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

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Matrix Decomposition:

• Singular Value Decomposition
• QR Factorization
• LU Factorization
• LDL Decomposition
• PLU Decomposition
• Eigenvalue Decomposition

Matrix Operations:

• Matrix-Matrix operations
• Matrix-Vector operations
• Transpose
• Determinant
• Minor
• Cofactor
• Adjugate
• 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