example_pkg.grid

A uniform 1D grid.

Classes

Name	Description
Grid	A uniform grid of cells on [start, stop].

Functions

Name	Description
linspace_midpoints	The midpoints of cells uniform cells on [start, stop].
midpoint_integral	The integral of f on [start, stop] by the composite midpoint rule.

Gridclassdataclass#

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class Grid(start: float, stop: float, cells: int)

A uniform grid of cells on [start, stop].

Attributes

Name	Type	Description
start	float	Left end of the domain.
stop	float	Right end of the domain.
cells	int	Number of cells.

cellsattributeinstance attribute#

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cells: int

startattributeinstance attribute#

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start: float

stopattributeinstance attribute#

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stop: float

spacingproperty#

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spacing: float

The cell width $h = (b - a) / n$.

nodesmethod#

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def nodes() -> np.ndarray

The cells + 1 cell boundaries.

Returns

np.ndarray

The node coordinates.

linspace_midpointsfunction#

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def linspace_midpoints(start: float, stop: float, cells: int) -> np.ndarray

The midpoints of cells uniform cells on [start, stop].

Parameters

Name	Type	Description
start	float	Left end of the domain.
stop	float	Right end of the domain.
cells	int	Number of cells.

Returns

np.ndarray

An array of length cells.

midpoint_integralfunction#

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def midpoint_integral(f, start: float, stop: float, cells: int) -> float

The integral of f on [start, stop] by the composite midpoint rule.

Parameters

Name	Type	Description
f		The integrand, called on an array of points.
start	float	Left end of the domain.
stop	float	Right end of the domain.
cells	int	Number of cells.

Returns

float

The approximate integral.

Numerics

rule

$$\int_a^b f(x)\,dx \approx h \sum_{i=0}^{n-1} f\left(\frac{x_i + x_{i+1}}{2}\right)$$

error

$$\left|E\right| \le \frac{(b - a) h^2}{24} \max_{[a, b]} |f''|$$

Order of accuracy: 2.

Assumptions:

• f twice continuously differentiable on [a, b]

References:

• Süli and Mayers, An Introduction to Numerical Analysis (2003), section 7.2