Equation Solving with Newton-Raphson Method

Date: 07/23/2004 at 14:04:00
From: Steve
Subject: Simplifying an Equation

Hello Dr. Math!

The equation for a future worth (F) given a uniform series payment 
(A), an interest rate per period (i), and (n) number of periods is:

  F = A*(((1+i)^n) - 1)/i

Is there a way to solve this equation for i, such that we have 
i = ...?
	
I am having trouble combining the two i's into one i.  I'm thinking of 
natural logs, but seem to get stuck there.

Date: 07/23/2004 at 14:56:58
From: Doctor Korsak
Subject: Re: Simplifying an Equation

Hello Steve,

There is no so-called "closed form" solution for i in terms of the 
other parameters; however, you can iterate towards a solution using 
something like the Newton's Method algorithm (often called Newton-
Raphson).  Define a function like

    f(i) = A*(((1+i)^n) - 1)/i  - F

and then solve the equation f(i) = 0 by starting out with a trial 
value like i = 0.1 and then iterating towards a solution to your 
desired accuracy.  Many hand-held calculators have a button for this 
kind of operation.

Take a look at a couple of the Dr. Math library archives on this 
subject:

  Cow Grazing Half the Circle: Newton-Raphson Method
    http://mathforum.org/library/drmath/view/53622.html 

  Determining Roots of a Function
    http://mathforum.org/library/drmath/view/62610.html 

Please write back if you need further help.

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