Crate lds_rs 

Low-Discrepancy Sequence (LDS) Generator

This library implements a set of low-discrepancy sequence generators, which are used to create sequences of numbers that are more evenly distributed than random numbers. These sequences are particularly useful in various fields such as computer graphics, numerical integration, and Monte Carlo simulations.

The library defines several structs, each representing a different type of low-discrepancy sequence generator. The main types of sequences implemented are:

1. van der Corput sequence
2. Halton sequence
3. Circle sequence
4. Disk sequence
5. Sphere sequence
6. 3-Sphere Hopf sequence
7. N-dimensional Halton sequence

Each generator takes specific inputs, usually in the form of base numbers or sequences of base numbers. These bases determine how the sequences are generated. The generators produce outputs in the form of floating-point numbers or vectors of floating-point numbers, depending on the dimensionality of the sequence.

The core algorithm used in most of these generators is the van der Corput sequence. This sequence is created by expressing integers in a given base, reversing the digits, and placing them after a decimal point. For example, in base 2, the sequence would start: 1/2, 1/4, 3/4, 1/8, 5/8, and so on.

The Halton sequence extends this concept to multiple dimensions by using a different base for each dimension. The Circle and Sphere sequences use trigonometric functions to map these low-discrepancy sequences onto circular or spherical surfaces.

The library also includes utility functions and constants to support these generators. For instance, there’s a list of prime numbers that can be used as bases for the sequences.

Each generator struct has methods to produce the next value in the sequence (pop()) and to reset the sequence to a specific starting point (reseed()). This allows for flexible use of the generators in various applications.

The purpose of this library is to provide a toolkit for generating well-distributed sequences of numbers, which can be used in place of random numbers in many applications to achieve more uniform coverage of a given space or surface. This can lead to more efficient and accurate results in tasks like sampling, integration, and optimization.

Figure: 500 points of a 2D Halton sequence with bases 2 and 3.

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Visual examples

Generator	Visualisation
Van der Corput (base 2) — 64 values
Circle (base 2) — 200 points
Disk (bases 2, 3) — 200 points
Halton 2D (bases 2, 3) — 500 points

Modules

ilds

Integer low-discrepancy sequence generators Integer Low-Discrepancy Sequence (ILDS) Generator

sphere_n

N-dimensional sphere sequence generators Generates points on n-dimensional spheres.

Structs

Circle

Unit Circle sequence generator

Disk

Unit Disk sequence generator

Halton

Halton sequence generator

HaltonN

N-dimensional Halton sequence generator

Sphere

Unit Sphere sequence generator

Sphere3Hopf

Sphere-3 sequence generator using Hopf coordinates

VdCorput

van der Corput sequence generator

Constants

MAX_DIGITS

Maximum number of digits for van der Corput sequence

PRIME_TABLE

First 1000 prime numbers

TWO_PI

Constant for 2π

Functions

vdc

van der Corput sequence function