# MeshLines
This repository is a work in progress. It will contain a set of algorithms for solving some problems in Linear Algebra.
Motivation, Goals, and Disclaimers
I started this project to educate myself on writing linear algebra routines and extending these algorithms to a compatible type. I intend to learn about matrix factorization and develop insights to elide operations given their type information e.g. computing $M \times I = M$ without a cost or $M\times M^{-1} = I$.
Note that there is no long-term initiative to support and resolve bugs on this project. The robustness of the codebase relies on the test coverage. You are welcome to submit an issue or drop a suggestion for this repo.
Design Principles
Design for genericity and specialize later on performance
- Add template mechanisms for interopt: it should be able to work with types that meets the operations required by Matrix semantics.
- Zero cost abstraction: Imposing rules and expectations should come without cost
- Minimal redundancies
Matrix Decomposition:
- [ ] Singular Value Decomposition
- [x] QR Factorization
- [x] LU Factorization
- [x] LDL Decomposition
- [x] PLU Decomposition
- [ ] Eigenvalue Decomposition
Matrix Operations:
- [x] Matrix-Matrix operations
- [x] Matrix-Vector operations
- [x] Transpose
- [x] Determinant
- [x] Minor
- [x] Cofactor
- [x] Adjugate
- [x] Matrix-Scalar operations
- [ ] Inverse
- [ ] Colspace
- [ ] Rowspace
- [ ] Nullspace
- [ ] Span
Matrix Decompositions
| Method | Description | Type of Matrix | Status |
|---|---|---|---|
| LU Decomposition | Factors a matrix into a product of a lower triangular matrix $L$ and an upper triangular matrix $U$. | Square | |
| QR Decomposition | Factors a matrix into a product of an orthogonal matrix $Q$ and an upper triangular matrix $R$. | Square, Rectangular | |
| Cholesky Decomposition | Factors a symmetric positive-definite matrix into a product of a lower triangular matrix $L$ and its transpose $𝐿^T$. | Symmetric Positive-Definite | |
| Singular Value Decomposition (SVD) | Factors a matrix into a product of three matrices: $U$ (orthogonal), $\sum$ (diagonal), and $VT$ (orthogonal). | Square, Rectangular | |
| Eigenvalue Decomposition | Decomposes a matrix into its eigenvalues and eigenvectors. | Square | |
| Schur Decomposition | Decomposes a matrix into a product of an orthogonal matrix $Q$ and an upper triangular matrix $T$. | Square | |
| Hessenberg Decomposition | Decomposes a matrix into a product of an orthogonal matrix $Q$ and a Hessenberg matrix $H$. | Square | |
| Jordan Decomposition | Decomposes a matrix into its Jordan normal form. | Square | |
| Polar Decomposition | Decomposes a matrix into a product of a unitary matrix $U$ and a positive semi-definite matrix $P$. | Square | |
| Bunch-Kaufman Decomposition | Factors a symmetric or Hermitian matrix into a product involving a permutation matrix, a lower triangular matrix, and a block diagonal matrix. | Symmetric, Hermitian |