3 - Numerical solution of equations pp. 17-22By Roger Cooke
Mathematical Time Capsules
Online ISBN: 9780883859841
Chapter DOI: http://dx.doi.org/10.5948/UPO9780883859841.004
Methods of solving polynomial equations lie at the heart of classical algebra. There are two interpretations of the problem of solving an equation, leading to two different approaches to its solution. In most courses, the emphasis is on the structure of the equation and finding a way to express the roots as a formula in terms of the coefficients. The simplest example of such a formula is the quadratic formula, which gives the solution of the equation ax2 + bx + c = 0 as
This approach is elegant and leads to some exceedingly profound mathematics. However, for one who actually needs to know a number that satisfies the equation, this approach leaves something to be desired. It works with maximum efficiency in the case of the quadratic equation, but even in that case, if the quantity under the radical is not the square of a rational number, one is forced to resort to approximations in order to get a usable number. For cubic and quartic equations, there are formulas, but they work even less well, since they often involve taking the cube root of a complex number, which is a problem just as complicated as the original equation was, if not more so. Once again, one is forced to resort to numerical approximations. Beyond the fourth degree, the only formulas involve non-algebraic expressions, and are of little practical use. Higher-degree equations are the realm of numerical methods. To understand how numerical methods work, it is useful to begin with the simplest cases and the simplest methods. That is what we are about to do.
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