Solving Nonlinear Equations

''solve $f(x) = 0$'' where $f(x)$ is a function of $x$. The values of $x$ that make $f(x) = 0$ are called the roots of the equation. They are a1so called the zeros of $f(x)$.
The following non-linear equation can compute the friction factor,$f$:

$$\frac{1}{\sqrt{f}}=\left(\frac{1}{k}\right)ln(RE\sqrt{f)}+\left(14-\frac{5.6}{k}\right)$$

where the parameter $k$ is known and RE, the so-called Reynold's number. The equation for $f$ is not solvable except by the numerical procedures of this chapter.

• Interval Halving (Bisection). Describes a method that is very simple and foolprof but is not very efficient. We examine how the error decreases as the method continues.
• Linear Interpolation Methods. Tells how approximating the function in the vicinity of the root with a straight line can find a root more efficiently. It has a better "rate of convergence".
• Newton's Method. Explains a still more efficient method that is very widely used but there are pitfalls that you should know about. Complex roots can be found if complex arithmetic is employed.
• Muller's Method. Approximates the function with a quadratic polynomial that fits to the function better than a straight line. This significantly improves the rate of convergence over linear interpolation.
• Fixed-Point Iteration: $x = g(x)$ Method. Uses a different approach: The function $f(x)$ is rearranged to an equivalent form, $x = g(x)$. A starting value, $x_0$, is substituted into $g(x)$ to give a new x-value, $x_1$. This in turn is used to get another x-value. If the function $g(x)$ is properly chosen, the successive values converge.