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Sinc.cpp
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/* -*- c++ -*-
* Copyright (c) 2012-2026 by the GalSim developers team on GitHub
* https://github.com/GalSim-developers
*
* This file is part of GalSim: The modular galaxy image simulation toolkit.
* https://github.com/GalSim-developers/GalSim
*
* GalSim is free software: redistribution and use in source and binary forms,
* with or without modification, are permitted provided that the following
* conditions are met:
*
* 1. Redistributions of source code must retain the above copyright notice, this
* list of conditions, and the disclaimer given in the accompanying LICENSE
* file.
* 2. Redistributions in binary form must reproduce the above copyright notice,
* this list of conditions, and the disclaimer given in the documentation
* and/or other materials provided with the distribution.
*/
#include <cmath>
#include "math/Sinc.h"
#include "math/Angle.h"
namespace galsim {
namespace math {
// sinc(x) is defined here as sin(Pi x) / (Pi x)
double sinc(double x)
{
if (std::abs(x) < 1.e-4) return 1.- (M_PI*M_PI/6.)*x*x;
else return std::sin(M_PI*x)/(M_PI*x);
}
double f_pade(double x)
{
// I used Maple to calculate a Chebyshev-Pade approximation of 1/sqrt(y) f(1/sqrt(y))
// from 0..1/4^2, which leads to the following formula for f(x). It is accurate to
// better than 1.e-16 for x > 4.
double y = 1./(x*x);
double f =
(1. +
y*(7.44437068161936700618e2 +
y*(1.96396372895146869801e5 +
y*(2.37750310125431834034e7 +
y*(1.43073403821274636888e9 +
y*(4.33736238870432522765e10 +
y*(6.40533830574022022911e11 +
y*(4.20968180571076940208e12 +
y*(1.00795182980368574617e13 +
y*(4.94816688199951963482e12 +
y*(-4.94701168645415959931e11)))))))))))
/ (x*(1. +
y*(7.46437068161927678031e2 +
y*(1.97865247031583951450e5 +
y*(2.41535670165126845144e7 +
y*(1.47478952192985464958e9 +
y*(4.58595115847765779830e10 +
y*(7.08501308149515401563e11 +
y*(5.06084464593475076774e12 +
y*(1.43468549171581016479e13 +
y*(1.11535493509914254097e13)))))))))));
return f;
}
double g_pade(double x)
{
// Similarly, a Chebyshev-Pade approximation of 1/y g(1/sqrt(y)) from 0..1/4^2
// leads to the following formula for g(x), which is also accurate to better than
// 1.e-16 for x > 4.
double y = 1./(x*x);
double g =
y*(1. +
y*(8.1359520115168615e2 +
y*(2.35239181626478200e5 +
y*(3.12557570795778731e7 +
y*(2.06297595146763354e9 +
y*(6.83052205423625007e10 +
y*(1.09049528450362786e12 +
y*(7.57664583257834349e12 +
y*(1.81004487464664575e13 +
y*(6.43291613143049485e12 +
y*(-1.36517137670871689e12)))))))))))
/ (1. +
y*(8.19595201151451564e2 +
y*(2.40036752835578777e5 +
y*(3.26026661647090822e7 +
y*(2.23355543278099360e9 +
y*(7.87465017341829930e10 +
y*(1.39866710696414565e12 +
y*(1.17164723371736605e13 +
y*(4.01839087307656620e13 +
y*(3.99653257887490811e13))))))))));
return g;
}
// Utility for calculating the integral of sin(t)/t from 0 to x. Note the official definition
// does not have pi multiplying t.
double Si(double x)
{
#if 0
// These are the version Gary had taken from Abramowitz & Stegun's formulae.
// Unfortunately, they don't seem to be quite accurate enough for our needs.
// They are accurate to better than 1.e-6, but since our calculation of the
// fourier transform of Lanczon(n,x) involves subtracting multiples of these
// from each other, there is a lot of cancelling, and the resulting relative
// accuracy for uval is much worse than 1.e-6.
double x2=x*x;
if(x2>=3.8) {
// Use rational approximation from Abramowitz & Stegun
// cf. Eqns. 5.2.38, 5.2.39, 5.2.8 - where it says it's good to <1e-6.
