Numerical Analysis

Numerical analysis is an area of mathematics that creates, analyzes, and implements algorithms for obtaining numerical solutions to problems involving continuous variables. The Newton-Raphson method to solve numerically the equation \(\tan x = x\) is one example. The Gauss quadrature method to numerically evaluate any definite integral \(\int _a^b f(x)dx\) is also one example. The finite difference method to solve ordinary and partial differential equations is yet another example. Numerical solutions are numbers not closed-form expressions. For example, it is possible to solve the quadratic equation \(ax^2+bx+c=0\) exactly to get the well-known closed-form solutions \(x_{1,2}=\nicefrac {-b\pm \sqrt{b^2-4ac}}{2a}\) . Such solutions do not exist for polynomial equations of fifth order or higher and for transcendental equations such as \(\tan x = x\) . However, the Newton-Raphson method can solve all the equations efficiently, but it only gives us numerical solutions. For example, applying to \(\tan x =x\) , it gives us 4.49340946. Once we have an algorithm the procedure to the solution becomes mechanical: we simply follow the algorithm step by step and voilà, there comes the solution. No genius is needed, and that’s why scientists and engineers love numerical methods.