https://github.com/benw126/matrix-decomposition-algorithms

MATLAB code about several matrix decomposition and linear system algorithms. (Note for MATH4602)
https://github.com/benw126/matrix-decomposition-algorithms

algorithms applied-mathematics computation matlab matrix-factorization

Last synced: about 1 year ago
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MATLAB code about several matrix decomposition and linear system algorithms. (Note for MATH4602)

• Host: GitHub
• URL: https://github.com/benw126/matrix-decomposition-algorithms
• Owner: benw126
• License: mit
• Created: 2024-06-07T18:05:18.000Z (about 2 years ago)
• Default Branch: main
• Last Pushed: 2024-07-04T12:24:05.000Z (almost 2 years ago)
• Last Synced: 2025-02-15T03:24:14.657Z (over 1 year ago)
• Topics: algorithms, applied-mathematics, computation, matlab, matrix-factorization
• Language: MATLAB
• Homepage:
• Size: 23.4 KB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

          
## Doolittle Factorization

Doolittle Factorization is a method used to decompose a square matrix $A$ into the product of a unit lower triangular matrix $L$ and an upper triangular matrix $U$, i.e., $A = LU$. Solving a linear system of the form $Ax = b$ using Gaussian elimination generally requires at least $O(n^3)$ computational cost, where $n$ is the size of the square matrix $A$. **For this factorization, decomposition itself requires $O(n^3)$ computations, but solving $LUx = b$ only requires the cost of $O(n^2)$ through forward and backward substitutions.**

 

Therefore, if we need to solve the linear system $Ax = b$ multiple times with different right-hand side vectors $b$, the computational cost is much lower by using the Doolittle Factorization, as we only need to **perform the $O(n^3)$ decomposition once**, and then the **$O(n^2)$ forward and backward substitutions for each new $b$**.

## Cholesky Factorization

Cholesky Factorization is a method used to decompose a Hermitian, positive-definite matrix $A$ into the product of a lower triangular matrix $L$ and its conjugate transpose, i.e., $A = LL^*$. The computational cost required in this decomposition is $O(n^3)$. **Every Hermitian positive-definite matrix admits a unique Cholesky decomposition**.

## Circulant Matrix System

A circulant matrix $C$ is a special type of square matrix where each row (or column) is a circular shift of the previous row (or column). We can take advantage of the fact that **every circulant matrix can be diagonalized by the Discrete Fast Fourier Transform (*DFT*) matrix $F_n$**, where $F_n C F_n^{-1} = D$ and $n$ is the size of the circulant matrix $C$. Furthermore, all the eigenvalues of $C$ can be obtained by applying the *DFT* to the first column of $C$.

The linear system $Cx = b$ is equivalent to $D(F_n)x = F_nb$. Matrix-vector multiplications of $F_nx$ and $F_n^{-1}x$ can be performed in $O(n\log n)$ computational cost. As a result, **the linear system $D(F_n)x = F_nb$ can be solved in $O(n\log n)$ time** using the following steps:

1. Apply the *DFT* to the first column of $C$ to obtain diagonal matrix $D$.

2. Apply the *DFT* to the right-hand side vector $b$, then divide the result elementwise by the diagonal elements of $D$.

3. Apply the inverse *DFT* to the resulting vector.

This efficient algorithm of the circulant matrix system requires only $O(n\log n)$ computational cost, making it a very useful technique in various applications.