ExtendedMath.h

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#ifndef EXTENDEDMATH_H
#define EXTENDEDMATH_H

#include <iostream> //std::istream and std::stringstream
#include <fstream> //std::fstream
#include <cmath>
#include <complex>
#include <limits>

#include "NumericalVector.h"
#include "Vector.h"
#include "Quaternion.h"

/*
 * \brief
 */
namespace constants
{
//Values from WolframAlpha
const Mdouble pi = 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068;
const Mdouble sqrt_pi = 1.772453850905516027298167483341145182797549456122387128213807789852911284591032181374950656738544665;
const Mdouble sqr_pi = 9.869604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232;
const Mdouble sqrt_2 = 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573;
const Mdouble sqrt_3 = 1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576;
const std::complex<Mdouble> i = {0.0, 1.0};
}

namespace mathsFunc
{
Mdouble gamma(Mdouble gamma_in);

Mdouble beta(Mdouble z, Mdouble w);


Mdouble chi_squared(Mdouble x, unsigned int k);

Mdouble chi_squared_prob(Mdouble x, unsigned int k);

Mdouble goldenSectionSearch(Mdouble (* function)(const Mdouble), Mdouble min, Mdouble cur, Mdouble max,
                            Mdouble endCondition, Mdouble curVal = std::numeric_limits<Mdouble>::quiet_NaN());

template<typename T>
int sign(T val)
{
    return (T(0) < val) - (val < T(0));
}

template<typename T>
T square(const T val)
{
    return val * val;
}

template<typename T>
T cubic(const T val)
{
    return val * val * val;
}

bool isEqual(Mdouble v1, Mdouble v2, Mdouble absError);

bool isEqual(Vec3D v1, Vec3D v2, Mdouble absError);

bool isEqual(Matrix3D m1, Matrix3D m2, Mdouble absError);

bool isEqual(MatrixSymmetric3D m1, MatrixSymmetric3D m2, Mdouble absError);

bool isEqual(Quaternion v1, Quaternion v2, double absError);

template<typename T>
constexpr T factorial(const T t)
{
    return (t == 0) ? 1 : t * factorial(t - 1);
}

//platform independent implementation of sine and cosine, taken from
// http://stackoverflow.com/questions/18662261/fastest-implementation-of-sine-cosine-and-square-root-in-c-doesnt-need-to-b
// (cosine was implemented wrongly on the website, here is a corrected version)

// sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...
Mdouble sin(Mdouble x);

// cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...
Mdouble cos(Mdouble x);

Mdouble exp(Mdouble Exponent);

Mdouble log(Mdouble Power);


// tan=sin/cos
template<typename T>
T tan(T x)
{
    return sin(x) / cos(x);
}


Mdouble chebyshev(Mdouble x, const Mdouble coef[], int N);

Mdouble I0_exp(Mdouble x);

Mdouble I0(Mdouble x);
    
}

/*
 * \brief Namespace for functions required to calculate spherical harmonics
 */

namespace sphericalHarmonics
{

//Compute all the associated LegenderePolynomials up to order n, and only positive order m at location x
NumericalVector<> associatedLegendrePolynomials(int n, Mdouble x);

//Compute all spherical harmonics up to order p, at angles theta and phi
NumericalVector<std::complex<Mdouble>> sphericalHarmonics(int p, Mdouble theta, Mdouble phi);

//Compute all squaredFactorials (see eqn 5.23 in a short course on fast multipole methods) up to order p
NumericalVector<> computeSquaredFactorialValues(int p);
}

#endif