---
name: "quasirandom"
section: "3tcl"
platform: "linux"
package: "tcllib"
version: "2.0+dfsg-4"
release: "Debian 13 (Trixie)"
platforms: ["linux"]
url: "https://new.carta.tech/quasirandom"
---

# quasirandom(3tcl)

Quasi-random points for integration and Monte Carlo type methods

```

```

## NAME

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

## SYNOPSIS

package require **Tcl 8.6 9**

package require **TclOO**

package require **math::quasirandom 1.1**

**::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