quasirandom(n)

1math::quasirandom(n)           Tcl Math Library           math::quasirandom(n)
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NAME

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

12       package require Tcl  8.6
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14       package require TclOO
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16       package require math::quasirandom  1
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18       ::math::quasirandom::qrpoint create NAME DIM ?ARGS?
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20       gen next
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22       gen set-start index
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24       gen set-evaluations number
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26       gen integral func minmax args
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DESCRIPTION

31       In many applications pseudo-random numbers and pseudo-random points  in
32       a  (limited)  sample  space play an important role. For instance in any
33       type of Monte Carlo simulation.  Pseudo-random numbers, however, may be
34       too  random  and  as a consequence a large number of data points is re‐
35       quired to reduce the error or fluctuation in the results to the desired
36       value.
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38       Quasi-random  numbers  can  be used as an alternative: instead of "com‐
39       pletely" arbitrary points, points are generated that are diverse enough
40       to  cover  the  entire sample space in a more or less uniform way. As a
41       consequence convergence to the limit can  be  much  faster,  when  such
42       quasi-random numbers are well-chosen.
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44       The  package defines a class "qrpoint" that creates a command to gener‐
45       ate quasi-random points in 1, 2 or more dimensions. The command can ei‐
46       ther  generate  separate points, so that they can be used in a user-de‐
47       fined algorithm or use these points to calculate integrals of functions
48       defined over 1, 2 or more dimensions.  It also holds several other com‐
49       mon algorithms. (NOTE: these are not implemented yet)
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51       One particular characteristic of the generators is that  there  are  no
52       tuning parameters involved, which makes the use particularly simple.
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COMMANDS

55       A quasi-random point generator is created using the qrpoint class:
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57       ::math::quasirandom::qrpoint create NAME DIM ?ARGS?
58              This command takes the following arguments:
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60              string NAME
61                     The name of the command to be created (alternatively: the
62                     new subcommand will generate a unique name)
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64              integer/string DIM
65                     The number of dimensions or  one  of:  "circle",  "disk",
66                     "sphere" or "ball"
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68              strings ARGS
69                     Zero or more key-value pairs. The supported options are:
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71                     •      -start  index:  The index for the next point to be
72                            generated (default: 1)
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74                     •      -evaluations number: The number of evaluations  to
75                            be used by default (default: 100)
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77       The  points that are returned lie in the hyperblock [0,1[^n (n the num‐
78       ber of dimensions) or on the unit circle, within the unit disk, on  the
79       unit sphere or within the unit ball.
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81       Each generator supports the following subcommands:
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83       gen next
84              Return the coordinates of the next quasi-random point
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87       gen set-start index
88              Reset  the index for the next quasi-random point. This is useful
89              to control which list of points is returned.  Returns the new or
90              the current value, if no value is given.
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92
93       gen set-evaluations number
94              Reset  the default number of evaluations in compound algorithms.
95              Note that the actual number is the  smallest  4-fold  larger  or
96              equal  to  the given number. (The 4-fold plays a role in the de‐
97              tailed integration routine.)
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100       gen integral func minmax args
101              Calculate the integral of the given function over the block  (or
102              the circle, sphere etc.)
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104              string func
105                     The name of the function to be integrated
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107              list minmax
108                     List  of  pairs  of minimum and maximum coordinates. This
109                     can be used to map the quasi-random  coordinates  to  the
110                     desired hyper-block.
111
112                     If  the  space  is a circle, disk etc. then this argument
113                     should be a single value, the radius.  The circle,  disk,
114                     etc. is centred at the origin. If this is not what is re‐
115                     quired, then a coordinate transformation should  be  made
116                     within the function.
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118              strings args
119                     Zero  or  more key-value pairs. The following options are
120                     supported:
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122                     •      -evaluations number: The number of evaluations  to
123                            be  used.  If not specified use the default of the
124                            generator object.
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TODO

127       Implement other algorithms and variants
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129       Implement more unit tests.
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131       Comparison to pseudo-random numbers for integration.
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REFERENCES

134       Various algorithms exist for generating quasi-random numbers. The  gen‐
135       erators  created  in  this  package  are based on: http://extremelearn‐
136       ing.com.au/unreasonable-effectiveness-of-quasirandom-sequences/
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KEYWORDS

139       mathematics, quasi-random
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