23 May 2017

Root Finding Methods

The bisection method in mathematics is a root-finding method that repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. The bisection method is a variation of the incremental search method in which the interval is always divided in half (in this case (1+2)/2 will be the midpoint and the range changes to subranges 1-1.5 and 1.5-2). If a function changes sign over an interval, the function value at the midpoint is evaluated. The location of the root is then determined as lying within the subinterval where the sign change occurs. The subinterval then becomes the interval for the next iteration. The process is repeated until the root is known to the required precision.

Perhaps the most widely used of all root-locating formulas is the Newton – Raphson method. If the initial guess at the root is xi,(in this case xi=1.5) a tangent can be extended from the point [xi , f (xi )]. The point where this tangent crosses the x axis usually represents an improved estimate of the root.

The secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite difference approximation of Newton’s method. Let x0 and x1 are two initial approximations for the root ‘s’ of f(x) = 0 and f(x0) & f(x1) respectively, are their function values. If x 2 is the point of intersection of x-axis and the line-joining the points (x0, f(x0)) and (x1, f(x1)) then x2 is closer to ‘s’ than x0 and x1. The equation relating x0, x1 and x2 is found by considering the slope ‘m’

The above figures show the secant methods behaviour for