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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.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.