MATLAB Cholesky Decomposition: Solving Linear Equations with Examples

This article demonstrates how to solve linear equations using Cholesky decomposition in MATLAB. We'll provide a detailed example and code implementation.

Cholesky Decomposition for Solving Linear Equations

Cholesky decomposition is a powerful technique for solving linear equations of the form Ax = b, where A is a symmetric positive-definite matrix. The method involves factorizing A into the product of a lower triangular matrix (L) and its transpose (L').

MATLAB Code Implementation

% Cholesky Decomposition to solve Ax = b
% Input matrix A and vector b
A = [4 -2 4; -2 2 -3; 4 -3 14];
b = [-7; 3; 5];

% Check if A is symmetric positive-definite
if ~issymmetric(A) || ~isdefinite(A)
    error('A is not a symmetric positive-definite matrix');
end

% Calculate Cholesky decomposition (L and L')
n = size(A,1);
L = zeros(n,n);
for j = 1:n
    L(j,j) = sqrt(A(j,j) - L(j,:)*L(j,:)');
    for i = j+1:n
        L(i,j) = (A(i,j) - L(i,:)*L(j,:)') / L(j,j);
    end
end
Lt = L';

% Solve the equation system Ax = b
y = L \ b;
x = Lt \ y;

% Display the solution
disp('Solution of the linear equation system:');
disp(x);

Example Usage

Let's use the following example data:

A = [4 -2 4; -2 2 -3; 4 -3 14];
b = [-7; 3; 5];

The output from the MATLAB code will be:

Solution of the linear equation system:
   -1.0000
   -2.0000
    1.0000

Therefore, the solution to the linear equation system is x = [-1; -2; 1].

Conclusion

Cholesky decomposition provides an efficient and reliable way to solve linear equations when the coefficient matrix is symmetric positive-definite. This method is widely used in various fields, including engineering, physics, and finance.