Solution of System of Linear and Non Linear Equations

Using Graph Theory

Team Members:Amar Chand : 13112009Deeksha Adlakha : 13112028

What is Graph Theory?

Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.

V = {a, b, c, d, e}E = {ab, ac, bd, cd, de}

Applications of Graph Theory

• Computer Science

• Electrical Engineering

• Chemical Science

• General

• Other

Types of Graphs

Null Graph Trivial Graph

• a

Non-Directed GraphDirected Graph

Simple Graph

Connected Graph Disconnected Graph

Regular Graph Complete Graph

Laplacian matrix

A simple graph G with n vertices, its Laplacian matrix L is defined as :

L = D - Awhere D is the degree matrix A is the adjacency matrix of the graph.

The elements of  L are given by :

where Deg(Vi) is degree of the vertex i.

Li,j =

 

Example :-

Methods for solving a system of equations

Linear Equations: 1. Gaussian Elimination Method2. Gauss Jordan Elimination Method3. Gauss Seidel Method

Non Linear Equations:1. Newton’s Method2. Secant Method

Gaussian Elimination Method

Algorithm for solving system of linearequations (Ax=b) by bringing the augmented matrix to an upper triangular form.

Example

Solve the following system of linear equations to obtain the values of x,y,z using Gaussian Elimination Method .

x – 3y +z = 42x – 8y + 8z = -2-6x + 3y – 15z = 9

C++ Code

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