static
double integrate(double val) {
double out;
if (val <= SEG1)
out = gauss_legendre(0., val);
else if (val <= SEG2)
out = BASE1 + gauss_legendre(SEG1, val);
else if (val <= SEG3)
out = BASE2 + gauss_legendre(SEG2, val);
else
out = BASE3 + gauss_legendre(SEG3, val);
return out;
}
/* Calculates the volkov density function */
extern
void volkov ( double * res, double * theta0, double * m0, int * J0, int * N0, double * total) {
double data[5]; /* Stores: n, ln, J, upper(=gam), theta */
data[2] = J0[0]; data[3] = m0[0]*(J0[0]-1)/(1-m0[0]); data[4] = theta0[0];
/* some preliminary calculations to speed up
lJm is the term that only depends on J and m */
double lJm = lgamma(data[2]+1) + lgamma(data[3]) - lgamma(data[2]+data[3]);
for (int i = 0; i < J0[0]; i++) {
data[0] = i + 1.0;
data[1] = lJm - lgamma(i+2) - lgamma(data[2]-i);
res[i] = data[4] * gauss_legendre(N0[0],f,data,0,data[3]);
total[0] += res[i];
}
for (int i = 0; i < J0[0]; i++) {
res[i] /= total[0];
}
}
/**
* Gauss-Legendre quadature.
* \param _x array of abscissa
* \param _w array of corresponding wights
*/
void gauss_legendre(const dvector& _x, const dvector& _w)
{
gauss_legendre(0,1,_x,_w);
}
main()
{
double ans, exact, err;
int m;
double eps = 1.0e-6;
double a = 0.0;
double b = PI;
int n_jac = 12;
std::vector<double> x_jac(n_jac);
std::vector<double> w_jac(n_jac);
gauss_jacobi(x_jac, w_jac, n_jac, 1.5, -0.5);
printf("\n\n Abscissas and wights from gauss_jacobi with alpha = 1.5, beta = -0.5.\n");
for (int i = 0; i < n_jac; ++i)
printf("\n x[%d] = %16.12f w[%d] = %16.12f",
i, x_jac[i], i, w_jac[i]);
int n_leg = 12;
std::vector<double> x_leg(n_leg);
std::vector<double> w_leg(n_leg);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
printf("\n\n Abscissas and wights from gauss_legendre.\n");
for (int i = 0; i < n_leg; ++i)
printf("\n x[%d] = %16.12f w[%d] = %16.12f",
i, x_leg[i] / PIO2 - 1.0, i, w_leg[i] / PIO2);
int n_jac2 = 12;
std::vector<double> x_jac2(n_jac2);
std::vector<double> w_jac2(n_jac2);
gauss_jacobi(x_jac2, w_jac2, n_jac2, 0.0, 0.0);
printf("\n\n Abscissas and wights from gauss_jacobi with alpha = beta = 0.0.\n");
for (int i = 0; i < n_jac2; ++i)
printf("\n x[%d] = %16.12f w[%d] = %16.12f",
i, x_jac2[i], i, w_jac2[i]);
int n_cheb = 12;
std::vector<double> x_cheb(n_cheb);
std::vector<double> w_cheb(n_cheb);
gauss_chebyshev(x_cheb, w_cheb, n_cheb);
printf("\n\n Abscissas and wights from gauss_chebyshev.\n");
for (int i = 0; i < (n_cheb + 1) / 2; ++i)
printf("\n x[%d] = -x[%d] = %16.12f w[%d] = w[%d] = %16.12f",
i, n_cheb + 1 - i , x_cheb[i], i, n_cheb + 1 - i, w_cheb[i]);
int n_jac3 = 12;
std::vector<double> x_jac3(n_jac3);
std::vector<double> w_jac3(n_jac3);
gauss_jacobi(x_jac3, w_jac3, n_jac3, -0.5, -0.5);
printf("\n\n Abscissas and wights from gauss_jacobi with alpha = beta = -0.5.\n");
for (int i = 0; i < n_jac3; ++i)
printf("\n x[%d] = %16.12f w[%d] = %16.12f",
i, x_jac3[i], i, w_jac3[i]);
int n_herm = 12;
std::vector<double> x_herm(n_herm);
std::vector<double> w_herm(n_herm);
gauss_hermite(x_herm, w_herm, n_herm);
printf("\n\n Abscissas and wights from gauss_hermite.\n");
for (int i = 0; i < (n_herm + 1) / 2; ++i)
printf("\n x[%d] = -x[%d] = %16.12f w[%d] = w[%d] = %16.12f",
i, n_herm + 1 - i , x_herm[i], i, n_herm + 1 - i, w_herm[i]);
n_lag = 12;
std::vector<double> x_lag(n_lag);
std::vector<double> w_lag(n_lag);
gauss_laguerre(x_lag, w_lag, n_lag, 1.0);
printf("\n\n Abscissas and wights from gauss_laguerre.\n");
for (int i = 0; i < n_lag; ++i)
printf("\n x[%d] = %16.12f w[%d] = %16.12f",
i , x_lag[i], i, w_lag[i]);
m = 40;
std::vector<double> c(m);
std::vector<double> cint(m);
std::vector<double> cder(m);
printf("\n\n\n\n Test of integration routines...");
printf("\n\n");
printf("\n\n %-40s %g", "Input requested error", eps);
printf("\n\n %-40s %d", "Input order of Gaussian quadrature", n_leg);
printf("\n\n %-40s %d", "Input order of Chebyshev fit", m);
printf("\n\n");
a = 0.0;
b = PI;
printf("\n\n Integrate cos(x) from a = %f to b = %f . . .", a, PI);
exact = 0.0;
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_trapezoid(std::cos, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Trapezoid rule", ans, err);
ans = quad_simpson(std::cos, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Simpson's rule", ans, err);
ans = quad_romberg(std::cos, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Romberg integration", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(std::cos, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
