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Newton-Raphson Method

 

We have find the solution of f (xy) = 0 and g (xy) = 0


First find an approximate solution of equation (x0y0) by graphical or trial and error method.


From this we find more approximate values (x1y1) such that
x1 = x0 h and y1 y0 + k


f (x0 + hy0 + k) = 0 and g (x0 + hy0 + k) = 0


From Taylor’s series we get,

 

495.png = 0 and 500.png = 0


Solving these equations we get h and k.


Thus we find x1 and y1

From this value of x1 and y1 we find x2 and y2 in similar way.

Bisection Method

 

For a non linear equation f (x) and for any two number a and b if f (a), f (b) > 0, then there is atleast one root for the equation f (x) in between a and b.


Now let x1 = a and x2 = b


We find the mid point of x1 and x2 i.e. x0

x0 = 505.png


If f (x0) = 0, x0 is the root

If f (x0f (x1) < 0, root is in between x0 and x1

If f (x0f (x2) < 0, root is in between x0 and x2 and this process in continued with either (x0x1) or with (x0x2).

 





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