GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command-line interface (CLI) for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB.
Mathematica is a mathematics software program used to help with science, math, and engineering fields. It can be used for industrial work, or for educational purposes. It allows integration of computation to many different types of fields and work flows.
CPLEX is an optimization software package, which helps users solve integer programming problems, linear programming problems, and many more.
The IBM ILOG CPLEX Optimizer solves integer programming problems, very large linear programming problems using either primal or dual variants of the simplex method or the barrier interior point method, convex and non-convex quadratic programming problems, and convex quadratically constrained problems (solved via second-order cone programming, or SOCP).
The CPLEX Optimizer has a modeling layer called Concert that provides interfaces to the C++, C#, and Java languages. There is a Python language interface based on the C interface. Additionally, connectors to Microsoft Excel and MATLAB are provided. Finally, a stand-alone Interactive Optimizer executable is provided for debugging and other purposes.
The CPLEX Optimizer is accessible through independent modeling systems such as AIMMS, AMPL, GAMS, MPL, OpenOpt, OptimJ and TOMLAB. In addition to that AMPL provides an interface to the CPLEX CP Optimizer.