Root-Finding Methods

Broadly speaking, the study of numerical methods is known as “numerical analysis,” but also as “scientific computing,” which includes several sub-areas such as sampling theory, matrix equations, numerical solution of differential equations, and optimisation. Numerical analysis does not seek exact answers, because exact answers rarely can be obtained in practice. Instead, much of numerical analysis aims at determining approximate solutions and at the same time keeping reasonable bounds on errors. In fact, computations with floatingpoint numbers are performed on a computer through approximations, instead of exact values, of real numbers, so that it is inevitable that some errors will creep in. Besides, there are frequently many different approaches to solve a particular numerical problem, being some methods faster, more accurate or requiring less memory than others.

The ever-increasing advances in computer science and technology have enabled us to apply numerical methods to simulate physical phenomena in science and engineering, but nowadays they are also found and applied to interesting scientific computations in life sciences and even arts. For example, ordinary differential equations are used in the study of the movement of heavenly bodies (planets, stars and galaxies); optimisation appears in portfolio management; numerical linear algebra plays an important role in quantitative psychology; stochastic differential equations and Markov chains are employed in simulating living cells for medicine and biology; and, the chaotic behaviour of numerical methods associated to colour theory in computer graphics can be used to generate art on computer.