05 - Extending ‘caracas’

library(caracas)

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 + jM_ij where M_ij is the determinant of the submatrix of A obtained by deleting the ith row and the jth 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()