🟢04 - Cholesky Decomposition Method (Algorithm)

Introduction

In this tutorial, we will learn how to solve a system of linear equations using the Cholesky Decomposition Method. This method is particularly useful for solving equations where the coefficient matrix is symmetric and positive definite. By the end of this guide, you will understand how to represent your system of equations, decompose the matrix, and find the solution step by step.

Step 1: Represent the System of Equations

First, we need to represent your system of linear equations in matrix form.

• Identify your coefficients and constants from your linear equations.
• Arrange them into the form Ax = b, where:
  • A is the matrix of coefficients.
  • x is the vector of variables.
  • b is the vector of constants.

Example: For the equations:

1. 2x + 3y = 5
2. 4x + 1y = 6

The matrix representation would be:

• A = (\begin{pmatrix} 2 & 3 \ 4 & 1 \end{pmatrix})
• x = (\begin{pmatrix} x \ y \end{pmatrix})
• b = (\begin{pmatrix} 5 \ 6 \end{pmatrix})

Step 2: Decompose the Matrix

Next, we decompose the matrix A into the product of a lower triangular matrix H and its transpose H^T.

• Start with the decomposition: A = HH^T
• H should be a lower triangular matrix with positive diagonal entries.

Steps to Perform Decomposition

1. Begin with the first row and column of A, setting the first entry of H to the square root of the first entry of A.
2. Use the formula to fill the entries of H:
   • For each element ( h_{ij} ) in H, calculate using: [ h_{ij} = \frac{1}{h_{jj}} \left( a_{ij} - \sum_{k=1}^{j-1} h_{ik}h_{jk} \right) ]
3. Repeat until all elements of H are filled.

Step 3: Solve the Intermediate Equation

Once you have H, we need to solve for an intermediate variable y in the equation H^T y = b.

• Substitute H^T and b into the equation.
• This can be solved using forward substitution since H is a lower triangular matrix.

Steps for Forward Substitution

1. Initialize the vector y with zeros.
2. For each row:
   • Calculate each entry using: [ y_i = \frac{1}{h_{ii}} \left( b_i - \sum_{j=1}^{i-1} h_{ij}y_j \right) ]

Step 4: Solve for the Original Variables

Now, we solve for the original variable vector x from the equation H x = y.

Steps for Backward Substitution

1. Initialize the vector x with zeros.
2. For each row in reverse order:
   • Calculate each entry using: [ x_i = y_i - \sum_{j=i+1}^{n} h_{ij}x_j ]

Conclusion

You have successfully implemented the Cholesky Decomposition Method to solve a system of linear equations. To summarize:

• Represent your equations in matrix form.
• Decompose the matrix into a lower triangular form.
• Solve for intermediate variables and then for the original variables.

Next Steps

• Practice with different systems of equations to become proficient.
• Explore other methods of solving linear equations for comparison, such as Gaussian elimination or LU decomposition.