Introduction

Nearly every scientific problem which is solvable on a computer can be represented by matrices. However the ease of solution can depend crucially on the types of matrices involved. There are 3 main classes of problems which we might want to solve:

1. Trivial Algebraic Manipulation such as addition, ${\bss A} + {\bss B}$ or multiplication, ${\bss A}{\bss B}$, of matrices.
2. Systems of equations: ${\bss A}{\bi x} = {\bi b}$ where ${\bss A}$ and ${\bi b}$ are known and we have to find ${\bi x}$. This also includes the case of finding the inverse, ${\bss A}^{-1}$. The standard example of such a problem is Poisson's Equation.
3. Eigenvalue Problems: ${\bss A}{\bi x} = \alpha {\bi x}$. This also includes the generalised eigenvalue problem: ${\bss A}{\bi x} = \alpha{\bss B}{\bi x}$. Here we will consider the time-independent Schrödinger equation.

In most cases there is no point in writing your own routine to solve such problems. There are many computer libraries, such as Numerical Algorithms Group (n.d.), Lapack Numerical Library (n.d.) (for linear algebra problems and eigenvalue problems). which contain well tried routines. In addition vendors of machines with specialised architectures often provide libraries of such routines as part of the basic software.