A New High-Order Stable Numerical Method for Matrix Inversion

Algorithm 1

An algorithm for constructing a rapid and robust initial approximation for 𝐴⁻¹.

(1) Given 𝐴∈ℂ^𝑚×𝑚

(2) construct the matrix 𝑉₀=1/(‖𝐴‖²_𝐹)𝐴^∗

(3) obtain the LU decomposition of the matrix 𝐴, for example LU = LinearSolve[A]

(4) for 𝑙=1,…,𝑗𝑗≪𝑛, and when 𝑒_𝑙 is the 𝑙th column of the identity matrix

(5) update the columns of 𝑉₀ by finding the columns of the real 𝐴⁻¹ as follows: 𝑣_𝑙=LU[𝑒_𝑙]

(6) end for.