In RobinHankin/Brobdingnag: Very Large Numbers in R

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Overview

The Brobdingnag package provides R-centric functionality for working with very large numbers. It represents a number $x$ as its logarithm $\log\left|x\right|$ together with a logical that indicates whether $x$ is positive. The vignette functions as a "hello world" for S4 methods.

The package has functionality to represent Brobdingnagian matrices, and complex numbers.

Installation

You can install the released version of Brobdingnag from CRAN with:

# install.packages("Brobdingnag")  # uncomment this to use the package
library("Brobdingnag")

The package is maintained on github.

The Brobdingnag package in use

Consider a googol, conventionally defined as $10^{100}$. In standard arithmetic we may evaluate this as follows:

10^100

But a googolplex, $10^{10^{100}}$, is beyond floating-point representation:

10^(10^100)

We may represent a googol straightforwardly using Brobdingnagian arithmetic, specifically by working with brob objects.

as.brob(10)
googol <- as.brob(10)^100
googol
10^googol

In the above, we coerce 10 to Brobdingnagian form (that is, an object of class brob) using as.brob(). Most arithmetic methods work transparently with Brobdingnagian numbers:

googol
googol + googol
googol/10
1/googol
sqrt(googol)
exp(googol)

Note that precision may be adversely affected for very large numbers:

gplex <- 10^googol
gplex
gplex - gplex/10

The last line looks odd [the return value is package idiom for zero] but symbolically we have $10^{10^{100}}/10=10^{10^{100}-1}\simeq 10^{10^{100}}$. The package is vectorised and allows Brobdingnagian arithmetic:

googol/(1:20)

Signed quantities operate as expected:

as.brob(5) - 1:10

It is a semi-nonobvious fact that, in addition to large numbers (that is, $x\colon\left|x\right|\gg 1$ and small numbers ($x\colon\left|x\right|\ll 1$), the package can deal with numbers very close to 1 (that is, $x\colon\left|x-1\right|\ll 1$). Suppose we wish to deal with $e^{10^{-100}}\simeq 1+10^{-100}$. In the package:

a <- brob(1e-100)
a
a^googol

The last line being numerical verification of the fact that $\left(1+10^{-100}\right)^{10^{100}}\simeq e=2.718+$. The package also includes some functionality for Brobdingnagian matrices:

brobmat(1:9,3,3)
brobmat(1:9,3,3) %*% 10^brobmat(9:1,3,3)

RobinHankin/Brobdingnag documentation built on Sept. 24, 2024, 11:44 a.m.