PREPARED BYKETAN NAYAK (140413117005)HARSH PARMAR (140413117006)SAWAN PARMAR (140413117007)Chandan pathAK(140413117008)ASHISH PATIL(140413117009)

INSTRUMENTATION and Control DEPT

NUMERICAL TECHINIQUES & STATISTICAL METHODS

LINEAR SYSTEM SOLUTION USING ITERATIVE METHODS

Sr. No TITLE

1. INTRODUCTION

2. ITERATION METHODS

3. JACOBI’S METHOD

4. GAUSS SEIDEL METHOD

5. APPLICATION OF ITERATION METHODS

INTRODUCTION

• In various applications of Engineering the Linear Algebraic Equations occurs quite often.

• The Solution of such system of the Equations can be obtained by Direct Methods or Iterative Method.

• Here we are Going to Talk about the VARIOUS ITERATIVE METHODS used in Engineering.

ITERATIVE METHODS

• Whenever the Algebraic Equations contains large no of equations at that time DIRECT METHOD becomes lengthy and Time Consuming Headache.

• In this Case the Iteration methods plays a Vital Role by being the Substitute of the Method.

• In Iterative methods we start from an Initial Approximation to actual Solution & obtain the better Approximation after Repeating the ITERATION. This method gives Desired Accuracy.

ITERATIVEMETHODS

JACOBIMETHOD

GAUSS SIEDELMETHOD

Jacobi Method

•The Jacobi method is a method of solving a matrix equation on a matrix that has no zeros along its main diagonal (Bronstein and Semendyayev 1997, p. 892).

• Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges.

• This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization.

• The Jacobi method is easily derived by examining each of the n equations in the linear system of equations Ax=b in isolation. If, in the ith equation

• sum_(j=1)^na_(ij)x_j=b_i,

GAUSS SIEDEL METHOD

• It is a iterative method used to solve a linear system of equation.

• It uses the latest value to find the value of next variable.

APPLICATIONS

• Analysis of the Circuit containing the Invariant Elements.

• Determination of Output of Chemical Plants.

• Study of the Civil Engineering Structures.

• Problems of Layered Media in Siesmology etc.