LU Decomposition LAB ACTIVITY This tool is provided by a third party. Though your activity may be accessed, page refresh may be needed to fill the banner. 2.12.1: MATLAB: LU Decomposition MATLAB: LU Decomposition In this activity, you will find the LU decomposition of a matrix, utilize the decomposition to solve a system of linear equations, and check the solution using the operator. Consider the linear system of equations: 3x + 4y – 5z = 2 -5x + 7y – 4z = -4 x + 3y – 4z = 3 Create the coefficient matrix [4 3 -5; -5 7 -4] and the column matrix of constants [2; -4; 3]. Use the lu() command to find the LU decomposition of the coefficient matrix. Solve the system of linear equations Cx = d using the LU decomposition. This two-step process involves solving the system Ly = d for y, and then solving the system Ux = y for x. Utilize the following linear system of equations for this activity: 2x + 1y = 3 -1x + 4y = 2 Script 0 Save Reset MATLAB Documentation Create the coefficient matrix and the column matrix of constants. Use the lu() command to find the LU decomposition of A, storing the lower and upper matrices in L and U, respectively. Solve the system of linear equations Ax = b using the LU decomposition. Store the intermediary solution in y. Check that the solution matches that found by directly using the backslash operator to solve the system Ax = b. Use the backslash operator to solve Ax = b in one step, storing the solution in x2. Run Script
The Correct Answer and Explanation is:
To solve the LU decomposition problem using MATLAB, follow these steps based on the given system of equations:
Given system of equations:
2x + 1y = 3
–1x + 4y = 2
MATLAB Code:
matlabCopyEdit% Define the coefficient matrix A and constant matrix b
A = [2 1; -1 4];
b = [3; 2];
% Perform LU decomposition
[L, U] = lu(A);
% Solve Ly = b using forward substitution
y = L \ b;
% Solve Ux = y using back substitution
x = U \ y;
% Use the backslash operator to solve Ax = b directly
x2 = A \ b;
% Display results
disp('LU Decomposition:');
disp('L ='); disp(L);
disp('U ='); disp(U);
disp('Solution using LU decomposition:');
disp('x ='); disp(x);
disp('Solution using backslash operator:');
disp('x2 ='); disp(x2);
Output:
You will get:
- Matrices L and U
- Vector y (intermediate result)
- Final solution vector x using LU decomposition
- Direct solution x2 using backslash
Explanation
LU decomposition is a mathematical method used to simplify the process of solving a system of linear equations. It breaks down a given matrix A into two simpler matrices: a lower triangular matrix L and an upper triangular matrix U, such that A = LU. Once A is decomposed, the system Ax = b can be solved in two simpler steps.
First, we solve Ly = b for y using forward substitution. Since L is a lower triangular matrix, this step is efficient and avoids dealing with the entire system at once. After finding y, we solve Ux = y using back substitution because U is an upper triangular matrix.
In MATLAB, the lu() function performs this decomposition and returns matrices L and U. To solve the equations, MATLAB’s backslash operator \ is used both for forward and backward substitution.
The backslash operator x2 = A\b is MATLAB’s direct method for solving systems and is highly optimized. By comparing the result of LU decomposition with the backslash operator’s result, we can verify the accuracy of our LU-based solution. Both should produce the same solution vector x.
This two-step LU method is particularly useful for large systems or when solving multiple systems with the same matrix A but different vectors b, because L and U only need to be computed once.
