Cases Where the Newton-Raphson Method Fails

Date: 06/30/2005 at 12:25:31
From: Ola
Subject: what are the problems with the Newton Raphson method

The Newton Raphson method works with certain equations, like 
f(x) = x^5 - 5x + 3, where a tangent is drawn and it is to find the 
root of the line between the interval.  But it doesn't work on an 
equation like y = ln(x+1) + 1.  When this graph is plotted it does not 
work, it shows overflow on the Newton Raphson iteration.

I find it difficult with the tangent where it converges sometimes and 
diverges if the initial value is not close to the root, or is near a 
turning point of y = f(x).

Date: 07/01/2005 at 06:58:38
From: Doctor George
Subject: Re: what are the problems with the Newton Raphson method

Hi Ola,

Thanks for writing to Doctor Math.

There are different ways in which Newton-Raphson can fail.  You 
already know that Newton-Raphson can fail because of turning points.  
It can also fail because of asymptotes.

In your case I suspect the failure is because your initial estimate is
too high compared to the actual solution.  This causes your next
iteration to evaluate ln(x) at a negative value.  ln(x) has an 
asymptote at zero and is not defined for negative values.

Try a lower initial estimate.

Does that make sense?  Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/04/2005 at 11:02:37
From: Ola
Subject: Thank you (what are the problems with the Newton Raphson method)

Thank you very much.  It is nice to know that there are people who are
willing to help teenagers like myself in mathematics apart from my
teacher.