# Iterative Methods

**iterative methods** are based on the idea of successive approximations. We start with an initial approximation to the solution vector x = x0, to solve the system of equations Ax = b, and obtain a sequence of approximate vectors x_{0}, x_{1}, ..., x_{k}, ..., which in the limit as k → ∞, converges to the exact solution vector x = A^{-1}b. A general linear iterative method for the solution of the system of equations Ax = b, can be written in matrix form as

x^{(k 1)}= Hx^{(k)} c, k = 0, 1, 2, …................1.1

where x^{(k 1)} and x^{(k)} are the approximations for x at the (k 1)_{th} and k_{th} iterations respectively. H is called the iteration matrix, which depends on A and c is a column vector, which depends on A and b.

When to stop the iteration We stop the iteration procedure when the magnitudes of the differences between the two successive iterates of all the variables are smaller than a given accuracy or error tolerance or an error bound ε, that is,

......................1.2

For example, if we require two decimal places of accuracy, then we iterate until

for all i

If we require three decimal places of accuracy, then we iterate until

for all i

Convergence property of an iterative method depends on the iteration matrix H.

Now, we derive two iterative methods for the solution of the system of algebraic equations

.........................1.3