Matrix Exponential Using the Pade Approximation Calculator

The matrix exponential e^A for a square matrix A is a fundamental concept in linear algebra, often used to solve systems of linear differential equations. It is defined by the power series:
e^A = I + A + A²/2! + A³/3! + ...

However, directly computing this series is numerically unstable for many matrices. The Pade approximation method provides a robust and efficient way to approximate e^A. It represents the matrix exponential as a rational function of the matrix A:
e^A ≈ R_pq(A) = D_pq(A)^-1N_pq(A),
where N_pq(A) and D_pq(A) are polynomial matrices. This method often involves scaling and squaring (i.e., e^A = (e^A/2s)^2s) to reduce the norm of the input matrix, improving accuracy and stability before applying the Pade approximation.