Numerical Methods

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 x0, x1, ..., xk, ..., which in the limit as k → ∞, converges to the exact solution vector x = A-1b. 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 kth 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