T bessel_jn(int n, T x, const Policy& pol)
{
T value(0), factor, current, prev, next;
BOOST_MATH_STD_USING
//
// Reflection has to come first:
//
if (n < 0)
{
factor = static_cast<T>((n & 0x1) ? -1 : 1); // J_{-n}(z) = (-1)^n J_n(z)
n = -n;
}
else
{
factor = 1;
}
if(x < 0)
{
factor *= (n & 0x1) ? -1 : 1; // J_{n}(-z) = (-1)^n J_n(z)
x = -x;
}
//
// Special cases:
//
if(asymptotic_bessel_large_x_limit(T(n), x))
return factor * asymptotic_bessel_j_large_x_2<T>(T(n), x);
if (n == 0)
{
return factor * bessel_j0(x);
}
if (n == 1)
{
return factor * bessel_j1(x);
}
if (x == 0) // n >= 2
{
return static_cast<T>(0);
}
BOOST_ASSERT(n > 1);
T scale = 1;
if (n < abs(x)) // forward recurrence
{
prev = bessel_j0(x);
current = bessel_j1(x);
policies::check_series_iterations<T>("boost::math::bessel_j_n<%1%>(%1%,%1%)", n, pol);
for (int k = 1; k < n; k++)
{
T fact = 2 * k / x;
//
// rescale if we would overflow or underflow:
//
if((fabs(fact) > 1) && ((tools::max_value<T>() - fabs(prev)) / fabs(fact) < fabs(current)))
{
scale /= current;
prev /= current;
current = 1;
}
value = fact * current - prev;
prev = current;
current = value;
}
}
else if((x < 1) || (n > x * x / 4) || (x < 5))
{
return factor * bessel_j_small_z_series(T(n), x, pol);
}
else // backward recurrence
{
T fn;
int s; // fn = J_(n+1) / J_n
// |x| <= n, fast convergence for continued fraction CF1
boost::math::detail::CF1_jy(static_cast<T>(n), x, &fn, &s, pol);
prev = fn;
current = 1;
// Check recursion won't go on too far:
policies::check_series_iterations<T>("boost::math::bessel_j_n<%1%>(%1%,%1%)", n, pol);
for (int k = n; k > 0; k--)
{
T fact = 2 * k / x;
if((fabs(fact) > 1) && ((tools::max_value<T>() - fabs(prev)) / fabs(fact) < fabs(current)))
{
prev /= current;
scale /= current;
current = 1;
}
next = fact * current - prev;
prev = current;
current = next;
}
value = bessel_j0(x) / current; // normalization
scale = 1 / scale;
}
value *= factor;
if(tools::max_value<T>() * scale < fabs(value))
return policies::raise_overflow_error<T>("boost::math::bessel_jn<%1%>(%1%,%1%)", 0, pol);
return value / scale;
}
T bessel_jn(int n, T x, const Policy& pol)
{
T value(0), factor, current, prev, next;
BOOST_MATH_STD_USING
//
// Reflection has to come first:
//
if (n < 0)
{
factor = (n & 0x1) ? -1 : 1; // J_{-n}(z) = (-1)^n J_n(z)
n = -n;
}
else
{
factor = 1;
}
//
// Special cases:
//
if (n == 0)
{
return factor * bessel_j0(x);
}
if (n == 1)
{
return factor * bessel_j1(x);
}
if (x == 0) // n >= 2
{
return static_cast<T>(0);
}
typedef typename bessel_asymptotic_tag<T, Policy>::type tag_type;
if(fabs(x) > asymptotic_bessel_j_limit<T>(n, tag_type()))
return factor * asymptotic_bessel_j_large_x_2<T>(n, x);
BOOST_ASSERT(n > 1);
if (n < abs(x)) // forward recurrence
{
prev = bessel_j0(x);
current = bessel_j1(x);
for (int k = 1; k < n; k++)
{
value = 2 * k * current / x - prev;
prev = current;
current = value;
}
}
else // backward recurrence
{
T fn; int s; // fn = J_(n+1) / J_n
// |x| <= n, fast convergence for continued fraction CF1
boost::math::detail::CF1_jy(static_cast<T>(n), x, &fn, &s, pol);
// tiny initial value to prevent overflow
T init = sqrt(tools::min_value<T>());
prev = fn * init;
current = init;
for (int k = n; k > 0; k--)
{
next = 2 * k * current / x - prev;
prev = current;
current = next;
}
T ratio = init / current; // scaling ratio
value = bessel_j0(x) * ratio; // normalization
}
value *= factor;
return value;
}
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");
}
