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 16-FEB-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.

Updates:

• The QR, Hessenburg and Schur decompositions have had a complete overall.
  • QR/Hessenburg/Schur: The way reflectors are computed and applied has been significantly improved in terms of performance, stability, and sensitivity to over(under)flow issues.
  • Schur: Some correctness issues have been ironed out and significant performance enhancements have been made. The decomposition uses an implicit double shifted QR algorithm. This means computing eigenvalues should be more accurate/correct and significantly faster in most cases.
• Tensor Solver: Added a solver which can solve tensor equations of the form $A \cdot X = B$ where $A, \ X$, and $B$ are tensors of arbitrary rank and shape as long as the shapes are conducive to computing the tensor dot product B = A.dot(X).
• Tensor Inverse: Added ability to compute the inverse of an arbitrary invertible tensor $A$, $A^{-1}$ relative to a specified tensor product.

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Features and Functionality

Algebraic Objects

• Complex Numbers
• Vectors
  • Real/Complex Dense Vector
  • Real/Complex Sparse Vector
• Matrices
  • Dense Matrices
    • Real/Complex Dense Matrix
  • Sparse Matrices
    • Real/Complex COO Matrix
    • Real/Complex CSR Matrix
    • Real/Complex Symmetric Tri-diagonal Matrix.
    • Permutation Matrix
• Tensors
  • Real/Complex Dense Tensor
  • Real/Complex 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
  • No pivoting
  • Partial Pivoting
  • Full Pivoting
• QR Decomposition
  • Householder Reflectors
  • Full/Reduced
• Cholesky Decomposition
• Hessenburg Decomposition
• Eigen/Schur Decompositions
• Singular Value Decomposition

Linear Solvers

• Exact solution for well determined matrix systems
  • General systems
  • Triangular systems
• Exact solution for well determined tensor 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.