Kao-math Library

The kao-math library is a 100% pure Java implementation of useful mathematical functions. It contains a Matrix and Vector implementation which all other functionality is built on.

Matrix/Vector Library

The Matrix/Vector library kao.math.matvec provides classes for a Matrix and Vector implementation. The library includes dense and sparse matrices, multi-threaded Matrix/Vector operations, and direct and iterative solvers of systems of linear equations. A full list of the key features is given below:

Dense Matrix
Sparse Matrix
- Compressed Row Format
Vector
Multi-threaded operations
Matrix/Vector can be defined with an arbitrary non-negative starting index
Find the inverse of a Matrix
QR Factorisation
LU Decompostion with backward and forward substitution
Matrix Preconditioning
Conjugate Gradient Solver
BiConjugate Gradient Solver
Gaussian Elimination
Gauss-Siedel Iterative Solver

See the documentation for detailed information of the mathematical operations.

QP Library

The QP library kao.math.QP provides classes for solving Quadratic Programs (QP). QPs have the following form:

$\min f(x) = \frac{1}{2} x^{T} G x + d_{0}^{T} x$

such that

$A_{eq}^{T}x - b_{eq} = 0$

$A_{ineq}^{T}x - b_{ineq} \geq 0$

where:

is a column vector of length
$d_{0}$ is a column vector of length
is a $n \times n$ positive definite matrix
$A_{eq}^{T}$ is a $q \times n$ matrix where each row defines an equality constraint
$A_{ineq}^{T}$ is a $p \times n$ matrix where each row defines an inequality constraint
$b_{eq}$ is a column vector of length
$b_{ineq}$ is a column vector of length

The method provided by the QP library is based on Goldfarb [1]. In addition a set of active constraints $A_{0}$ calculated from an initial guess can be passed into the algorithm.

[1] D. Goldfarb and A. Idnani A numerically stable dual method for solving strictly convex quadratic problems, Mathematical Programming 27, 1983, 1-33.