Solving Nonlinear Algebraic Equations

Abstract

As a reader of this book you are probably well into mathematics and often ‘‘accused’’ of being particularly good at ‘‘solving equations’’ (a typical comment at family dinners!). However, is it really true that you, with pen and paper, can solve many types of equations? Restricting our attention to algebraic equations in one unknown x, you can certainly do linear equations: ax+b=0, and quadratic ones: ax2+bx+c=0. You may also know that there are formulas for the roots of cubic and quartic equations too. Maybe you can do the special trigonometric equation sinx+cosx=1sinx+cosx=1\sin x+\cos x=1 as well, but there it (probably) stops. Equations that are not reducible to one of the mentioned cannot be solved by general analytical techniques, which means that most algebraic equations arising in applications cannot be treated with pen and paper! If we exchange the traditional idea of finding exact solutions to equations with the idea of rather finding approximate solutions, a whole new world of possibilities opens up. With such an approach, we can in principle solve any algebraic equation. Let us start by introducing a common generic form for any algebraic equation: f(x)=0.f(x)=0\thinspace. Here, f(x)f(x) is some prescribed formula involving x. For example, the equation e−xsinx=cosxe^{-x}\sin x=\cos x has f(x)=e−xsinx−cosx.f(x)=e^{-x}\sin x-\cos x\thinspace. Just move all terms to the left-hand side and then the formula to the left of the equality sign is f(x)f(x).