Tuesday, September 10, 2013

Newton-Raphson Method

Welcome everyone to my blog. Today I am going to write on Newton-Raphson method, which is in Numerical Analysis called the Newton's method widely in brief.

Newton-Raphson method is used widely in finding better approximation of roots of real-valued function. 

For doing this calculation we can take two easy ways, one with a graph that will help us through, or directly start calculating.

In the graphical method, all we need is to guess a point on the function f(x), draw its Tangent. Find out where it intersects x, again draw a new tangent using that intersecting point, we will get a new point on x. If it is closer to the function than the previous one, it is a better approximation. More iteration may result in even better approximation. Remember it may not work all the time.





Next one is the straight-forward calculating using calculator or something. In straight forward calculation, we need the formula. 




As you can see the formula is very simple.You have to make sure a few things though. 

1. f(x) must be equal to 0. That means it should be f(x)=0. If you have an equation like X^3 +2X=2, You have to change it to X^3+2X-2=0 this form.

2. Finding out f'(x) symbolically. According to the formula you have to find out the first derivative of the function that is given. You have to manually calculate this.

3. Choose an initial guess Xo, some people may call it X1 and the next one is X2 and so on, its totally up to you, you can write anything you want as long as you are writing the formula correctly.

4. Use formula to find out next value.

5. Now the next value will be the new guess, put it in the formula and get the next value.

6. You can keep on the iteration to find out better or closer results.

Absolute Relative Approximate Error 


Sometimes pre-specified tolerance is given, you have to use this approximate error to see how much error is in the calculation. To minimize error or get in the tolerance level more iterations might be needed.

Example 

1.

x^3 = 20, we have to find out a root of this function. 

First of all it needs to be converted to a form like this f(x)=0, so we can write x^3-20=0.

Now finding out the derivative. x^3 -20 will result in 3x^2.

Assume a guess of x=3.

Now putting that in theorem we get 

X1 = Xo -(f(Xo)/f'(Xo))

that is X1 = 3 - ((3^3-20)/(3*3^2))

X1 = 2.740740741

To find out X2, we will just replace the position of Xo with X1 and will get X2.

From there we can check what if the new result satisfies the condition or not. If not iteration will go on.

We can find approximate error if pre-specified error tolerance is given. We will just use the formula shown in earlier figure and will find out error. 

Using Calculator

If you want to speed up the calculation you can use a scientific calculator. Scientific calculators saves the last value as Ans. Just type the initial guess press = sign. Initial guess will be stored in Ans. Now use appropriate operators to establish the example 1. Now each time you press = it will show the next value, as Ans value changes with every click it will actually give you the results of onward iterations.


Any way that was all for now. Pretty basic stuff. Hope it will help.

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