Implementing the Matrix Exponential in Julia

As a Research Ecologist, I recently discovered an interesting connection between Wien's/Planck's blackbody radiation responses and tree population growth. This led me to implement a crucial mathematical operation, the matrix exponential times vector, in Julia.

I started by implementing the function using the built-in `exp` function from Julia's standard library. However, I soon realized that this approach had limitations. For instance, I wanted my function to work on both dense and sparse matrices without having to write runtime checks or use manual conversions. This led me to explore multiple dispatch in Julia.

The idea is to define a method for `exp` that can handle various types of matrices, including abstract supertypes like `(Bi/Tri)diagonal`. This way, developers can easily overload the method with their own implementations without having to write separate code for each type.

In Julia, we don't have automatic promotion or conversion when calling functions. This means that if we want `exp` to work on a sparse matrix, we need to define a specific method for it. One possible approach is to use the `collect` function to convert the sparse matrix to a dense one and then apply the `exp` function.

However, this approach has limitations. For example, `exp(Int(1))` returns a float, which made me expect that `exp(sparsematrix)` would return a dense matrix if necessary. Unfortunately, Julia doesn't provide such a default behavior.

In conclusion, implementing the matrix exponential times vector in Julia requires careful consideration of type handling and multiple dispatch. By defining specific methods for various types of matrices, we can create robust and efficient functions that work seamlessly with Julia's type system.

Matlab matrix exponential times vector