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math::quasirandom(n) Tcl Math Library math::quasirandom(n) 





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










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:







  

  
The name of the command to be created (alternatively: the new
      subcommand will generate a unique name)

  

  
The number of dimensions or one of: "circle", "disk",
      "sphere" or "ball"

  

  
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:



  

  
Return the coordinates of the next quasi-random point
    

  

  

  
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.
    

  

  

  
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.)
    

  

  

  
Calculate the integral of the given function over the block (or the
      circle, sphere etc.)







  

  
The name of the function to be integrated

  

  
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.

  

  

  
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







  

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