crout s method for solving system of linear equations

Numerical Matrix methods

for solving the System of Linear

algebraic equations

By Poonam DeshpandeTeam 5 - RC 1229

Pre-requisites for this topic

Students should have the knowledge of

• Definition of a Matrix

• Different types of matrices

• Upper and lower triangular matrices

• Matrix algebra like addition, subtraction and

multiplication of matrices

• System of Linear Algebraic Equations

Learning Objectives: • To understand how to write a System of Linear Algebraic

Equations in the matrix equation form.

• To enable students to understand how to solve the large system of Linear algebraic equations using iterative numerical methods and how to write a programing code for these matrix methods

• To master the numerical methods like Gauss-Jordan method, Crout’s Method, Iterative Method, and Gauss-Seidel Method for solving the System of Linear Algebraic Equations

• To develop the analytical ability to apply these learnings to the real world problems

Learning Outcomes• Students will be able to understand what is the System of

Linear Algebraic Equations and how to write a System of Linear Algebraic Equations in the matrix equation form

• Students will be able to understand and master the numerical methods like Gauss-Jordan method, Crout’s Method, Iterative Method, and Gauss-Seidal Method for solving the large System of Linear Algebraic Equations

• Students will be able to write a programing code for these matrix methods

• Students will develop the analytical ability to apply these learnings to the real world problems

System of linear algebraic equations

Consider the system of linear algebraic equations given by

.

.

Which can be written in the matrix equation form as

Here • A is the Co-efficient matrix • X the solution matrix (which to be calculated) and• B is the constant matrix.

Numerical Matrix methods

• Gauss-Jordan Method

• Crout’s Method

• Iterative Method

• Gauss-Seidel method

Crout’s MethodConsider the matrix equation of the system of 3 equations in

3 unknowns

We write matrix A as a product of an Upper and Lower

Triangular matrices[1]

Where,

[1] http://ktuce.ktu.edu.tr/~pehlivan/numerical_analysis/chap02/Cholesky.pdf

Crout’s Method (cont.)

Since (1)

Gives (2)

Let us take (3)

is some unknown matrix which is to be evaluated

Then (4)

Therefore to find the solution of the system (1) we will have to solve (4) and then (3), but before that we will have to evaluate the values of L and U

Algorithm for Crout’s Method

Use the following steps to solve the System of Linear algebraic equations. • Step 1: Write

• Step 2: Calculate the Product of L and U

Algorithm for Crout’s Method (cont.)

• Step 3: write and

• Step 4: Solve by forward substitution

• Step 5: Solve by backward substitution

Thank You

ExampleSolve the following system of equations by Crout’s Method

Thank You