https://github.com/aostrun/matrix-lup
Matrix implementation that includes LU/LUP decomposition and solving basic linear equations
https://github.com/aostrun/matrix-lup
linear-equations lu-decomposition matrix-decompositions matrix-factorization python
Last synced: about 1 year ago
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Matrix implementation that includes LU/LUP decomposition and solving basic linear equations
- Host: GitHub
- URL: https://github.com/aostrun/matrix-lup
- Owner: aostrun
- License: mit
- Created: 2018-02-11T13:13:34.000Z (over 8 years ago)
- Default Branch: master
- Last Pushed: 2018-02-11T14:33:01.000Z (over 8 years ago)
- Last Synced: 2025-03-29T11:11:26.847Z (about 1 year ago)
- Topics: linear-equations, lu-decomposition, matrix-decompositions, matrix-factorization, python
- Language: Python
- Size: 17.6 KB
- Stars: 3
- Watchers: 1
- Forks: 2
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# Matrix LUP Decomposition
---
Python implementation of matrices that implements features such as LU/LUP decomposition and
forward and backward substitutions which are used for solving linear equation **Ax = B**
and solving matrix inverse trough LU decomposition.
## LU and LUP
LU decomposition refers to the factorization of square matrix A into two factors,
a unit in lower triangular matrix L and an upper triangular matrix U, **A = LU**.
More on the theory behind LU decomposition can be found [here](https://en.wikipedia.org/wiki/LU_decomposition).
This implementation allows for LU and LUP decompositions. Difference between the is that LU is the pure form of
decomposition and LUP uses partial pivoting (permutations in rows) which solves problem of selecting pivot with value 0.
## Linear equations
We can represent linear equation with matrix form Ax = B where we want to solve the equation for x given A and B.
Firstly we need to decompose matrix A to **A = LU** with LU decomposition. After that our linear equations turns to
**LUx = B**. We can insert substitution y = Ux and get equation **Ly = B**. Now we can easily solve this equation with
forward substitution and find the y. After finding y we solve **Ux = y** with backward substitution and in the end we
find the missing x. If we use LUP insted of LU we need to do matrix multiplication with
P (permutation matrix of LUP) and B, resulting with equation **Ly = PB** and **Ux = y**.
## Examples
See [testScript.py](testScript.py) for simple examples of LU/LUP and solving linear equations using LU decomposition.
## Authors
* **Andrijan Ostrun** - [aostrun](https://github.com/aostrun)
## License
This project is licensed under the MIT License - see the [LICENSE.md](LICENSE.md) file for details