# Tcl Library Source Code

Documentation
Bounty program for improvements to Tcl and certain Tcl packages.

# 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

# 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/

Mathematics