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Flag4j

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Fast linear algebra for Java

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# Flag4j - Fast Linear Algebra for Java
Flag4j is a fast and easy to use linear algebra library for Java and provides operations and linear algebra routines for real/complex sparse/dense tensors, matrices, and vectors.

## State of Project as of 7-DEC-2024
Flag4j is currently in the final steps of development before an initial beta release. 
Nearly all features have been fully implemented or beta implementations.

### Recent Updates:
- I/O Improvements:
  - All tensors, vectors, and matrices may be serialized.
  - Added support for reading/writing from/to Matrix Market Exchange file formats.
    - Currently only supports a subset including real/complex/integer dense general matrices 
    and real/complex/integer coordinate general matrices
  - Added support for reading/writing from/to csv files for real and complex matrices (i.e. `Matrix` and `CMatrix`).
- Improved API: There have been significant changes to the API specifically with the inheritance hierarchy of arrays (tensors, 
  matrices, vectors). This was in part to serve the creation of general field/ring/semiring tensors.
- New Algebraic Structures and Generic Tensors:
  - Added `Field`, `Ring`, and `Semiring` interfaces representing the mathematical objects.
    - Complex numbers are now implemented as a `Field`. There are now 64-bit and 128-bit variants of complex numbers.
    - Several reference implementations of the interfaces are provided, e.g. `Bool`, `RealInt16`, `Complex128`.
  - Added generic `FieldMatrix<T extends Field<T>>` for creating matrices for a generic field. This allows users
  to easily create a matrix for a custom field which implements the `Field` interface. Support for generic 
  ring and semiring matrices are planed.
___

## Features and Functionality

### Algebraic Objects
- Complex numbers, fields, rings, and semirings.
- Vectors
  - Real/complex/field dense vector
  - Real/complex/field sparse vector
- Matrices
    - Dense Matrices
        - Real/complex/field dense dense matrix
    - Sparse Matrices
        - Real/complex/field COO matrix
        - Real/complex/field CSR matrix
        - Real/complex/field symmetric tri-diagonal matrix. 
        - Permutation matrix
- Tensors
    - Real/complex/field dense tensor
    - Real/complex/field sparse COO tensor

### Operations
- Basic Arithmetic Operations: add, subtract, scalar/element multiply, scalar/element divide, etc.
- Basic Properties: tensor/matrix/vector shape, non-zero entries, etc.
- Basic Manipulations: insert, join/stack/augment, extract, etc.
- Basic Comparisons: equal, same shape, etc.
- Vector Operations:
  - Arithmetic: inner/outer/cross product, vector norms, etc.
  - Comparisons: Parallel, orthogonal, etc.
- Matrix Operations:
  - Arithmetic: Matrix multiplication, transpose, matrix norms, inverse, pseudo-inverse, etc.
  - Features: Matrix rank, symmetric, definiteness, eigenvalues and vectors, singularity, triangular, etc.
  - Comparisons: Similar, etc. 
- Tensor Operations:
  - Arithmetic: Tensor dot product, tensor transpose, tensor inverse, etc.
  - Comparisons: Tensor rank/dimension, etc.

### Matrix Decompositions
- LU Decompositions (real/complex)
  - No pivoting
  - Partial Pivoting
  - Full Pivoting
- QR Decomposition (real/complex)
  - Householder Reflectors
  - Full/Reduced
- Cholesky Decomposition (real/complex)
- Hessenburg Decomposition (real/complex)
  - General and specialized symmetric and Hermitian implementations are provided.
- Eigen/Schur Decompositions (real/complex)
- Singular Value Decomposition (real/complex)

### Linear Solvers
- Exact solution for well determined matrix systems
  - General systems
  - Triangular systems
- Exact solution for well determined tensor or matrix equations
- Least Squares solution

### Linear and Homography Transformations
- Scale
- Shift
- Rotate
- Affine
- Projections
  - Orthographic
  - Perspective

### Random
- Random Complex Numbers
- Random Matrices
  - Orthogonal, Unitary, symmetric, triangular, etc.
- Random Vectors
- Random Tensors

### I/O: Read/write tensors, matrices, and vectors to/from a file.
- Serialization (all types)
- Matrix Market Exchange Format (all real/complex dense and sparse matrix types)
- CSV format (only supported for `Matrix` and `CMatrix`)

Flag4j

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