Newton-Method / Newton-Raphson Method

 1. What is Newton-Method?

Since early education in Mathematics, we have been exposed to technique of finding roots. For instance when we need to find which value of \(y\) that can fulfills the following equation

$$ y^2 = 25,$$

we can use root finding method to solve it. Analytically, we can then proceed to solve the above equation as follows

$$ y^2 - 25 = 0$$

$$ y^2 - 5^2 = 0$$

$$ (y-5)(y+5) = 0.$$

Thus, the \(y\)'s that can become the part of our solution are 5 and -5.

This simple procedure is a hard problem when the degree of polynomial increases or involving complex functions such as trigonometric function. To make it worse, the difficulties magnified when we want to set/automate the root finding procedure into the computer. Thus, the Newton-Method which is among the earliest algorithm suggested by Joseph Raphson to approximate the solution of the roots can be used.

2. How to perform Newton Method?

We will first look at how this algorithm is derived. Recalled that in calculus, the derivative of a function \(f(x)\) is defined as,

$$ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{(x+h)-x}.$$

Our goal is to develop an algorithm that will proposed a new \(x\) in each iteration and the value of \(f(x) \rightarrow 0 \) (a root) as the iteration continues. But how do we generate the new \(x\)?

This is obvious if we manipulate the definition above as follows.

Let \(x_{n+1} = x+h\) be the new value that we want to propose and \(x_n = x\) be the former value that we already know its \(f(x_n)\) value. Then it is noted that,

$$ f'(x_n) \approx \frac{f(x_{n+1}) - f(x_n)}{x_{n+1} - x_n}.$$ 

$$ x_{n+1} - x_n \approx \frac{f(x_{n+1}) - f(x_n)}{f'(x_n)}$$

$$ x_{n+1} \approx x_n + \frac{f(x_{n+1}) - f(x_n)}{f'(x_n)}.$$

Since we are expecting the new \(x_{n+1}\) to cause \(f(x_{n+1})\) approaching 0, thus the last equation is updated into,

$$ x_{n+1} \approx x_n + \frac{0 - f(x_n)}{f'(x_n)} = x_n - \frac{f(x_n)}{f'(x_n)}.$$

Thus completing our algorithm. 

EG:

a) Find a positive \(y\) that satisfy \(y^2 = 25\).

Solution: Denote that \(y^2 - 25 = 0\). We can use the root finding algorithm discussed above. Information that we need are as follows,

$$ f(y) = y^2 - 25$$

$$ f'(y) = 2y.$$

R script to run Newton-Raphson method:

f <- function(y) y^2 - 25
f.prime <- function(y) 2*y

y <- 3 # set initial y (close to answer is better)
tol <- 100 # initialize threshold to exit loop when f(y) is 0
n.iter <- 0 # store iteration

while(abs(tol) > 1e-5 && n.iter < 500){
  tol <- f(y)
  y <- y - f(y)/f.prime(y)
  n.iter = n.iter + 1
}

print(paste0("With ", n.iter, " iterations, y = ",y))

## [1] "With 5 iterations, y = 5"

b) Find all solutions to \(2x^3 - 4x = 1\).

Graph below shows the curve for this question.

Solution: There are many R packages that can perform root finding algorithms. Among the package that accept Newton-Method is the 'rootSolve' package.

R script below is an example on how to find multiple roots.

f <- function(x) 2*x^3 - 4*x - 1
rootSolve::uniroot.all(f, c(-5,5))

## [1] -1.2669017 -0.2586498  1.5256917

3. When it does not work?

Although this algorithm is simple to be implemented, we still need to be careful when using it. As we have seen that this procedure rely on the computation of the first derivative and selection of the initial point, this also could become the reason of why the Newton-Method might fail. 

For instance, it is normal to see function which requires cumbersome computation for its first derivative. For this case, approximating the derivative using a line slope is more practical. Such technique can be found through the secant method.

Selecting initial point also could lead to a problem when the solution is far from the selected point or have a close roots. This would lead to a slow convergence. 

4. Conclusion. 

Newton-Method is among the earliest technique used in root finding. Although there are other superior techniques been developed to solve problems on root finding, the Newton-Method act as an initial platform to embark the journey in numerical analysis.