## table of contents

math::quasirandom(3tcl) | Tcl Math Library | math::quasirandom(3tcl) |

# NAME¶

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

# SYNOPSIS¶

package require **Tcl 8.5**

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

1 | tcllib |