math::quasirandom - Tcl Math Library

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# math::quasirandom(n) 1 tcllib "Tcl Math Library"

## Name

math::quasirandom - Quasi-random points for integration and Monte Carlo type methods

## Synopsis

- package require Tcl 8.6
- package require TclOO
- package require math::quasirandom 1

- ::math::quasirandom::qrpoint create NAME DIM ?ARGS?
- gen next
- gen set-start index
- gen set-evaluations number
- gen integral func minmax args

## Description

In many applications pseudo-random numbers and pseudo-random points in a (limited) sample space play an important role. For instance in any type of Monte Carlo simulation. Pseudo-random numbers, however, may be too random and as a consequence a large number of data points is required to reduce the error or fluctuation in the results to the desired value.

Quasi-random numbers can be used as an alternative: instead of "completely" arbitrary points, points are generated that are diverse enough to cover the entire sample space in a more or less uniform way. As a consequence convergence to the limit can be much faster, when such quasi-random numbers are well-chosen.

The package defines a class "qrpoint" that creates a command to generate quasi-random points in 1, 2 or more dimensions. The command can either generate separate points, so that they can be used in a user-defined algorithm or use these points to calculate integrals of functions defined over 1, 2 or more dimensions. It also holds several other common algorithms. (NOTE: these are not implemented yet)

One particular characteristic of the generators is that there are no tuning parameters involved, which makes the use particularly simple.

## COMMANDS

A quasi-random point generator is created using the qrpoint class:

::math::quasirandom::qrpoint create NAME DIM ?ARGS?

  This command takes the following arguments:

  string NAME

    The name of the command to be created (alternatively: the new subcommand will generate a unique name)

  integer/string DIM

    The number of dimensions or one of: "circle", "disk", "sphere" or "ball"

  strings ARGS

    Zero or more key-value pairs. The supported options are:

    - -start index: The index for the next point to be generated (default: 1)

    - -evaluations number: The number of evaluations to be used by default (default: 100)

The points that are returned lie in the hyperblock [0,1[^n (n the number of dimensions) or on the unit circle, within the unit disk, on the unit sphere or within the unit ball.

Each generator supports the following subcommands:

gen next

  Return the coordinates of the next quasi-random point

gen set-start index

  Reset the index for the next quasi-random point. This is useful to control which list of points is returned. Returns the new or the current value, if no value is given.

gen set-evaluations number

  Reset the default number of evaluations in compound algorithms. Note that the actual number is the smallest 4-fold larger or equal to the given number. (The 4-fold plays a role in the detailed integration routine.)

gen integral func minmax args

  Calculate the integral of the given function over the block (or the circle, sphere etc.)

  string func

    The name of the function to be integrated

  list minmax

    List of pairs of minimum and maximum coordinates. This can be used to map the quasi-random coordinates to the desired hyper-block.

    If the space is a circle, disk etc. then this argument should be a single value, the radius. The circle, disk, etc. is centred at the origin. If this is not what is required, then a coordinate transformation should be made within the function.

  strings args

    Zero or more key-value pairs. The following options are supported:

    - -evaluations number: The number of evaluations to be used. If not specified use the default of the generator object.

## TODO

Implement other algorithms and variants

Implement more unit tests.

Comparison to pseudo-random numbers for integration.

## References

Various algorithms exist for generating quasi-random numbers. The generators created in this package are based on: http://extremelearning.com.au/unreasonable-effectiveness-of-quasirandom-sequences/

## Keywords

mathematics, quasi-random

## Category

Mathematics