In stla/expansions: Expansions of integer and real numbers

library(expansions)
knitr::opts_chunk$set(echo = TRUE, collapse=TRUE)

Expansions of integers

Conversion of decimal integers to integer base

This is the role of the intAtBase function:

intAtBase(32, base=3)

This result means that $$ 32 = \color{red}{2} \times 1 + \color{red}{1} \times 3^1 + \color{red}{0} \times 3^2 + \color{red}{1} \times 3^3 $$

2*1 + 1*3 + 0*3^2 + 1*3^3

The intAtBasePower function returns the expansion of the integer $m^p$ where $m$ and $p$ are supplied by the user:

intAtBasePower(2, 5, base=3)
intAtBase(2^5, base=3)

The advantage is the possibility to convert huge numbers $m^p$:

# 2^100 is too big for intAtBase:
intAtBase(2^100, base=3)
# then use intAtBasePower:
( digits <- intAtBasePower(2, 100, base=3) )
sum(digits * 3^(seq_along(digits)-1)) == 2^100

The intAtBaseFib function returns the expansion of a Fibonacci number given by its index. For example, the $6$-th Fibonacci number is $8$:

intAtBaseFib(6, base=3)
intAtBase(8, base=3)

The $50$-th Fibonacci number is $12586269025$:

digits <- intAtBaseFib(50, base=3)
sum(digits * 3^(seq_along(digits)-1))

Cantor expansions, or "ary" expansions

The intToAry function performs the Cantor expansion of an integer.

For example, the $(3,4,7)$-ary expansion of $34$:

intToAry(34, sizes=c(3,4,7))

This means that $$ 34 = \color{red}{1} \times 1 + \color{red}{3} \times 3 + \color{red}{2} \times (3 \times 4). $$

1*1 + 3*3 + 2*(3*4)

Each $(3,4,7)$-ary expansion of an integer corresponds to a path on the following graph:

There are $84$ paths on the above graph. That means that the $(3,4,7)$-ary expansion is possible for integers from $0$ to $83$.

The last integer of the sizes vector is not seemingly involved in the development $d_1 \times 1 + d_2 \times 3 + d_3 \times (3 \times 4)$. Actually it is: the digits satisfy $d_1 < 3$, $d_2 < 4$ and $d_3 < 7$. Thus, indeed, the maximal integer allowing a $(3,4,7)$-ary expansion is $\color{red}{2}\times 1 +\color{red}{3}\times 3+\color{red}{6}\times(3\times 4)=83$:

intToAry(83, sizes=c(3,4,7))
intToAry(84, sizes=c(3,4,7))

As an illustration, the intToAry function allows to get a Cartesian product of sets:

# Cartesian product {0,1,2}x{0,1,2,3}:
sapply(0:11, function(x) intToAry(x, sizes=c(3,4)))
t(expand.grid(c(0,1,2), c(0,1,2,3)))

This function is implemented with Rcpp (source code), and it is fast.

It can be applied to avoid nested loops.

Float expansions

Expand a number between $0$ and $1$

The floatExpand01 function returns the expansion of a real number between $0$ and $1$ in a given integer base. For example:

floatExpand01(0.625, base=2)

It means that $0.625 =\color{red}{1} \times \frac{1}{2} + \color{red}{0} \times \frac{1}{2^2} + \color{red}{1} \times \frac{1}{2^3}$.

Expansion of a positive number

The floatExpand function returns a list representing the expansion of a positive number in scientific notation: $$ x = 0.d_1d_2\ldots d_n \times \text{base}^e. $$ The digits $d_1$, $\ldots$, $d_n$ are given in the first component of the list and the exponent $e$ is given in the second one.

For example:

floatExpand(1.125, base=2)
# check:
1.125 == (1*1/2 + 0*1/2^2 + 0*1/2^3 + 1*1/2^4) * 2^1

Odometers and addition of adic integers

The odometer is the action $x \mapsto x+1$ on the group of $b$-adic integers. In other words, $x$ is the expansion in an integer base $b$ of a real number between $0$ and $1$, and the odometer maps $x$ to $x + (1, 0, 0, \ldots)$ where "$+$" is the addition modulo $b$ with carry to the right.

odometer(c(1,0,1), base=2)

By transporting the $3$-adic integers to $[0,1)$ with the float expansion in base $3$, let's have a look at the graph of the $3$-adic odometer:

ternary2num <- function(t) sum(t/3^seq_along(t))
num2ternary <- function(u) floatExpand01(u, base=3)
par(mar=c(4,4,2,2))
u <- seq(0, 0.995, by=0.0025)
Ou <- sapply(u, function(u) ternary2num(odometer(num2ternary(u), base=3)))
plot(u, Ou, xlab="u", ylab="O(u)", xlim=c(0,1), ylim=c(0,1),
     pch=19, cex=.25, pty="s", xaxs="i", yaxs="i")
grid(nx=9)

The expansions package also provides the function sumadic to perform the sum of two adic integers:

sumadic(c(0,1), c(1,1,1), base=2)
identical(sumadic(c(0,1), 1, base=2), odometer(c(0,1), base=2))

The function odometer_iterated iterates the odometer function:

d <- c(1,0,2)
b <- 3
identical(odometer(odometer(d, base=b), base=b), odometer_iterated(d, n=2, base=b))

Note these two ways to get the same result:

n <- 13
odometer_iterated(d, n=n, base=b)
sumadic(d, intAtBase(n, base=b), base=b)

The second one is faster:

n <- 2000000
system.time(odometer_iterated(d, n=n, base=b))
system.time(sumadic(d, intAtBase(n, base=b), base=b))

knitr::knit_exit()

Vignettes are long form documentation commonly included in packages. Because they are part of the distribution of the package, they need to be as compact as possible. The html_vignette output type provides a custom style sheet (and tweaks some options) to ensure that the resulting html is as small as possible. The html_vignette format:

• Never uses retina figures
• Has a smaller default figure size
• Uses a custom CSS stylesheet instead of the default Twitter Bootstrap style

Vignette Info

Note the various macros within the vignette section of the metadata block above. These are required in order to instruct R how to build the vignette. Note that you should change the title field and the \VignetteIndexEntry to match the title of your vignette.

Styles

The html_vignette template includes a basic CSS theme. To override this theme you can specify your own CSS in the document metadata as follows:

output: 
  rmarkdown::html_vignette:
    css: mystyles.css

Figures

The figure sizes have been customised so that you can easily put two images side-by-side.

plot(1:10)
plot(10:1)

You can enable figure captions by fig_caption: yes in YAML:

output:
  rmarkdown::html_vignette:
    fig_caption: yes

Then you can use the chunk option fig.cap = "Your figure caption." in knitr.

More Examples

You can write math expressions, e.g. $Y = X\beta + \epsilon$, footnotes^[A footnote here.], and tables, e.g. using knitr::kable().

knitr::kable(head(mtcars, 10))

Also a quote using >:

"He who gives up [code] safety for [code] speed deserves neither." (via)

stla/expansions documentation built on May 30, 2019, 5:46 p.m.