] << ")";
std::cout << std::endl;
}
#endif
// Starting from iteration k on, use polynomial interpolation
if (m_iter >= k) {
ss = polint(&h[m_iter-k], &s[m_iter-k], k, 0.0, &dss);
if (std::abs(dss) <= m_eps * std::abs(ss)) {
converged = true;
result = ss;
break;
}
}
// Reduce step size
h[m_iter+1]= 0.25 * h[m_iter];
} // endfor: iterative loop
// Free temporal storage
delete [] s;
delete [] h;
// Dump warning
if (!m_silent) {
if (!converged) {
std::cout << "*** WARNING: GIntegral::romb: ";
std::cout << "Integration did not converge ";
std::cout << "(iter=" << m_iter;
std::cout << ", result=" << ss;
std::cout << ", d=" << std::abs(dss);
std::cout << " > " << m_eps * std::abs(ss) << ")";
std::cout << std::endl;
//throw;
}
}
} // endif: integration range was valid
// Return result
return result;
}