Power Method

A is an n x n matrix.

Choose a starting vector u arbitrarily (usually components of vector all equal 1).

1.       Calculate Au

2.       Normalise resulting vector by dividing each component by the largest in magnitude.

3.       Repeat steps 1 and 2 until change in normalising factor is negligible.

Resulting normalising vector is the dominant eigenvalue and the final vector is the eigenvector.

Gerschgorin's Circle Theorem

Let A be an n x n matrix.
The circle in the complex plane is given by:

Eigenvalues of A are contained in the union of the circles.

Economizing a Power Series

where

refers to the n-th degree Chebyshev polynomial

Chebyshev Polynomials

Pade Approximation

Runge-Kutta-Fehlberg

Fourth order Runge Kutta Method

Second order Runge Kutta Methods

Modified Euler Method

Euler Method

Taylor method of order n

where,