Numerical analysis and mathematical modeling are essential in many areas of modern life. Sophisticated numerical analysis software is commonly embedded in popular software packages (e.g., spreadsheet programs) and allows fairly detailed models to be evaluated, even when the user is unaware of the underlying mathematics. Attaining this level of user transparency requires reliable, efficient, and accurate numerical analysis software, and it requires problem-solving environments (PSE) in which it is relatively easy to model a given situation. PSEs are usually based on excellent theoretical mathematical models, made available to the user through a convenient graphical user interface.

Computer-aided engineering (CAE) is an important subject within engineering, and some quite sophisticated PSEs have been developed for this field. A wide variety of numerical analysis techniques is involved in solving such mathematical models. The models follow the basic Newtonian laws of mechanics, but there is a variety of possible specific models, and research continues on their design. One important CAE topic is that of modeling the dynamics of moving mechanical systems, a technique that involves both ordinary differential equations and algebraic equations (generally nonlinear). The numerical analysis of these mixed systems, called differential-algebraic systems, is quite difficult but necessary in order to model moving mechanical systems. Building simulators for cars, planes, and other vehicles requires solving differential-algebraic systems in real time.

Another important application is atmospheric modeling. In addition to improving weather forecasts, such models are crucial for understanding the possible effects of human activities on the Earth’s climate. In order to create a useful model, many variables must be introduced. Fundamental among these are the velocity V(x, y, z, t), pressure P(x, y, z, t), and temperature T(x, y, z, t), all given at position (x, y, z) and time t. In addition, various chemicals exist in the atmosphere, including ozone, certain chemical pollutants, carbon dioxide, and other gases and particulates, and their interactions have to be considered. The underlying equations for studying V(x, y, z, t), P(x, y, z, t), and T(x, y, z, t) are partial differential equations; and the interactions of the various chemicals are described using some quite difficult ordinary differential equations. Many types of numerical analysis procedures are used in atmospheric modeling, including computational fluid mechanics and the numerical solution of differential equations. Researchers strive to include ever finer detail in atmospheric models, primarily by incorporating data over smaller and smaller local regions in the atmosphere and implementing their models on highly parallel supercomputers.

Modern businesses rely on optimization methods to decide how to allocate resources most efficiently. For example, optimization methods are used for inventory control, scheduling, determining the best location for manufacturing and storage facilities, and investment strategies.

Software to implement common numerical analysis procedures must be reliable, accurate, and efficient. Moreover, it must be written so as to be easily portable between different computer systems. Since about 1970, a number of government-sponsored research efforts have produced specialized, high-quality numerical analysis software.

The most popular programming language for implementing numerical analysis methods is Fortran, a language developed in the 1950s that continues to be updated to meet changing needs. Other languages, such as C, C++, and Java, are also used for numerical analysis. Another approach for basic problems involves creating higher level PSEs, which often contain quite sophisticated numerical analysis, programming, and graphical tools. Best known of these PSEs is MATLAB, a commercial package that is arguably the most popular way to do numerical computing. Two popular computer programs for handling algebraic-analytic mathematics (manipulating and displaying formulas) are Maple and Mathematica.