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LinearLeastSquareHH

LinearLeastSquareHH is a MATLAB function that solves the linear least squares problem using Householder transformation. This method is particularly useful for numerical linear algebra tasks where stability and efficiency are important.

Overview

The function LinearLeastSquareHH computes the solution to a linear least squares problem of the form $A \mathbf{x} = \mathbf{b}$ by applying Householder transformations. The Householder method is used to orthogonalize the matrix $A$ and transform it into an upper triangular matrix $R$, making it easier to solve the system.

Function Signature

function x = LinearLeastSquareHH(A, b)

Inputs

  • A : A matrix of size $m \times n$ where $m$ is the number of equations and $n$ is the number of unknowns.
  • b : A vector of size $m \times 1$ representing the right-hand side of the linear system.

Output

  • x : A vector of size $n \times 1$ that represents the least squares solution to the system $A \mathbf{x} = \mathbf{b}$.

Description

The function performs the following steps:

  1. Initialization: The function initializes variables and sets up the Householder matrix to reflect the first column of $A$.

  2. Householder Transformations: Iteratively applies Householder reflections to transform matrix $A$ into an upper triangular matrix $R$ while updating the right-hand side vector $b$.

  3. Solution: Once $A$ is transformed into $R$, the function solves the upper triangular system to find the least squares solution $x$.

Example Usage

Here's an example of how to use the LinearLeastSquareHH function:

% Define matrix A and vector b
A = [1 2; 3 4; 5 6];
b = [7; 8; 9];

% Compute the least squares solution
x = LinearLeastSquareHH(A, b);

% Display the result
disp('Least squares solution:');
disp(x);

Dependencies

  • MATLAB (tested with versions R2020a and later)

License

This code is licensed under the MIT License. See the LICENSE file for more details.

Author

Sorawit Chokphantavee