julenum/integrate

Package to compute an integral given a specific list of evaluations.

Index

fn Gaussian[T: integer | float](x: []T, f: []T): T
fn Romberg[T: integer | float](f: []T, dx: T): T
fn Simpsons[T: integer | float](x: []T, f: []T): T
fn Trapezoidal[T: integer | float](x: []T, f: []T): T

Gaussian

fn Gaussian[T: integer | float](x: []T, f: []T): T

Returns an approximate value of the integral

\int_a^b f(x) dx

computed using the 3-point Composite Gauss-Legendre Quadrature rule.

The slice x represents the segment boundaries defining subintervals [x[i], x[i+1]]. The slice f contains the function samples evaluated at 3 Gauss nodes within each subinterval:

For subinterval i, the 3 nodes in physical space are: x_i,0 = mid - half * sqrt(3/5) x_i,1 = mid x_i,2 = mid + half * sqrt(3/5) where mid = (x[i+1] + x[i])/2 and half = (x[i+1] - x[i])/2.

Thus, len(f) must be equal to 3 * (len(x) - 1). The slice x must be sorted in strictly increasing order and len(x) >= 2.

Computations are performed using 64-bit floating-point precision. The result is returned as type T, which may cause rounding errors or loss of precision. To preserve exact results, use 64-bit floating-point type.

Romberg

fn Romberg[T: integer | float](f: []T, dx: T): T

Returns an approximate value of the integral

\int_a^b f(x)dx

computed using the Romberg's method. The function f is given as a slice of equally-spaced samples, that is,

f[i] = f(a + i*dx)

and dx is the spacing between the samples.

The length of f must be 2^k + 1, where k is a positive integer, and dx must be positive.

See https://en.wikipedia.org/wiki/Romberg%27s_method for a description of the algorithm.

Computations are performed using 64-bit floating-point precision. The result is returned as type T, which may cause rounding errors or loss of precision. To preserve exact results, use 64-bit floating-point type.

Simpsons

fn Simpsons[T: integer | float](x: []T, f: []T): T

Returns an approximate value of the integral

\int_a^b f(x)dx

computed using the Simpsons's method. The function f is given as a slice of samples evaluated at locations in x, that is,

f[i] = f(x[i]), x[0] = a, x[len(x)-1] = b

The slice x must be sorted in strictly increasing order. x and f must be of equal length and the length must be at least 3.

See https://en.wikipedia.org/wiki/Simpson%27s_rule#Composite_Simpson's_rule_for_irregularly_spaced_data for more information.

Computations are performed using 64-bit floating-point precision. The result is returned as type T, which may cause rounding errors or loss of precision. To preserve exact results, use 64-bit floating-point type.

Trapezoidal

fn Trapezoidal[T: integer | float](x: []T, f: []T): T

Returns an approximate value of the integral

\int_a^b f(x) dx

computed using the trapezoidal rule. The function f is given as a slice of samples evaluated at locations in x, that is,

f[i] = f(x[i]), x[0] = a, x[len(x)-1] = b

The slice x must be sorted in strictly increasing order. x and f must be of equal length and the length must be at least 2.

The trapezoidal rule approximates f by a piecewise linear function and estimates

\int_x[i]^x[i+1] f(x) dx

as

(x[i+1] - x[i]) * (f[i] + f[i+1])/2

More details on the trapezoidal rule can be found at: https://en.wikipedia.org/wiki/Trapezoidal_rule

Computations are performed using 64-bit floating-point precision. The result is returned as type T, which may cause rounding errors or loss of precision. To preserve exact results, use 64-bit floating-point type.