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NAME
math::quasirandom  Quasirandom points for integration and Monte Carlo type methods
Table Of Contents
SYNOPSIS
package require Tcl 8.6 9
package require TclOO
package require math::quasirandom 1
::math::quasirandom::qrpoint create NAME DIM ?ARGS?
gen next
gen setstart index
gen setevaluations number
gen integral func minmax args
DESCRIPTION
In many applications pseudorandom numbers and pseudorandom points in a (limited) sample space play an important role. For instance in any type of Monte Carlo simulation. Pseudorandom 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.
Quasirandom 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 quasirandom numbers are wellchosen.
The package defines a class "qrpoint" that creates a command to generate quasirandom points in 1, 2 or more dimensions. The command can either generate separate points, so that they can be used in a userdefined 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 quasirandom 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 keyvalue 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:

Return the coordinates of the next quasirandom point

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

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

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 quasirandom coordinates to the desired hyperblock.
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 keyvalue 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 pseudorandom numbers for integration.
References
Various algorithms exist for generating quasirandom numbers. The generators created in this package are based on: http://extremelearning.com.au/unreasonableeffectivenessofquasirandomsequences/
KEYWORDS
CATEGORY
Mathematics