Basic numerical methods - Numerical algebra

This second dossier on basic numerical methods covers linear and nonlinear numerical algebra.

The first paragraph is devoted to iterative methods for calculating the roots of a non-linear equation with one unknown (or, what amounts to the same thing, the fixed points of a function). This is followed by the special case of finding the roots of a polynomial. The paragraph concludes with methods for solving systems of nonlinear equations.

We then study numerical methods for solving systems of linear equations. These methods fall into two classes: direct methods, which provide the exact solution in a finite number of arithmetic operations (assuming zero errors due to the computer's arithmetic), and iterative methods, which generate a sequence of vectors converging (under certain conditions) to the exact solution. For very large systems, it is imperative to use an iterative method.

In the final paragraph, we move on to numerical methods for calculating the eigenvalues and eigenvectors of a matrix. These methods are all iterative.