INTRODUCTION TO DIRECT AND ITERATIVE METHOD

Many important practical problems give rise to systems of linear equations written as the matrix equation

Ax = c,

where A is a given n × nnonsingular matrix and c is an n-dimensional vector; the

problem is to find an n-dimensional vector x satisfying equation .

Such systems of linear equations arise mainly from discrete approximations of partial

differential equations. To solve them, two types of methods are normally used: direct

methods and iterative methods.

Directapproximate the solution after a finite number of floating point operations.

Since computer floating point operations can only be obtained to a given

precision, the computed solution is usually different from the exact solution. When a

square matrix A is large and sparse, solving Ax = c by direct methods can be impractical,

and iterative methods become a viable alternative.

Iterative methods, based on splitting A into A = M−N, compute successive approximations

x(t) to obtain more accurate solutions to a linear system at each iteration

step t. This process can be written in the form of the matrix equation

x(t) = Gx(t−1) + g,

where an n × n matrix G = M−1N is the iteration matrix. The iteration process

is stopped when some predefined criterion is satisfied; the obtained vector x(t) is an

approximation to the solution. Iterative methods of this form are called linear stationary

iterative methods of the first degree. The method is of the first degree because x(t)

depends explicitly only on x(t−1) and not on x(t−2), . . . , x(0). The method is linear

because neither G nor g depends on x(t−1), and it is stationary because neither G nor g

depends on t. In this book, we also consider linear stationary iterative methods of the

second degree, represented by the matrix equation

x(t) = Mx(t−1) − Nx(t−2) + h.

HISTORY OF DIRECT AND ITERATIVE METHOD

Ø Direct methods to solve linear systems

Direct methods for solving the linear systems with the Gauss elimination method is given byCarl Friedrich Gauss (1777-1855). Thereafter the Choleski gives method for symmetric positive definite matrices.

Ø Iterative methods for non-linear equations