2 Solutions of Equations in One Variable
One of the oldest problems of numerical analysis is to find a root, or zero, of an equation \(f(x) = 0\): a number \(p\) with \(f(p) = 0\). Except for special equations (linear and quadratic ones, for instance) there is no formula for the root, and it must be approximated by a sequence \(p_0, p_1, p_2,\dots \) that converges to it. This section develops the main methods, proves when and how fast they converge, and compares them on the same problems.
Throughout the section we use the equation \begin {equation} \label {eq:test} f(x) = x^3+4x^2-10 = 0 \end {equation} as a common test problem. Since \(f(1) = -5<0\) and \(f(2) = 14>0\), the Intermediate Value Theorem gives a root in \((1,2)\); and since \(f'(x) = 3x^2+8x>0\) there, the root is unique. Its value is \(p = 1.365230013\) to ten significant digits.
2.2 The method of false position
2.3 Fixed-point iteration
2.4 Newton’s method and the secant method
2.5 Multiple roots and accelerating convergence
2.6 Practice Problems
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