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

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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

tcllib                                                  1                                math::quasirandom(3tcl)