05 - Extending 'caracas

In caracas: Computer Algebra

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(caracas)

inline_code <- function(x) {
  x
}

if (!has_sympy()) {
  # SymPy not available, so the chunks shall not be evaluated
  knitr::opts_chunk$set(eval = FALSE)

  inline_code <- function(x) {
    deparse(substitute(x))
  }
}

It is relatively easy to extend caracas by calling SymPy functions directly.

This can be achived using sympy_func(x, fun, ...) that calls a member function on the object provided, i.e. x$fun(...), or if that fails it calls a function from the global namespace fun(x, ...).

As an example consider inverting a regular matrix $A$: Let $B$ be the inverse of $A$. Then, using cofactors, $B_{ij} =C_{ji} / det(A)$. The cofactor $C_{ij}$ is given as $C_{ij}=(-1)^{i+j}M_{ij}$ where $M_{ij}$ is the determinant of the submatrix of $A$ obtained by deleting the $i$th row and the $j$th column of $A$.

A quick search https://docs.sympy.org/latest/modules/matrices/matrices.html shows that there are two relevant functions in SymPy: cofactor and cofactor_matrix.

If these functions are not available in caracas they can be made so using sympy_func:

cofactor_matrix <- function(x) {
  sympy_func(x, "cofactor_matrix")
}

cofactor <- function(x, i, j) {
  # Python indexing starts at 0 - thus subtract 1 to convert from R indexing
  # to Python indexing
  sympy_func(x, "cofactor", i - 1, j - 1)
}

A <- matrix_sym(3, 3, "a")

CC <- cofactor_matrix(A)
CC
cc <- cofactor(A, 1, 1)
cc

We get the right answer

B <- t(CC) / det(A)
P <- A %*% B
P %>% simplify()

caracas documentation built on Oct. 17, 2023, 5:08 p.m.