What is Numerical Analysis and Why Should You Care?

Numerical analysis, also known as scientific computing, refers to the use of computer programs to perform math and science problems that would be extremely complicated to do manually or through mathematical formulae alone. Today’s numerical analysis tools are even able to handle problems that computers in the past would have been unable to deal with, and they’re constantly evolving to become more advanced and efficient. But before you can use these tools, you need to understand what numerical analysis is and why it’s important.

An Introduction to Numerical Analysis

Before you embark on a numerical analysis assignment, it’s important to understand exactly what that entails. The term numerical analysis refers to a wide range of techniques for modeling, analyzing, and solving quantitative problems. As a broad-based field of study, numerical analysis integrates many disciplines—including applied mathematics, computer science, engineering, physics, and statistics—to approach problems using computational methods in lieu of (or in addition to) analytical ones.

The History of Numerical Analysis

To properly understand numerical analysis, it’s first important to realize how we got here. The roots of numeric analysis can be traced back to 16th century, but it wasn’t until Isaac Newton published his Philosophiae Naturalis Principia Mathematica in 1687 that computation became part of math.

Time-Stepping Methods

Time stepping methods are used for computational simulation of systems governed by partial differential equations with strong time-dependence. A differential equation is a mathematical equation which relates functions of one or several variables to their derivatives (and, hence, to themselves). It deals with rates of change, which means that its solutions change over time. One of its greatest applications has been in physics: it allows us to model and understand physical phenomena as well as to predict how they will behave under different conditions.

Interpolation Schemes

Also known as numerical integration, these are methods for approximating an unknown function (like, say, how much money you’ll make after taxes) from discrete samples of that function (the two pieces of data we know: how much money you made before taxes and what your tax rate was). The simplest scheme, linear interpolation, gives a very rough approximation; more complex techniques like regression analysis or Newton’s method give more accurate results.

Linear Algebraic Equations

Solving linear equations, finding a root to an equation, solutions that are multiple solutions etc. are very basic in numerical analysis. Linear algebraic equations are used extensively in science and engineering. For example you may have to find out what happens to your motor if it overheats. The simplest way of doing it is by solving linear algebraic equations.

Finite Difference Methods

The finite difference method of solving a differential equation is to approximate its solution by an algebraic formula. The general finite difference method uses second-order derivatives for approximating derivatives and first-order derivatives for approximating solutions. Finite difference methods are used in modeling different problems in physics, engineering, chemistry, economics, biology, among others. They are also widely used in computational fluid dynamics and molecular dynamics simulations.

Direct Solution Methods

There are three primary types of solution methods for differential equations: direct methods, iterative methods, and transform methods. The difference between these can be confusing, but essentially direct methods are used when a system of differential equations can be solved with ease. For example, if you knew that \(y'=-y\) then you could use a forward Euler method to solve it.

Finite Element Methods

In finite element analysis, engineers typically divide a physical object into different segments. These segments are then broken down into even smaller blocks called elements. An engineer can use finite element analysis to calculate an object’s stresses in any situation where stresses vary over a surface—that means almost any engineering application. Finite element analysis also makes it possible to design an object so that it won’t fail when stressed in a given way—the Holy Grail of stress-analysis design tools!

Verification and Validation Tools

When you’re evaluating software and systems, there are many verification methods at your disposal. The specific method you use depends on a variety of factors, including: how big your team is, how much time you have to test, what testing phase you’re in (the early or late phases), etc. This post covers some of the most common tools used for verification/validation—from white box tests to static analysis to live testing.