What is the best algorithm to find the inverse of matrix $A$ I'm going to build a C-library so I can do linear algebra at embedded systems. It's most for machine learning. https://github.com/DanielMartensson/EmbeddedAlgebra/
Anyway, I need to compute inverse of matrix $A$. What is the absolute best algorithm to use for that? Remember it must be a numerical algorithm.
Which one should I use?
Analytic solution
$\mathbf{A}^{-1}={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}}\mathbf{C}^{\mathrm{T}}={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}}
\begin{pmatrix}
\mathbf{C}_{11} & \mathbf{C}_{21} & \cdots & \mathbf{C}_{n1} \\
\mathbf{C}_{12} & \mathbf{C}_{22} & \cdots & \mathbf{C}_{n2} \\
\vdots & \vdots & \ddots & \vdots \\
\mathbf{C}_{1n} & \mathbf{C}_{2n} & \cdots & \mathbf{C}_{nn} \\
\end{pmatrix}$
Cholesky decomposition
$\mathbf{A}^{-1} = (\mathbf{L}^{*})^{-1} \mathbf{L}^{-1}$
Eigendecomposition
$\mathbf{A}^{-1}=\mathbf{Q}\mathbf{\Lambda}^{-1}\mathbf{Q}^{-1}$
Cayley–Hamilton method
$\mathbf{A}^{-1} = \frac{1}{\det (\mathbf{A})}\sum_{s=0}^{n-1}A^s$
Newton's method
$X_{k+1} = 2X_k - X_k A X_k$
Neuman series
$\lim_{n \to \infty} (\mathbf I - \mathbf A)^n = 0$
Gaussian elimination
$A =
\begin{bmatrix}
2 & -1 & 0 \\
-1 & 2 & -1 \\
0 & -1 & 2
\end{bmatrix}$
To find the inverse of this matrix, one takes the following matrix augmented by the identity, and row reduces it as a 3 × 6 matrix:
$ 
[ A | I ] = 
\left[ \begin{array}{rrr|rrr}
2 & -1 & 0 & 1 & 0 & 0\\
-1 & 2 & -1 & 0 & 1 & 0\\
0 & -1 & 2 & 0 & 0 & 1
\end{array} \right]$
By performing row operations, one can check that the reduced row echelon form of this augmented matrix is:
$[ I | B ] = 
\left[ \begin{array}{rrr|rrr}
1 & 0 & 0 & \frac34 & \frac12 & \frac14\\[3pt]
0 & 1 & 0 & \frac12 & 1 & \frac12\\[3pt]
0 & 0 & 1 & \frac14 & \frac12 & \frac34
\end{array} \right]$

It truly depends on the type of matrix you're going to compute the inverse from. Some methods are better for some classes of matrices than other.
But more importantly, why do you want to invert matrices? In many problems, you don't need to invert matrices, but only need to apply the inverse to some vectors. The latter problem is much easier to tackle, especially from a computational complexity standpoint (e.g. if your matrix is very large) and stability point of view (a matrix could have bad conditioning, be numerically close to non-invertible etc...). Does your application require that compute the matrix inverse?
