Post on 27-Jan-2017
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