legendre_polynomials.cc File Reference

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Namespaces
namespace	hummingbird

Functions
double	hummingbird::LegendrePolynomial (const int n, const double x)
	Computes the Legendre polynomial of degree n at a point x using the formula:
double	hummingbird::LegendrePolynomialPrime (const int n, const double x)
	Computes the first derivative of the Legendre polynomial of degree n at a point x using the recurrence relation:
double	hummingbird::LegendrePolynomialPrimePrime (const int n, const double x)
	Computes the second derivative of the Legendre polynomial of degree n using Legendre's differential equation:
std::vector< double >	hummingbird::AllLegendreRoots (const int n)
	Compute all roots of the Legendre polynomial of order n.
double	hummingbird::LegendreRoot (const int n, const int k)
	Computes the k-th root of the n-th order Legendre polynomial by first approximating the root with ApproximateLegendreRoot, then using Newton's method to converge to the root.
double	hummingbird::ApproximateLegendreRoot (const int n, const int k)
	Approximates the k-th root of the n-th order Legendre polynomial with:
std::vector< double >	hummingbird::AllLegendrePrimeRoots (const int n)
	Computes all roots of the \(P'_n(x)\) polynomial. A total of n-1 roots will be computed and returned.
double	hummingbird::LegendrePrimeRoot (const int n, const int k)
	Computes the k-th root of the n-th order first derivative of the Legendre polynomial by first approximating with ApproximateLegendrePrimeRoot, then using Newton's method to converge.
double	hummingbird::ApproximateLegendrePrimeRoot (const int n, const int k)
	Approximates the k-th root of the first derivative of the n-th degree Legendre polynomial using the average of the approximations (ApproximateLegendreRoot()) of the surrounding roots of the n-th degree Legendre polynomial. Note that zero-indexing is used, so the roots of the \(P'_4 (x)\) polynomial are \(k=0,1,2\). The approximations of these are given by: