A computer algebra system (CAS) is mathematical software that can manipulate mathematical formulae in a way similar to the traditional manual computations of mathematicians and scientists. This type of system supports a wide range of mathematics including linear algebra, calculus, and algebraic and ordinary differential equations.
A CAS offers a rigorous environment for defining and working with structures such as groups, rings, fields, modules, algebras, schemes, curves, graphs, designs, codes and many others.
They have been extensively used in higher education.
The main features of a CAS include:
- Numerical Computations: The software can determine numerical approximations of solutions, derivatives, integrals, differential equations, etc. Solve, manipulate, and plot functions without needing to generate numeric data. Often problems that cannot be solved explicitly can be solved numerically, and often only a numerical answer is sufficient.
- Data Analysis: Having data is not sufficient; we need to extract useful information from it. There are many algorithms designed for data analysis, most of which involve too much work to be done by manual computations. CAS’s put these algorithms in one place, and offer an environment where the algorithms are easy to implement.
- Data Visualization: CAS’s can graph 2D and 3D functions in a variety of ways. They are also designed to graph vector fields and solutions to differential equations.
- Symbolic Computations: Most of the CAS’s can perform symbolic manipulation of expressions: reducing, expanding, simplifying, derivatives, antiderivatives, etc. Unlike numerical computations, which can exhibit floating-point errors, symbolic computations are determined exactly. They can therefore provide the exact answer to an equation (as opposed to a decimal approximation), and they can express results in terms of a wide variety of previously defined functions.
A CAS automates tedious and sometimes difficult algebraic manipulation tasks. The principal difference between a CAS and a traditional calculator is the ability to deal with equations symbolically rather than numerically.
The chart below offers our rating for each software. Some of the software is very specialized, designed to fill a particular niche. This makes comparisons difficult.
To provide an insight into the quality of software that is available, we have compiled a list of 13 impressive algebra systems. There’s general purposes systems as well as specialist software solutions. All of them are open source software.
Let’s explore the 13 algebra systems at hand. For each application we have compiled its own portal page, a full description with an in-depth analysis of its features, screenshots, together with links to relevant resources.
| Computer Algebra Systems | |
|---|---|
| Maxima | System for the manipulation of symbolic and numerical expressions |
| PARI/GP | Widely used algebra system designed for fast computations in number theory |
| SymPy | Python library for symbolic mathematics |
| Scilab | Numerical computational package |
| SageMath | Open source alternative to Magma, Maple, Mathematica and Matlab |
| Octave | Powerful programming language with built-in plotting and visualization tools |
| Axiom | General purpose Computer Algebra system |
| SINGULAR | Computer Algebra System for polynomial computations |
| GAP | System for computational discrete algebra |
| CoCoA | System for doing computations in commutative algebra |
| Cadabra | Symbolic computer algebra system for field theory problems |
| Macaulay2 | Software system for research in algebraic geometry |
| FriCAS | Fork of Axiom |