chebyshev_fit(a, b, c, m, cos);
chebyshev_integ(a, b, c, cint, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = clenshaw_curtis_quad(a, b, c, m, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Clenshaw - Curtis quadrature", ans, err);
a = 0.0;
b = PI;
printf("\n\n Integrate sin(x) from a = %f to b = %f . . .", a, b);
exact = 2.0;
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_trapezoid(sin, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Trapezoid rule", ans, err);
ans = quad_simpson(sin, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Simpson's rule", ans, err);
ans = quad_romberg(sin, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Romberg integration", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(sin, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
chebyshev_fit(a, b, c, m, sin);
chebyshev_integ(a, b, c, cint, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = clenshaw_curtis_quad(a, b, c, m, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Clenshaw - Curtis quadrature", ans, err);
a = 0.0;
b = PI;
printf("\n\n Integrate cos^2(x) from a = %f to b = %f . . .", a, b);
exact = PI/2.0;
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_trapezoid(cos2, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Trapezoid rule", ans, err);
ans = quad_simpson(cos2, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Simpson's rule", ans, err);
ans = quad_romberg(cos2, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Romberg integration", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(cos2, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
chebyshev_fit(a, b, c, m, cos2);
chebyshev_integ(a, b, c, cint, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = clenshaw_curtis_quad(a, b, c, m, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Clenshaw - Curtis quadrature", ans, err);
a = 0.0;
b = PI;
printf("\n\n Integrate sin^2(x) from a = %f to b = %f . . .", a, b);
exact = PI/2.0;
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_trapezoid(sin2, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Trapezoid rule", ans, err);
ans = quad_simpson(sin2, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Simpson's integral", ans, err);
ans = quad_romberg(sin2, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Romberg integration", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(sin2, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
chebyshev_fit(a, b, c, m, sin2);
chebyshev_integ(a, b, c, cint, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = clenshaw_curtis_quad(a, b, c, m, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Clenshaw - Curtis quadrature", ans, err);
a = 0.0;
b = PI;
printf("\n\n Integrate J_1(x) from a = %f to b = %f . . .", a, b);
exact = bessel_j0(0.0) - bessel_j0(PI);
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_trapezoid(bessel_j1, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Trapezoid rule", ans, err);
ans = quad_simpson(bessel_j1, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Simpson's integral", ans, err);
ans = quad_romberg(bessel_j1, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Romberg integration", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(bessel_j1, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
chebyshev_fit(a, b, c, m, bessel_j1);
chebyshev_integ(a, b, c, cint, m);
chebyshev_deriv(a, b, c, cder, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = clenshaw_curtis_quad(a, b, c, m, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Clenshaw - Curtis quadrature", ans, err);
a = 0.0;
b = 10.0*PI;
printf("\n\n Integrate foo(x) = (1 - x)exp(-x/2) from a = %f to b = %f . . .", a, b);
exact = 2.0*(1.0 + b)*exp(-b/2.0) - 2.0*(1.0 + a)*exp(-a/2.0);
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_trapezoid(foo, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Trapezoid rule", ans, err);
ans = quad_simpson(foo, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Simpson's rule", ans, err);
ans = quad_romberg_open(foo, a, b, eps, midpoint_exp);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Open Romberg integration", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(foo, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
gauss_laguerre(x_lag, w_lag, n_lag, 0.0);
ans = 0.0;
for (int i = 0; i < n_lag; ++i)
ans += 2 * w_lag[i] * foonum(2.0 * x_lag[i]);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Laguerre quadrature", ans, err);
