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Numerical root finding methods use iteration, producing a sequence of numbers that hopefully converge towards a limits which is a root. In this post, only focus four basic algorithm on root finding, and covers bisection method, fixed point method, Newton-Raphson method, and secant method.

The simplest root finding algorithms is the bisection method. It works when f is a continuous function and it requires previous knowledge of two initial gueeses, u and v, such that f(u) and f(v) have opposite signs. This method is reliable, but converges slowly. For detail, see http://ygc.name/2008/11/25/bisect-to-solve-equation/ .

Root finding can be reduced to the problem of finding fixed points of the function g(x) = c*f(x) +x, where c is a non-zero constant. It is clearly that f(a) = 0 if and only if g(a) = a. This is the so called fixed point algorithm.

fixedpoint <- function(fun, x0, tol=1e-07, niter=500){ ## fixed-point algorithm to find x such that fun(x) == x ## assume that fun is a function of a single variable ## x0 is the initial guess at the fixed point xold <- x0 xnew <- fun(xold) for (i in 1:niter) { xold <- xnew xnew <- fun(xold) if ( abs((xnew-xold)) < tol ) return(xnew) } stop("exceeded allowed number of iterations") }

> f <- function(x) log(x) - exp(-x) > gfun <- function(x) x - log(x) + exp(-x) > fixedpoint(gfun, 2) [1] 1.309800 > x=fixedpoint(gfun, 2) > f(x) [1] 3.260597e-09

The fixed point algorithm is not reliable, since it cannot guaranteed to converge. Another disavantage of fixed point method is relatively slow.

Newtom-Raphson method converge more quickly than bisection method and fixed point method. It assumes the function f to have a continuous derivative. For detail, see http://ygc.name/2007/06/02/newton-raphson-method/ .

The secant method does not require the computation of a derivative, it only requires that the function f is continuous. The secant method is based on a linear approximation to the function f. The convergence properties of the secant method are similar to those of the Newton-Raphson method.

> f <- function(x) log(x) - exp(-x) > secant(f, x0=1, x1=2) [1] 1.309800

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