What Is Numerical Software

Digital computer was originally invented to do mathematics. The mission of the first digital computer, Electronic Numerical Integrator and Computer (ENIAC, 1945), was to quickly obtain artillery range tables. Running at the electronic speed, the computer performed a tremendous amount of calculation.

We use digital computers to crunch more and more numbers. Computer code follows the numerical methods, which are developed based on the mathematical formulations. Sometimes the mathematics has an associated physical problem. But sometimes, it’s just mathematics. The applications are endless, but here list some famous packages to give you an idea:

Despite the versatility, numerical software shares common traits:

  • Not visually pleasant, oftentimes no graphical user interface
  • Knowledge-intensive, unintuitive to code
  • Computation-intensive, often incorporating parallelism, distributed computing, and special hardware

Numerical software is developed to solve problems in science and engineering. It always has an application domain attached, and cannot be handled solely in computer science. Of course, since it is computer software, it cannot exist without computer science. Naturally it is cross-discipline and demands knowledge and skills in two or more fields from the practitioners.

Why Develop Numerical Software

Numerical software is developed to solve problems that are either impracticable without it.

Numerical software simply enables the solution so that we can study the impracticable problems, e.g., those in the fields of fluid dynamics and astrophysics. The software may also significantly reduce the cost to solution, so that industrial products can be derived from the solution, e.g., machine learning, visualization, communication, etc.


Like developing any software, the true driver must be identified so that the system can be properly specified.

A pattern is usually found in the development of numerical software:

  1. Observation
  2. Generalize to a theory in math
  3. Obtain analytical solutions for simple setup
  4. Get stuck with complex setup
  5. Numerical analysis comes to rescue
  6. a lot of code development
  7. Release a software package

Hybrid Architecture

A hybrid architecture is employed to achieve the highest-possible performance and the most flexible computation that are simultaneously required by numerical software. The system is composed of a fast, low-level computing engine and an easier-to-use, high-level scripting layer. It is usually developed as a platform, working like a library that provides data structures and helpers for problem solving. The users will use a scripting engine it provides to build applications. Assembly is allowed in the low-level computing engine to utilize every drop of hardware: multi-core, multi-threading, cache, vector processing, etc.


Performance and flexibility are usually mutually exclusive. The hybrid architecture achieves both in the same time by sacrificing ease of use.

As such, numerical software rarely provides decent graphical user interface (GUI). Using numerical software is like using a library, rather than a off-the-shelf software product. Even if you do purchase a commercial software package for numerical calculation, the full capabilities need to be accessed by scripting.

A general description of the architecture is like the following layers, from high-level to low-level:

  • External result
    • This is presented in a non-technical way to people outside the problem-solving team. They can be stakeholders for business or general public. The result has to be generated in some way, which may or may not be included in the numerical software we make.
  • Problem presentation: physics, math, or equations
    • Users use the software or associated tools to present the technical result.
  • Scripting or configuration
    • Users follow the example scripts to configure the problems to solve. Configuration files may also be used.
  • Library interface
    • This defines the application programming interface (API) for the numerical software. Scripts should not touch anything below this layer.
  • Library structure
    • This is where we architect the software. Good book-keeping code is here to separate the interface and the computing kernel. Data structures are designed at this layer to make sure no time is wasted in copying or converting data.
  • Computing kernel
    • This is the place the does the heavy-lifting, and where we do most of the optimization.

Pattern 1: Research Code

For a research code, the boundary between external result, problem presentation, and scripting, and that between library interface, library structure, and computing kernel, may be less clear. The architecture is usually like:

  • Problem presentation: high-level description, physics, and scripting / code configuration
  • Library implementation

But sometimes if we don’t pay attention to architecture, there may be no boundary between anything.

Pattern 2: Full-Fledged Application

For a commercial grade package, each of the layers will include more sub-layers. It is a challenge to prevent those layers or sub-layers from interweaving. From users’ point of view, the sophistication appears in the problem presentation and the scripting layers. Developers, on the other hand, take care of everything below problem presentation, so that users can focus on problem solving.

Pattern 3: Scripting for Modularization

At this point, it should be clear that the scripting layer is the key glue in the system architecture. The high-level users, who use the code for problem solving, wouldn’t want to spend time in the low-level implementation. Instead, they will specify the performance of the API exposed in the scripting layer. The performance may be about the quality of result and runtime (including memory).

The scripting layer can separate the programming work between the high-level problem presentation and the low-level library implementation. A scripting language is usually dynamically typed, while for speed, the low-level implementation language uses static typing system. In the dynamic scripting language, unit-testing is required for robustness. In a statically typed language like C++, the compiler and static analyzers are very good at detecting errors before runtime. But the great job done by the compiler makes it clumsy to use C++ to quickly write highly flexible code for problem presentation.

It is tempting to invent one programming language to rule them all. That approach needs to convince both the high-level problem solvers and the low-level implementers to give up the tools they are familiar with. The new language will also need to provide two distinct styles for both use cases. It will be quite challenging, and before anyone succeeds with the one-language approach, we still need to live with a world of hybrid systems.

Numerical Software = C++ + Python

The key to a successful numerical software system is make it uncompromisingly fast and extremely flexible. It should be flexible enough so that users, i.e., scientists and engineers, can easily write lengthy programs to control everything. It should be noted that, although the users program in the system, they by no means know about computer science.

Not all programming languages can meet the expectation. To this point, the most suitable scripting language is Python, and the most suitable low-level language may be C++. The choice of C++ can be controversial, but considering the support it received from the industry, it’s probably difficult to find another language of higher acceptance. Our purpose here is to introduce the skills for developing numerical software, not to analyze programming languages. We will focus on C++ and Python.

More Reasons to Use Python

  • Python provides a better way to describe the physical or mathematical problem.
  • Python can easily build an even higher-level application, using GUI, scripting, or both.
  • Is there alternative for C++? No. For Python? Yes. But Python is the easiest choice for its versatility and simplicity.
  • A numerical software developer sees through the abstraction stack:
    • The highest-level application is presented as a Python script.
    • The Python script drives the number-crunching C++ library.
    • C++ is the syntactic sugar for the machine code.