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.

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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)