![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](https://image.slidesharecdn.com/croutsmethod-160708063035/85/Crout-s-method-for-solving-system-of-linear-equations-7-320.jpg)
Crout s method for solving system 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