math::quasirandom(n)

✨ math::quasirandom(n) ✨

okay bestie, so this is a page from the Tcl Math Library manual, and it's literally about quasi-random points for integration and Monte Carlo type methods 💖 like, no cap, it's giving "math but make it cover the whole space evenly."

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🌸 SYNOPSIS 🌸

package require Tcl 8.6 9

package require TclOO

package require math::quasirandom 1.1

::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 🦋

so picture this: in so many applications, pseudo-random numbers and pseudo-random points in a (limited) sample space are doing the absolute most. Like, any type of Monte Carlo simulation? Yeah, they're there. Pseudo-random numbers, however, can be too random, and the consequence is kinda tragic — you need a huge number of data points just to reduce the error or the fluctuation in the results down to the value you actually want 😭

enter quasi-random numbers: the alternative we've all been waiting for. Instead of "completely" arbitrary points, you generate points that are diverse enough to cover the entire sample space in a more or less uniform way. And as a consequence? Convergence to the limit can be much faster when such quasi-random numbers are well-chosen. slay.

This 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. iconic behaviour, honestly.

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👑 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.

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💅 TODO 💅

• Implement other algorithms and variants
• Implement more unit tests.
• Comparison to pseudo-random numbers for integration.

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🎀 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/

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🌈 KEYWORDS 🌈

mathematics, quasi-random

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⭐ CATEGORY ⭐

Mathematics

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and that's the whole vibe check, bestie 🫶 this lil masterpiece comes to you from tcllib, version 1.1, and it lives its best life as the manual page math::quasirandom(n). go integrate something gorgeous 💖✨