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library(Ryacas)
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"
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.