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
𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ … … + 𝑎1𝑛 𝑥 𝑛 = 𝑏1
𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ … … + 𝑎2𝑛 𝑥 𝑛 = 𝑏2
.
.
𝑎 𝑚1 𝑥1 + 𝑎 𝑚2 𝑥2 + ⋯ … … + 𝑎 𝑚𝑛 𝑥 𝑛 = 𝑏 𝑚
Which can be written in the matrix equation form as
𝐴𝑋 = 𝐵
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
𝐴𝑋 = 𝐵
We write matrix A as a product of an Upper and Lower
Triangular matrices[1]
𝐴 = 𝐿𝑈
Where, 𝐿 =
𝑙11 0 0
𝑙21 𝑙22 0
𝑙31 𝑙32 𝑙33
𝑎𝑛𝑑 𝑈 =
1 𝑢12 𝑢13
0 1 𝑢23
0 0 1
[1] http://ktuce.ktu.edu.tr/~pehlivan/numerical_analysis/chap02/Cholesky.pdf

8. Crout’s Method (cont.)
Since 𝑨 = 𝑳𝑼 ∴ 𝑨𝑿 = 𝑩 (1)
Gives 𝑳𝑼𝑿 = 𝑩 (2)
Let us take 𝑼𝑿 = 𝒀 (3)
𝑌 is some unknown matrix which is to be evaluated
Then 𝐋𝐘 = 𝐁 (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 𝐴 = 𝐿𝑈 =
𝑙11 0 0
𝑙21 𝑙22 0
𝑙31 𝑙32 𝑙33
1 𝑢12 𝑢13
0 1 𝑢23
0 0 1
• Step 2: Calculate the Product of L and U
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
=
𝑙11 𝑙11 𝑢12 𝑙11 𝑢13
𝑙21 𝑙21 𝑢12 + 𝑙22 𝑙21 𝑢13 + 𝑙22 𝑢23
𝑙31 𝑙31 𝑢12 + 𝑙32 𝑙31 𝑢13 + 𝑙32 𝑢23 + 𝑙33

10. Algorithm for Crout’s Method (cont.)
• Step 3: write 𝐿 and 𝑈
• Step 4: Solve 𝐿𝑌 = 𝐵 by forward substitution
• Step 5: Solve 𝑈𝑋 = 𝑌 by backward substitution

11. Thank You

12. Example
Solve the following system of equations by Crout’s Method
2𝑥1 − 4𝑥2 + 𝑥3 = 4,
6𝑥1 + 2𝑥2 − 𝑥3 = 10,
−2𝑥1 + 6𝑥2 − 2𝑥3 = −6

15. Thank You