Add docs for matrix functions and exponential (#251)

* add docs

Add documentation for general matrix functions, specifically for `exponential`.

* Update docs/src/user_interface/matrix_functions.md

Co-authored-by: Lukas Devos <ldevos98@gmail.com>

* separate decompositions and matrix functions

* change docs

* Update docs/src/user_interface/matrix_functions.md

Co-authored-by: Jutho <Jutho@users.noreply.github.com>

* Update docs/src/user_interface/matrix_functions.md

Co-authored-by: Jutho <Jutho@users.noreply.github.com>

---------

Co-authored-by: Lukas Devos <ldevos98@gmail.com>
Co-authored-by: Jutho <Jutho@users.noreply.github.com>

Author: 3 people

docs/src/user_interface/algorithms.md

@@ -80,9 +80,11 @@ DefaultAlgorithm
 
 ## Available Algorithm Types
 
-The following high-level algorithm types are available.
+There are several high-level algorithm types available.
 They all accept an optional `driver` keyword to select the computational backend; see [Driver Selection](@ref sec_driverselection) for details.
 
+The following algorithms for matrix decompositions are available.
+
 | Algorithm | Applicable decompositions | Key keyword arguments |
 |:----------|:--------------------------|:----------------------|
 | [`Householder`](@ref) | QR, LQ | `positive`, `pivoted`, `blocksize` |
@@ -96,6 +98,14 @@ They all accept an optional `driver` keyword to select the computational backend
 | [`PolarViaSVD`](@ref) | polar | positional `svd_alg` argument |
 | [`PolarNewton`](@ref) | polar | `maxiter`, `tol` |
 
+The following algorithms for matrix functions are available.
+
+| Algorithm | Applicable matrix functions | Key keyword arguments |
+|:----------|:--------------------------|:----------------------|
+| [`MatrixFunctionViaLA`](@ref) | exponential | |
+| [`MatrixFunctionViaEig`](@ref) | exponential | `eig_alg` |
+| [`MatrixFunctionViaEigh`](@ref) | exponential | `eigh_alg` |
+
 For full docstring details on each algorithm type, see the corresponding section in [Decompositions](@ref).
 
 ## [Driver Selection](@id sec_driverselection)

docs/src/user_interface/matrix_functions.md

@@ -1,3 +1,37 @@
+```@meta
+CurrentModule = MatrixAlgebraKit
+CollapsedDocStrings = true
+```
+
 # Matrix functions
 
-Coming soon...
+Another class of matrix algebra methods consists of calculating some function of a single input `A`.
+In order to streamline these functions, they all follow a similar common code pattern.
+For a given function `f`, this consists of the following methods:
+
+```julia
+f(A; kwargs...) -> F...
+f!(A, [F]; kwargs...) -> F...
+```
+
+Here, the input matrix is always the first argument, and optionally the output can be provided as well.
+The keywords are algorithm-specific, and can be used to influence the behavior of the algorithms.
+For a full description of how to select and configure algorithms, see [Algorithm Selection](@ref sec_algorithmselection).
+Importantly, for generic code patterns it is recommended to always use the output `F` explicitly, since some implementations may not be able to reuse the provided memory.
+Additionally, the `f!` method typically assumes that it is allowed to destroy the input `A`, and making use of the contents of `A` afterwards should be deemed as undefined behavior.
+
+## Exponential
+
+The [exponential](https://en.wikipedia.org/wiki/Matrix_exponential) of a square matrix `A` is used in many scientific applications, as it arises in the solution of an autonomous linear differential equation.
+An implementation for the matrix exponential based on a Padé approximation is available in `LinearAlgebra`, and can be accessed by the algorithm [`MatrixFunctionViaLA`](@ref).
+For more generic data types, the exponential can be calculated by first calculating the (hermitian) eigenvalue decomposition, and then computing
+the scalar exponential of the diagonal elements.
+This strategy is implemented via the algorithms [`MatrixFunctionViaEig`](@ref) and [`MatrixFunctionViaEigh`](@ref), and call `eig_full` and `eigh_full`, respectively.
+Additionally, in order to calculate `exp(τ * A)`, the function `exponential` can be called with `(τ, A)`, using the same algorithms as before.
+
+```@docs; canonical=false
+exponential
+MatrixAlgebraKit.MatrixFunctionViaLA
+MatrixAlgebraKit.MatrixFunctionViaEig
+MatrixAlgebraKit.MatrixFunctionViaEigh
+```