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.

secant <- function(fun, x0, x1, tol=1e-07, niter=500){
for ( i in 1:niter ) {
x2 <- x1-fun(x1)*(x1-x0)/(fun(x1)-fun(x0))
if (abs(fun(x2)) < tol)
return(x2)
x0 <- x1
x1 <- x2
}
stop("exceeded allowed number of iteractions")
}

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

#### Related Posts

Nice post

Here's a crude approach using R which I've found to be quite handy when precision and speed aren't a big concern.

Reply

ygc Reply:

December 6th, 2010 at 10:12 am

Thanks for sharing.

It's much straightforward to do it by:

> fun = function(x) x/(1+x) -(5-x)/5

> x = seq(0, 5, length=5000)

> which.min(abs(fun(x)))

[1] 1792

> fun(x[1792])

[1] 2.312344e-05

You need not to split the function to f=x/(1+x) and g=(5-x)/5. Many functions, such as cos(x), cannot be splitted.

Reply

Paul Reply:

December 7th, 2010 at 7:12 am

Very true, using which.min() would be more efficient. With some tinkering, you can always deal with multiple roots by do something like...

> x=seq(0,15,length=5000)

> curve(cos(x),0,15);

> points(x[which((abs(cos(x))<2e-3))],

cos(x[which((abs(cos(x))

Reply

ygc Reply:

December 7th, 2010 at 9:55 am

Absolutely right.

Typo error occur in last line.

should be:

points(x[which((abs(cos(x))<2e-3))],cos(x[which((abs(cos(x))<2e-3))]))

Reply