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 \(C_{ij}=(-1)^{i+j}M_{ij}\) 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
#> c: ⎡a₂₂⋅a₃₃ - a₂₃⋅a₃₂   -a₂₁⋅a₃₃ + a₂₃⋅a₃₁  a₂₁⋅a₃₂ - a₂₂⋅a₃₁ ⎤
#>    ⎢                                                          ⎥
#>    ⎢-a₁₂⋅a₃₃ + a₁₃⋅a₃₂  a₁₁⋅a₃₃ - a₁₃⋅a₃₁   -a₁₁⋅a₃₂ + a₁₂⋅a₃₁⎥
#>    ⎢                                                          ⎥
#>    ⎣a₁₂⋅a₂₃ - a₁₃⋅a₂₂   -a₁₁⋅a₂₃ + a₁₃⋅a₂₁  a₁₁⋅a₂₂ - a₁₂⋅a₂₁ ⎦
cc <- cofactor(A, 1, 1)
cc
#> c: a₂₂⋅a₃₃ - a₂₃⋅a₃₂

We get the right answer

B <- t(CC) / det(A)
P <- A %*% B
P %>% simplify()
#> c: ⎡1  0  0⎤
#>    ⎢       ⎥
#>    ⎢0  1  0⎥
#>    ⎢       ⎥
#>    ⎣0  0  1⎦