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: $$ax^{2}+bx+c=0$$. 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 $$\sin x+\cos x=1$$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!