mathtools.h File Reference

Basic mathematic tools for common tasks. More...

#include <cstddef>

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Functions
void	GaussChebyshevI_unit (size_t N, double *nodes, double *weights)
	Get the Gauss-Chebyshev quadrature based on Chebyshev points of the first type (kind) More...
void	GaussChebyshevII_unit (size_t N, double *nodes, double *weights)
	Get the Gauss-Chebyshev quadrature based on Chebyshev points of the second type (kind) More...
void	GaussLegendre_unit (size_t N, double *nodes, double *weights)
	Compute N-point Gauss-Legendre grids in [-1, 1];. More...
void	LegendreP_order_N (int N, double x, double &y, double &dy)
	Compute Legendre polynomial of order N at x in [-1, 1];. More...
void	transform_GaussQuad_unit2x0inf (double x0, size_t N, double *nodes, double *weights)
	Transform Gauss grids from [-1, 1] to [x0, infty]. More...
void	transform_GaussQuad_unit2x0x1 (double x0, double x1, size_t N, double *nodes, double *weights)
	Transform Gauss grids from [-1, 1] to [x0, x1]. More...
void	transform_GaussQuad_unit2inf (size_t N, double *nodes, double *weights)
	Transform Gauss grids from [-1, 1] to [-infty, infty]. More...
void	transform_GaussQuad_unit2minfx0 (double x0, size_t N, double *nodes, double *weights)
	Transform Gauss grids from [-1, 1] to [-infty, x0]. More...

Detailed Description

Basic mathematic tools for common tasks.

Note

Most functions are adapted from Gauss_Quadrature.cpp

Function Documentation

GaussChebyshevI_unit()

void GaussChebyshevI_unit	(	size_t	N,
		double *	nodes,
		double *	weights
	)

Get the Gauss-Chebyshev quadrature based on Chebyshev points of the first type (kind)

Parameters

N	Number of grid points
nodes	Locations of grid points
weights	Weights of grid points

The nodes are zeros of order-N Chebyshev polynomials of first kind \(T_N(x)\) in \([-1,1]\):

\[ x_i = \cos(\frac{i + 0.5}{N}\pi) \]

The weight function is

\[ w(x) = \frac{1}{\sqrt{1-x^2}}. \]

with \(i = 0, 1, ..., N-1\).

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GaussChebyshevII_unit()

void GaussChebyshevII_unit	(	size_t	N,
		double *	nodes,
		double *	weights
	)

Get the Gauss-Chebyshev quadrature based on Chebyshev points of the second type (kind)

Parameters

N	Number of grid points
nodes	Locations of grid points
weights	Weights of grid points

The nodes are the local extrema of T_N(x) in [-1,1]: x_i = cos( i*PI /(N+1) ) The weght function w(x_i) = PI/(N+1) *sin^2 ( x^2_i ), with i = 1, ..., N.

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GaussLegendre_unit()

void GaussLegendre_unit	(	size_t	N,
		double *	nodes,
		double *	weights
	)

Compute N-point Gauss-Legendre grids in [-1, 1];.

Parameters

N	Number of grid points
nodes	Locations of grid points
weights	Weights of grid points

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LegendreP_order_N()

void LegendreP_order_N	(	int	N,
		double	x,
		double &	y,
		double &	dy
	)

Compute Legendre polynomial of order N at x in [-1, 1];.

Parameters

N	Order of polynomial
x	Grid point to evaluate the polynomial
y	Value of the polynomial at x
dy	value of the derivative of polynomial at x

The following recurrence relation is used: for n = 1, 2, ..., N, nL_n(x) = (2n-1)xL_{n-1}(x) - (n-1)L_{n-2}(x) (x^2-1)L'n(x) = nxL_n(x) - nL{n-1}(x)

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transform_GaussQuad_unit2inf()

void transform_GaussQuad_unit2inf	(	size_t	N,
		double *	nodes,
		double *	weights
	)

Transform Gauss grids from [-1, 1] to [-infty, infty].

Parameters

N	Number of grid points
nodes	Locations of grid points
weights	Weights of grid points

Transform from [-1,1] to [-inf,inf] x' = x/(1-x^2) wx' = (1+x^2)/(1-x^2)^2 *wx

transform_GaussQuad_unit2minfx0()

void transform_GaussQuad_unit2minfx0	(	double	x0,
		size_t	N,
		double *	nodes,
		double *	weights
	)

Transform Gauss grids from [-1, 1] to [-infty, x0].

Parameters

x0	right end of the targeted region
N	Number of grid points
nodes	Locations of grid points
weights	Weights of grid points

Transform from [-1,1] to [-inf,x0] wx' = 1 / (1+x)^2 *wx x' = 0.5 * (x-1)/(1+x) + x0

transform_GaussQuad_unit2x0inf()

void transform_GaussQuad_unit2x0inf	(	double	x0,
		size_t	N,
		double *	nodes,
		double *	weights
	)

Transform Gauss grids from [-1, 1] to [x0, infty].

Parameters

x0	Left end of the targeted region
N	Number of grid points
nodes	Locations of grid points
weights	Weights of grid points

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transform_GaussQuad_unit2x0x1()

void transform_GaussQuad_unit2x0x1	(	double	x0,
		double	x1,
		size_t	N,
		double *	nodes,
		double *	weights
	)

Transform Gauss grids from [-1, 1] to [x0, x1].

Parameters

x0	Left end of the targeted region
x1	Right end of the targeted region
N	Number of grid points
nodes	Locations of grid points
weights	Weights of grid points

Transform from [-1,1] to [x0,x1] x' = (x1-x0)/2 *x + (x0+x1)/2 wx' = (x1-x0)/2 *wx