MuonCalib::Legendre Namespace Reference

Functions
constexpr double	coeff (const unsigned l, const unsigned k)
	Calculates the n-th coefficient of the legendre polynomial series. More...
template<unsigned l, unsigned k>
constexpr double	polySum (const double x)
	Assembles the sum of the legendre monomials. More...
template<unsigned l, unsigned k, unsigned d>
constexpr double	derivativeSum (const double x)
	Assembles the n-th derivative of the legendre polynomial. More...
constexpr double	derivativeSum (const unsigned l, const unsigned d, const double x)
	Assembles the n-th derivative of a legendre polynomial at run time. More...
constexpr double	polySum (const unsigned l, const double x)
	Assembles the legendre polynomial at run-time. More...
template<unsigned l>
constexpr double	poly (const double x)
	Evaluates the n-th Legendre polynomial at x. More...
template<unsigned l, unsigned d>
constexpr double	derivative (const double x)
	Evaluates the d-th derivative of the n-th Legendre polynomial at x. More...

Function Documentation

coeff()

constexpr double MuonCalib::Legendre::coeff ( const unsigned  l, const unsigned  k  )

constexpr

Calculates the n-th coefficient of the legendre polynomial series.

Parameters

l	Order of the legendre polynomial
k	Coefficient insides the polynomial representation

Definition at line 92 of file LegendrePoly.h.

                                                                   {
            if (k %2 != l %2) {
                return 0.;
            } else if (k > 1) {
                const double a_k = -(1.*(l- k +2)*(l+ k-1)) / (1.*(k * (k-1))) * coeff(l, k-2);
                return a_k;
            } else {
                unsigned fl =  (l - l %2) /2;
                unsigned long binom = binomial(l,fl) * binomial(2*l - 2*fl,l);
                return (fl % 2 ? -1. : 1.) * pow(0.5, l) * (1.*binom);
            }
        }

derivative()

template<unsigned l, unsigned d>

constexpr double MuonCalib::Legendre::derivative ( const double  x )

constexpr

Evaluates the d-th derivative of the n-th Legendre polynomial at x.

Parameters

l	Order of the Legendre polynomial
d	Order of the respective derivative
x	Point of evaluation [-1;1]

Definition at line 164 of file LegendrePoly.h.

                                                        {
                return derivativeSum<l,l,d>(x);
            }

derivativeSum() [1/2]

template<unsigned l, unsigned k, unsigned d>

constexpr double MuonCalib::Legendre::derivativeSum ( const double  x )

constexpr

Assembles the n-th derivative of the legendre polynomial.

Parameters

l	Order of the legendre polynomial
k	Term in the polynomial to add
d	Order of the derivative
x	Point of evaluation [-1,1]

Definition at line 123 of file LegendrePoly.h.

                                                           {
                if constexpr(k <= l && k>=d) {
                    constexpr unsigned long powFac = factorial(k) / factorial(k-d);
                    const double a_n = coeff(l,k) * powFac;
                    return a_n *pow<k-d>(x) + derivativeSum<l, k-2, d>(x);
                } else {
                     return 0.;
                }
            }

derivativeSum() [2/2]

constexpr double MuonCalib::Legendre::derivativeSum ( const unsigned  l, const unsigned  d, const double  x  )

constexpr

Assembles the n-th derivative of a legendre polynomial at run time.

Parameters

l	Order of the legendre polynomial
d	Order of the derivative
x	Point of evaluation [-1,1]

Definition at line 136 of file LegendrePoly.h.

                                                                                           {
            double sum{0.};
            for (int k=l; k>= static_cast<int>(d) && k >=0; k-=2) {
                const double a_n = coeff(l,k) * factorial(k) / factorial(k-d);
                sum += a_n * pow(x,k-d);
            }
            return sum;
        }

poly()

template<unsigned l>

constexpr double MuonCalib::Legendre::poly ( const double  x )

constexpr

Evaluates the n-th Legendre polynomial at x.

Parameters

l	Order of the Legendre polynoimal
x	Point of evaluation [-1;1]

Definition at line 156 of file LegendrePoly.h.

                                                  {
                return polySum<l,l>(x);
        }

polySum() [1/2]

template<unsigned l, unsigned k>

constexpr double MuonCalib::Legendre::polySum ( const double  x )

constexpr

Assembles the sum of the legendre monomials.

Parameters

l	Order of the legendre polynomial
k	Term in the polynomial to add to the sum
x	Point of evaluation [-1,1]

Definition at line 109 of file LegendrePoly.h.

                                                     {
                const double a_n = coeff(l,k);
                if constexpr (k > 1) {                    
                    return a_n* pow<k>(x) + polySum<l, k-2>(x);
                } else{
                    return a_n*pow<k>(x);
                }
            }

polySum() [2/2]

constexpr double MuonCalib::Legendre::polySum ( const unsigned  l, const double  x  )

constexpr

Assembles the legendre polynomial at run-time.

Definition at line 145 of file LegendrePoly.h.

                                                                   {
            double sum{0.};
            for (unsigned k = l%2; k<=l; k+=2) {
                sum += pow(x,k) * coeff(l,k);
            }
            return sum;
        }