# User-defined yacas rules

#### Mikkel Meyer Andersen and Søren Højsgaard

#### 2023-01-16

    library(Ryacas)

# Included rules

`yacas` comes with a number of rules all defined in the `yacas` directory of the installed package:

    system.file(package = "Ryacas", "yacas")

    ## [1] "/tmp/Rtmp8hYNaH/Rinst909f59779d54/Ryacas/yacas"

For example in the `sums.rep` folder, a number of rules for sums are defined in the `code.ys` file.

As an example, the fact that \[ \sum_{k = 1}^n (2k-1) = n^2 \] is defined in `yacas` as

    SumFunc(_k,1,_n,2*_k-1, n^2 );

and the geometric sum is defined as

    SumFunc(_k,0,_n,(r_IsFreeOf(k))^(_k), (1-r^(n+1))/(1-r) );

These can be verified:

    yac_str("Sum(i, 1, m, 2*i-1)")

    ## [1] "m^2"

    yac_str("Sum(i, 0, m, 2^i)")

    ## [1] "2^(m+1)-1"

There are also rules in `yacas` that are able to let the user change some limits of some sums, e.g. for the geometric sum:

    yac_str("Sum(i, 1, m, 2^i)")

    ## [1] "2^(m+1)-2"

# Custom rules

But what about changing the limit of the first sum? I.e. instead of \[ \sum_{k = 1}^n (2k-1) = n^2 \] then know that \[ \sum_{k = 0}^n (2k-1) = -1 + \sum_{k = 1}^n (2k-1) = n^2 - 1 . \] But what does `yacas` say?

    yac_str("Sum(i, 0, m, 2*i-1)")

    ## [1] "Sum(i,0,m,2*i-1)"

We can then add our own rule by:

    yac_silent("SumFunc(_k,0,_n,2*_k-1, n^2 - 1)")

And then try again:

    yac_str("Sum(i, 0, m, 2*i-1)")

    ## [1] "m^2-1"

A good source of inspiration for writing custom rules is reading the included rules, but there is a lot to programming in `yacas` and we refer to `yacas`’s documentation, specifically the chapter Programming in Yacas.