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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

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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