// ain't this pretty?
return (M_PI/2.)*((x>0.)?1.:-1.)
- (38.102495+x2*(335.677320+x2*(265.187033+x2*(38.027264+x2))))
/ (x* (157.105423+x2*(570.236280+x2*(322.624911+x2*(40.021433+x2)))) )*std::cos(x)
- (21.821899+x2*(352.018498+x2*(302.757865+x2*(42.242855+x2))))
/ (x2*(449.690326+x2*(1114.978885+x2*(482.485984+x2*(48.196927+x2)))))*std::sin(x);
} else {
// x2<3.8: the series expansion is the better approximation, A&S 5.2.14
dbg<<"Calculate Si(x) for x = "<<x<<std::endl;
double n1=1.;
double n2=1.;
double tt=x;
double t=0;
for(int i=1; i<7; i++) {
t += tt/(n1*n2);
tt = -tt*x2;
n1 = 2.*double(i)+1.;
n2*= n1*2.*double(i);
}
return t;
}
#else
double x2 = x*x;
if (x2 > 16.) {
// For |x| > 4, we use the asymptotic formula:
//
// Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x)
//
// where f(x) = int(sin(t)/(x+t),t=0..inf)
// g(x) = int(cos(t)/(x+t),t=0..inf)
//
// (By asymptotic, I mean that f and g approach 1/x and 1/x^2 respectively as x -> inf.
// The formula as given is exact.)
//
double f = f_pade(x);
double g = g_pade(x);
double sinx,cosx;
math::sincos(x, sinx, cosx);
return ((x>0.)?(M_PI/2.):(-M_PI/2.)) - f*cosx - g*sinx;
} else {
// Here I used Maple to calculate the Pade approximation for Si(x), which is accurate
// to better than 1.e-16 for x < 4:
return
x*(1. +
x2*(-4.54393409816329991e-2 +
x2*(1.15457225751016682e-3 +
x2*(-1.41018536821330254e-5 +
x2*(9.43280809438713025e-8 +
x2*(-3.53201978997168357e-10 +
x2*(7.08240282274875911e-13 +
x2*(-6.05338212010422477e-16))))))))
/ (1. +
x2*(1.01162145739225565e-2 +
x2*(4.99175116169755106e-5 +
x2*(1.55654986308745614e-7 +
x2*(3.28067571055789734e-10 +
x2*(4.5049097575386581e-13 +
x2*(3.21107051193712168e-16)))))));
}
// Note: I also put these formulae on wikipedia, so other people can use them.
// http://en.wikipedia.org/wiki/Trigonometric_integral
// There was a notable lack of information online about how to efficiently calculate
// Si(x), so hopefully this will help people in the future to not have to reproduce
// my work. -MJ
#endif
}
// We don't use Ci for anything, but just in case it's useful for someone,
// here is that function using the same technique as above.
double Ci(double x)
{
double x2 = x*x;
if (x2 > 16.) {
// For |x| > 4, we use the asymptotic formula:
//
// Ci(x) = f(x) sin(x) - g(x) cos(x)
//
// where f(x) and g(x) are as above.
double f = f_pade(x);
double g = g_pade(x);
double sinx,cosx;
math::sincos(x, sinx, cosx);
return f*sinx - g*cosx;
} else {
// Here I used Maple to calculate the Pade approximation for Ci(x), which is accurate
// to better than 1.e-16 for x < 4:
double euler_gamma = 0.5772156649015328606;
return
euler_gamma + std::log(std::abs(x)) +
x2*(-0.25 +
x2*(7.51851524438898291e-3 +
x2*(-1.27528342240267686e-4 +
x2*(1.05297363846239184e-6 +
x2*(-4.68889508144848019e-9 +
x2*(1.06480802891189243e-11 +
x2*(-9.93728488857585407e-15)))))))
/ (1. +
x2*(1.1592605689110735e-2 +
x2*(6.72126800814254432e-5 +
x2*(2.55533277086129636e-7 +
x2*(6.97071295760958946e-10 +
x2*(1.38536352772778619e-12 +
x2*(1.89106054713059759e-15 +
x2*(1.39759616731376855e-18))))))));
}
}
}
}