antononcube / Raku-Math-NIntegrate

Raku library for numerical integration.

Math::NIntegrate

Introduction

This repository has the Raku code of a library for numerical integration. The design and architecture of library's main function, NIntegrate, resembles that of Mathematica's NIntegrate, [WRI1, WRI2].

While working for Wolfram Research Inc., during the years 2003-2006 I designed and implemented Mathematica's "new" NIntegrate and extensively documented it. (I wrote [WRI2].)

The Raku programming language has some built-in features that give the ability to:

• Program in an object-oriented manner

• Do bignum arithmetic computations

• Do symbolic computations

These are also programming language features on which Mathematica's NIntegrate is based upon.

Hence, it seems natural to think that an implementation of a powerful numerical integration framework in Raku that has unique features is achievable, worthy, and rewarding.

I plan to write a book that describes in detail software architecture designs and decisions for the development of numerical integration frameworks. That book is going to use this library.

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Design (in brief)

The NIntegrate framework is based on Object-Oriented Programming (OOP) Design Patterns (DP). NIntegrate uses the software design patterns Strategy, Composite, Decorator, and others.

Here is a description using the OOP DP language:

• The basic objects of NIntegrate are integration regions.
• Each region has its own integration function and integration rule.
• Conceptually, there are two main types of algorithms: integration strategies and integration rules.
• The integration strategies use the Template method for their "logic."
• The integration regions use Strategy for the computation of integral and error estimates.
• The integration regions can utilize singularity handler objects.
• Creations of integration rules generally use Builder.
• Symbolic preprocessing is done through Decorator.
• User specifications are translated into creations of numerical integration algorithm objects through Interpreter.
• The creation of the "final" integration algorithm object uses Abstract factory.

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Usage example

Basic usage examples:

use Math::NIntegrate;

say NIntegrate( -> $x { 1/sqrt($x) }, x => (0, 2) )

say NIntegrate( -> $x { 1/$x^2 }, x => (1, 2), Method => 'LocalAdaptive', WorkingPrecision => 40 );

say NIntegrate( -> $x, $y { $x + $y^2 }, x => (0, 2), y => (0, 12), Method => ('GlobalAdaptive' Method => ('GaussKronrod', Points => 5) );

say NIntegrate( -> $x, $y, $z { $x + $y^2 + 1/$z }, x => (0, 2), y => (0, 12), z => (1, 4), Method => 'AdaptiveMonteCarlo' )

Utilization through a DSL specification:

integrate 1/x^2 over the range [1,2] with a local adaptive strategy

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References

[WRI1] Wolfram Research (1988), NIntegrate, Wolfram Language function, https://reference.wolfram.com/language/ref/NIntegrate.html (updated 2014).

[WRI2] Wolfram Research (2006), Advanced Numerical Integration in the Wolfram Language, Wolfram Monograph, https://reference.wolfram.com/language/tutorial/NIntegrateOverview.html.

[AAr1] Anton Antonov, NIntegrate - The Missing Manual, (2019), GitHub/antononcube.

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Anton Antonov
Windermere, Florida, USA
2021-04-05