sem_gaussian_1d Namespace Reference

Functions
Union[float, NDArray[np.float64]]	gaussian (Union[float, NDArray[np.float64]] y, float sigma)
Union[float, NDArray[np.float64]]	power_law (Union[float, NDArray[np.float64]] k, float nu)
NDArray[np.float64]	finite_differences_coefficients (int num_points, int derivative_order=1)
float	finite_differences (Callable[[float], float] func, float x_val, int order=1, int grid_points=3, float step_size=1e-5)
float	binomial (int a, int b)
int	hermite_number (int n)
float	gaussian_derivative (float y, float sigma, int order)
float	lattice_contribution (float x_val, float nu, float sigma, int order)
float	sum_func (float x_val, float nu, float sigma)
float	integral (float x_val, float nu, float sigma)
float	sem (float x_val, float nu, float sigma, int order)
Union[float, NDArray[np.float64]]	min_abs_rel_error (Union[float, NDArray[np.float64]] ref, Union[float, NDArray[np.float64]] approx)
None	plot_left (Axes ax, NDArray[np.float64] x_values, dict[str, NDArray[np.float64]] plot_data_values, float nu)
None	plot_right (Axes ax, NDArray[np.float64] x_values, NDArray[np.float64] diff1, Union[NDArray[np.float64], None] diff2)

Variables
int	EPS_TAYLOR = 1e-8
int	EPS_IS_CLOSE = 1e-12
	parser
	type
	default
	help
	args = parser.parse_args()
	NU0 = args.nu
int	SIGMA0 = 100
	x = np.linspace(-2 * SIGMA0, 2 * SIGMA0, 51)
	ax1
	ax2
	figsize
	integral_values = np.array([integral(xx, NU0, SIGMA0) for xx in x])
	sum_func_values = np.array([sum_func(xx, NU0, SIGMA0) for xx in x])
	sem_order_0_values = np.array([sem(xx, NU0, SIGMA0, 0) for xx in x])
	diff_order_0
Union	diff_order_2 = None
	sem_order_2_values = np.array([sem(xx, NU0, SIGMA0, 2) for xx in x])
dict	plot_data

Detailed Description

Singular Euler-Maclaurin (SEM) expansion for a Gaussian function in 1D.

This script demonstrates the application of the Singular Euler-Maclaurin
expansion to a Gaussian function in one dimension. It calculates and compares
the integral, sum, and SEM approximations for various orders.

Features:
- Calculates and plots integral, sum, and SEM approximations
- Compares SEM approximations of different orders
- Provides error analysis between sum and SEM approximations

Usage:
    python sem_gaussian_1d.py [--nu NU]

Arguments:
    --nu NU    Optional. Set the value of nu for calculations. Default is 1.5.
               Example: python sem_gaussian_1d.py --nu 1

The script generates plots comparing the different approximations and their
errors for the specified nu value.

Function Documentation

binomial()

float sem_gaussian_1d.binomial	(	int	a,
		int	b )

Binomial coefficients

finite_differences()

float sem_gaussian_1d.finite_differences	(	Callable[[float], float]	func,
		float	x_val,
		int	order = 1,
		int	grid_points = 3,
		float	step_size = 1e-5 )

Calculate finite differences for given function and parameters.

finite_differences_coefficients()

NDArray[np.float64] sem_gaussian_1d.finite_differences_coefficients	(	int	num_points,
		int	derivative_order = 1 )

Calculate finite differences coefficients for Taylor methods.

gaussian()

Union[float, NDArray[np.float64]] sem_gaussian_1d.gaussian	(	Union[float, NDArray[np.float64]]	y,
		float	sigma )

Gaussian function with input y and standard deviation sigma.

gaussian_derivative()

float sem_gaussian_1d.gaussian_derivative	(	float	y,
		float	sigma,
		int	order )

Calculate the Gaussian derivative for given input, sigma, and order.

hermite_number()

int sem_gaussian_1d.hermite_number

(

int

n

)

Value of the n'th Hermite polynomial at 0

integral()

float sem_gaussian_1d.integral	(	float	x_val,
		float	nu,
		float	sigma )

Analytic representation of the integral over the summands.

lattice_contribution()

float sem_gaussian_1d.lattice_contribution	(	float	x_val,
		float	nu,
		float	sigma,
		int	order )

Calculate the lattice contribution for given parameters.

min_abs_rel_error()

Union[float, NDArray[np.float64]] sem_gaussian_1d.min_abs_rel_error	(	Union[float, NDArray[np.float64]]	ref,
		Union[float, NDArray[np.float64]]	approx )

Calculate the minimum between the absolute and relative error of two numbers.

plot_left()

None sem_gaussian_1d.plot_left	(	Axes	ax,
		NDArray[np.float64]	x_values,
		dict[str, NDArray[np.float64]]	plot_data_values,
		float	nu )

Plot the left subplot with integral, sum, and SEM (order 0) values.

plot_right()

None sem_gaussian_1d.plot_right	(	Axes	ax,
		NDArray[np.float64]	x_values,
		NDArray[np.float64]	diff1,
		Union[NDArray[np.float64], None]	diff2 )

Plot the right subplot with error analysis.

power_law()

Union[float, NDArray[np.float64]] sem_gaussian_1d.power_law	(	Union[float, NDArray[np.float64]]	k,
		float	nu )

Power law function with input k and exponent nu.

sem()

float sem_gaussian_1d.sem	(	float	x_val,
		float	nu,
		float	sigma,
		int	order )

Calculate the SEM function for given parameters.

sum_func()

float sem_gaussian_1d.sum_func	(	float	x_val,
		float	nu,
		float	sigma )

Calculate the sum function for given x, nu, and sigma.

Variable Documentation

diff_order_0

sem_gaussian_1d.diff_order_0

Initial value:

=  np.array(
        [
            min_abs_rel_error(s, s0)
            for s, s0 in zip(sum_func_values, sem_order_0_values)
        ]
    )

parser

sem_gaussian_1d.parser

Initial value:

=  argparse.ArgumentParser(
        description="Calculate SEM function with custom nu value."
    )

plot_data

dict sem_gaussian_1d.plot_data

Initial value:

=  {
        "integral": integral_values,
        "sum_func": sum_func_values,
        "sem_order_0": sem_order_0_values,
    }