, STATISTICAL CONSULTANT/ANALYST/TUTOR/CIVIL ENGINEER /MATHEMATICIAN/SUBJECT MATTER EXPERT
at CEO @ TG TUTORIALS / S.M.E @ TUTOR VISTA AND STUDENT OF FORTUNE ,
1.
TARUN GEHLOT (B.E, CIVIL, HONOURS)
The Newton-Raphson Method
Already the Babylonians knew how to approximate square roots. Let's consider the
example of how they found approximations to .
Let's start with a close approximation, say x1=3/2=1.5. If we square x1=3/2, we obtain
9/4, which is bigger than 2. Consequently . If we now consider 2/x1=4/3, its
square 16/9 is of course smaller than 2, so .
We will do better if we take their average:
If we square x2=17/12, we obtain 289/144, which is bigger than 2.
Consequently . If we now consider 2/x2=24/17, its square 576/289 is of
course smaller than 2, so .
Let's take their average again:
x3 is a pretty good rational approximation to the square root of 2:
but if this is not good enough, we can just repeat the procedure again and again.Newton
and Raphson used ideas of the Calculus to generalize this ancient method to find the
zeros of an arbitrary equation
Their underlying idea is the approximation of the graph of the function f(x) by the tangent
lines, which we discussed in detail in the previous pages.Let r be a root (also called a
"zero") of f(x), that is f(r) =0. Assume that . Let x1 be a number close
to r (which may be obtained by looking at the graph of f(x)). The tangent line to the graph
of f(x) at(x1,f(x1)) has x2 as its x-intercept.
2.
TARUN GEHLOT (B.E, CIVIL, HONOURS)
From the above picture, we see that x2 is getting closer to r. Easy calculations give
Since we assumed , we will not have problems with the denominator being
equal to 0. We continue this process and find x3 through the equation
This process will generate a sequence of numbers which approximates r.This
technique of successive approximations of real zeros is called Newton's method, or
the Newton-Raphson Method.
Example. Let us find an approximation to to ten decimal places.
Note that is an irrational number. Therefore the sequence of decimals which
defines will not stop. Clearly is the only zero of f(x) = x2
- 5 on the interval
[1,3]. See the Picture.
3.
TARUN GEHLOT (B.E, CIVIL, HONOURS)
Let be the successive approximations obtained through Newton's method. We
have
Let us start this process by taking x1 = 2.
It is quite remarkable that the results stabilize for more than ten decimal places after only
5 iterations!
Example. Let us approximate the only solution to the equation
In fact, looking at the graphs we can see that this equation has one solution.
4.
TARUN GEHLOT (B.E, CIVIL, HONOURS)
This solution is also the only zero of the function . So now we see
how Newton's method may be used to approximate r. Since r is between 0 and , we
set x1 = 1. The rest of the sequence is generated through the formula
We have
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