chebyshev_fit(a, b, c, m, foo);
chebyshev_integ(a, b, c, cint, m);
chebyshev_deriv(a, b, c, cder, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = clenshaw_curtis_quad(a, b, c, m, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Clenshaw - Curtis quadrature", ans, err);
a = 0.0;
b = PI;
printf("\n\n Integrate funk1(x) = cos(x)/sqrt(x(PI - x)) from a = %f to b = %f . . .", a, b);
exact = 0.0;
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_romberg_open(funk1, a, b, eps, midpoint);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Open Romberg quadrature with midpoint", ans, err);
ans = quad_romberg_open(funk1, a, (a+b)/2, eps, midpoint_inv_sqrt_lower)
+ quad_romberg_open(funk1, (a+b)/2, b, eps, midpoint_inv_sqrt_upper);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Open Romberg with inverse sqrt step", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(funk1, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
ans = quad_gauss(funk1num, a, b, x_cheb, w_cheb, n_cheb);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Chebyshev quadrature", ans, err);
ans = quad_gauss(funk1num, a, b, x_jac, w_jac, n_jac);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Jacobi quadrature", ans, err);
chebyshev_fit(a, b, c, m, funk1);
chebyshev_integ(a, b, c, cint, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = dumb_gauss_crap(funk1num, a, b, 8);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Chebyshev quadrature", ans, err);
printf("\n\n");
plot_func(funk1, a+0.1, b-0.1, "", "", "", "");
a = 0.0;
b = PI;
printf("\n\n Integrate funk2(x) = (2.0+sin(x))/sqrt(x(PI - x)) from a = %f to b = %f . . .", a, b);
exact = 0.0;
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_romberg_open(funk2, a, b, eps, midpoint);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Open Romberg quadrature with midpoint", ans, err);
ans = quad_romberg_open(funk2, a, (a+b)/2, eps, midpoint_inv_sqrt_lower)
+ quad_romberg_open(funk2, (a+b)/2, b, eps, midpoint_inv_sqrt_upper);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Open Romberg with inverse sqrt step", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(funk2, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
ans = quad_gauss(funk2num, a, b, x_cheb, w_cheb, n_cheb);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Chebyshev quadrature", ans, err);
ans = quad_gauss(funk2num, a, b, x_jac, w_jac, n_jac);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Jacobi quadrature", ans, err);
chebyshev_fit(a, b, c, m, funk2);
chebyshev_integ(a, b, c, cint, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = dumb_gauss_crap(funk2num, a, b, 8);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Adaptive Gauss - Chebyshev quadrature", ans, err);
printf("\n\n");
plot_func(funk2, a+0.1, b-0.1, "", "", "", "");
printf("\n\n");
}
int main()
{
// generate a sequence of quadrature tables
size_t maxorder = 200;
size_t ii,jj,kk,ll;
double * coeffs = calloc_double((maxorder/2)*(maxorder/2)*maxorder);
struct OrthPoly * leg = init_leg_poly();
size_t nquad = 3*maxorder+2;
double * pt = calloc_double(nquad);
double * wt = calloc_double(nquad);
gauss_legendre(nquad,pt,wt);
double * evals = calloc_double(nquad * (maxorder+1));
for (kk = 0; kk < maxorder; kk++){
for (ll = 0; ll < nquad; ll++){
if (kk == 0){
evals[ll + kk*nquad] = orth_poly_eval(leg,0,pt[ll]);
}
else if (kk == 1){
evals[ll + kk*nquad] = orth_poly_eval(leg,1,pt[ll]);
}
else {
evals[ll + kk*nquad] = eval_orth_poly_wp(leg,evals[ll + (kk-2)*nquad],
evals[ll + (kk-1)*nquad],kk, pt[ll]);
}
}
}
size_t total = 0;
size_t nonzero = 0;
double zerothresh = 1e-14;
for (ii = 0; ii < maxorder/2; ii++){
for (jj = 0; jj < maxorder/2; jj++){
for (kk = 0; kk < maxorder; kk++){
printf("ii,jj,kk = %zu,%zu,%zu\n",ii,jj,kk);
coeffs[ii+jj*maxorder/2+kk*maxorder*maxorder/4] = 0.0;
for (ll = 0; ll < nquad; ll++){
double evali = evals[ll + ii*nquad];
double evalj = evals[ll + jj*nquad];
double evalk = evals[ll + kk*nquad];
coeffs[ii+jj*maxorder/2+kk*maxorder*maxorder/4] += wt[ll]*evali*evalj*evalk;
}
coeffs[ii+jj*maxorder/2+kk*maxorder*maxorder/4] /= leg->norm(kk);
if (fabs(coeffs[ii+jj*maxorder/2+kk*maxorder*maxorder/4]) > zerothresh){
nonzero++;
}
total += 1;
}
}
}
printf("done and printing\n");
printf("density level = %G\n", (double) nonzero/total);
int success = darray_save(maxorder*maxorder*maxorder/4,1,coeffs,"legpolytens.dat",1);
assert ( success == 1);
free_orth_poly(leg);
free(pt);
free(wt);
free(coeffs);
return 0;
}
