math::quasirandom(3tcl)

✨ math::quasirandom ✨

okay bestie, welcome to the Tcl Math Library's quasi-random era 💖 this is the manual page for math::quasirandom — quasi-random points for integration and Monte Carlo type methods, no cap.

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🌸 NAME 🌸

math::quasirandom — quasi-random points for integration and Monte Carlo type methods 🎲✨

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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 like, in so many applications, pseudo-random numbers and pseudo-random points in a (limited) sample space are doing the absolute most. think any type of Monte Carlo simulation, bestie. but here's the tea 🍵 — pseudo-random numbers can be too random, and because of that you need a whole giant pile of data points just to squash the error or fluctuation in your results down to the value you actually wanted. exhausting, honestly 😭

Quasi-random numbers are the alternative, and they ate. 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 🌈 as a consequence, convergence to the limit can be much faster when such quasi-random numbers are well-chosen. periodt 💅

The package defines a class "qrpoint" that creates a command to generate quasi-random points in 1, 2 or more dimensions ✨ That command can either generate separate points — so you can use them in your own user-defined algorithm — or it can use those 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 🫶 no fiddling, no drama, we love that for us.

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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 🌐💖

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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 page, bestie 🫶 this one came from tcllib, version 1.1, and it's the math::quasirandom(3tcl) manual — go generate some unreasonably effective points for me, mwah 💋✨🎀