Importance of numerical methods

Numerical methods are techniques that give approximate numerical answers to mathematical problems that are hard or impossible to solve exactly (analytically). Computers are used to carry out the many repeated calculations involved. Their importance comes from the fact that most real engineering problems have no closed-form solution.

Why they are needed

• Many equations (e.g. x·eˣ = cos x, sin x = 1/x) cannot be solved by algebra.
• Data from experiments is given as a table, not a formula, so integration, differentiation and curve fitting must be done numerically.
• Large systems of equations (hundreds of unknowns) cannot be solved by hand.
• Exact methods are often too slow or too complex; an approximate answer with a known error is enough.

Applications in science and engineering

• Civil / Mechanical: structural analysis, finite element method, heat transfer, fluid flow (solving large linear systems and differential equations).
• Electrical / Electronics / Computer: circuit analysis (nodal equations), signal processing (FFT, interpolation), solving non-linear device equations, control system simulation.
• Computer science: computer graphics (curves, splines), machine learning (optimisation, least squares), scientific computing and simulation.
• Physics and chemistry: solving ordinary and partial differential equations such as motion, vibration, diffusion and reaction rate problems.
• Weather, economics, medicine: forecasting models, curve fitting of observed data, eigenvalue problems for stability.

Advantages

• Can handle complicated geometry and non-linear problems.
• Easily programmed and repeated, so results are fast and reproducible.
• Error can be estimated and controlled by choosing step size or number of iterations.