Crout s method for solving system of linear equations

This presentation shows the application of Crout's method for solution of systems of linear equations

1. Numerical Matrix methods for solving the System of Linear algebraic equations By Poonam Deshpande Team 5 - RC 1229

2. 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

3. 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

4. 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

5. 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.  

6. Numerical Matrix methods Gauss-Jordan Method Crout’s Method Iterative Method Gauss-Seidel method

7. Crout’s Method Consider 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  

8. Crout’s Method (cont.) Since (1) (1) Gives (2) (2) Let us take (3) is some unknown matrix which is to be evaluated (3) 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 (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

9. 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  

10. Algorithm for Crout’s Method (cont.) Step 3 : write and Step 4 : Solve by forward substitution Step 5 : Solve by backward substitution  

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12. Example Solve the following system of equations by Crout’s Method  

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