Linear Interpolation (False Position)

• A way to avoid such pathology is to ensure that the root is bracketed between the two starting values and remains between the successive pairs.
• When this is done, the method is known as linear interpolation (regula falsi).
• This technique is similar to bisection except the next iterate is taken at the intersection of a line between the pair of x-values and the x-axis rather than at the midpoint.
• Doing so gives faster convergence than does bisection, but at the expense of a more complicated algorithm.
• An algorithm for the method of false position:
  
• If $f(x)$ is not continuous, the method may fail.

Table 3.3: Comparison of methods, $f(x)=3x + sin(x) - e^x$, starting from $x_0=0,x_1=1$.

• Table 3.3 compares the results of three methods-interval halving (bisection), linear interpolation, and the secant method for $f(x) = 3x + sin(x) - e^x=0$
• Observe that the speed of convergence is best for the secant method, poorest for interval halving, and intermediate for false position.