The ideal programming environment for computational mathematics is one that enjoys the following characteristics:
It must be based on a computer language that allows the user to work quickly, and integrate many systems effectively. Ideally, the underlying computer language should run on all different platforms (Windows, Mac OS X, Linux, Unix, iOS, Android, and so on.). This is key to fostering cooperation among scientists with different resources, as well as accessibility.
It must contain a powerful set of libraries that allow the acquisition, storing, and handling of big datasets in a simple and effective way. This is key to allowing simulation and the employment of numerical computations at large scale.
Smooth integration with other computer languages, as well as third-party software.
Besides the usual running of compiled code, the programming environment should allow the possibility of interactive sessions, as well as scripting capabilities, for quick experimentation.
Different coding paradigms should be supported; imperative, object-oriented, or functional coding styles should all be available to the user.
It should be an open-source software; the user should be allowed to access the raw code of the libraries, and modify the basic algorithms if so desired. With commercial software, the inclusion of the improved algorithms is applied at the discretion of the seller, and it usually comes at a cost of the user. In the open-source universe, someone in the community usually performs these improvements, as they are published—at no cost.
The set of applications should not be restricted to mere numerical computations; it should be powerful enough to allow symbolic computations as well.
Among the best-known environments for numerical computations used by the scientific community, we have the powerful MATLAB® and Scilab® systems (although both of them are commercial, expensive, and do not allow any tampering with the code). Maple® and Mathematica® are more geared towards symbolic computation, although they can match many of the numerical computations from MATLAB®. As the previous two, these are also commercial, expensive, and closed to modifications. A decent alternative to MATLAB®, based on similar mathematical engine, is the GNU Octave system. Most of the MATLAB® code is easily portable in Octave. It also has the advantage of being open source. Unfortunately, the underlying programming environment is not very user friendly. It is also restricted to numerical computations.
The one environment that combines the best of all worlds is indeed the combination of Python with the NumPy and SciPy libraries. The first property that attracts the user to Python is, without a doubt, its code readability. The syntax is extremely clear and expressive. It has the advantage of supporting code written in different paradigms – object oriented, functional, or old school imperative. It allows the compilation of code for running standalone executable programs, but it can also be used interactively, or as a scripting language. This is a great advantage if the user needs to develop tools for symbolic computation. Python has been used in this sense as the basis of a firm competitor to Maple® and Mathematica®: the open-source mathematics software Sage (System for Algebra and Geometry Experimentation).
NumPy is an open-source extension to Python that adds support for multidimensional arrays of large sizes. This support allows the desired acquisition, storage, and complex manipulation of data mentioned previously. NumPy alone is a great tool to solve many numerical computations.
On top of NumPy, we have yet another open-source library, SciPy. This library contains algorithms and mathematical tools to manipulate NumPy objects, with very definite scientific and engineering objectives.
The combination of Python, NumPy, and SciPy (which henceforth should be coined "SciPy" for brevity) has been the environment of choice of many applied mathematicians for years; we work on a daily basis with both the pure mathematicians and with the hard-core engineers. One of the challenges of this trade is to bring to a single workstation the scientific production of professionals with different visions, techniques, tools, and software. SciPy is the perfect solution for coordinating everything together in a smooth, reliable, and coherent way.
Any day of the week, we are required to produce scripts in which, for example, there are combinations of experiments written and performed in SciPy itself, C/C++, Fortran, or MATLAB®. We often receive extremely large amounts of data from some signal acquisition devices. From all this heterogeneous material, we employ Python to retrieve the data, manipulate and, once finished with the analysis, produce high-quality documentation with professional-looking diagrams and visualization aids. SciPy allows performing all these tasks with ease.
This is partly because many dedicated software tools easily extend the core features of SciPy. For example, although any graphing and plotting is usually done with the Python libraries of matplotlib, there are also other packages, such as Biggles (biggles.sourceforge.net), Chaco (pypi.python.org/pypi/chaco), HippoDraw (github.com/plasmodic/hippodraw), MayaVi for 3D rendering (mayavi.sourceforge.net), or the Python Imaging Library or PIL (pythonware.com/products/pil).
Interfacing with non-Python packages is also possible. For example, the interaction of SciPy with the R statistical package can be done with RPy (rpy.sourceforge.net/rpy2.html). This allows for much more robust data analysis.