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Chapter3Numerical Algebra — Approximating Roots

In this chapter we want to solve algebraic equations using a computer. Consider the equation \(\ell(x) = r(x)\) (where \(\ell\) and \(r\) stand for left and right respectively). To solve this equation we can first rewrite it by subtracting the right-hand side from the left to get \begin{equation*} \ell(x) - r(x) = 0. \end{equation*}

For example, if we want to solve the algebraic equation \(\sin(x) + 9 = x^2 - \tan(x)\) then this is the same as solving \((\sin(x) + 9 ) - (x^2 - \tan(x)) = 0\) (please don't try to solve this one by hand!). Hence, we can let the function \(f(x)\) be defined as \(f(x)=\ell(x)-r(x)\) and observe that every algebraic equation can be written as \begin{equation*} \text{ if } f(x) = 0 \text{ find } x. \end{equation*} We now have one way to view every algebraic equation-solving problem. As we'll see in this chapter, if \(f(x)\) has certain properties then different numerical techniques for solving the equation will apply.