newton raphson method

1. Newton raphson method By- Yogesh bhargawa M .Sc. 4 th sem. Roll no. 4086

2. Introduction As we know from school days , and still we have studied about t he solutions of equations like Quadratic equations , cubical e quations and polynomial equations and having roots in the f orm of x= where a, b , c are the coefficient of equ . But nowadays it is very difficult to remember formulas for higher degree polynomial equations . Hence to remove these difficulties there are few numerical methods, one of them is Newton Raphson Method . Which has to be discussed in this power point presentation.  

3. Newton Raphson Method : Newton Raphson method is a numerical technique which is used to find the roots of Algebraic & transcendental Equations . Algebraic Equations : An equation of the form of quadratic or polynomial. e.g. + +1=0 -1 =0 -2x -5=0  

4. Transcendental equation : An equation which contains some transcendental functions Such as exponential or trigonometric functions. e.g. sin , cos , tan , , , log etc. 3x-cosx-1=0 l ogx+2x=0 -3x=0 Sinx+10x-7=38  

5. Newton Raphson Method : Let us consider an equation f(x)=0 having graphical representation as

6. f(x) =0 ,is an given equation Starting from an initial point Determine the slope of f(x) at x= Call it f’( Slope =tan Ѳ = From here we get f ’( Hence ; = Newton R aphson formula  

7. Algorithm for f(x)=0 Calculate f’(x) symbolically. Choose an initial guess as given below let [ a,b ] be any interval such that f(a)<0 and f(b)>0 , then = . then Similarly Then by repetition of this process we can find At last we reach at a stage where we find . Then we will stop. Hence will be the required root of given equation.  

8. NRM by Taylor series : if we have given an equation f(x)=0. be the approximated root of given equation. Let ( h) be the actual root where ‘h’ is very small such that f ( )=0 From T aylor series expansion on expanding to f( ) f( )=f + hf’( )+ f’’( ) + f’’’( ) +……. Now on neglecting higher powers of h f ( ) + hf’( ) =0 From above h= -   Cont..

9. Hence first approximation =( ; = - Second approximation ; = - On repeating this process We get = This is the required newton Raphson method.  

10. How to solve an example : F(x)= - 2x – 5 F’(x) = 3 -2 Now checking for initial point F(1) =-6 F(2)= -1 F(3) =16 Hence root lies between (2,3) Initial point ( ) = =2.5 From NRM formula = ,putting all these above values in this formula  

11. = On putting initial value We get first approximate root; =2.164179104 Similarly: =2.097135356 =2.094555232 =2.094551482 =2.094551482 Hence = Hence 2.094551482 is the required root of given equation.  

12. Application of NRM To find the square root of any no. To find the inverse To find inverse square root. Root of any given equation.

13. Limitations of NRM F’(x)=0 is real disaster for this method F’’(x)=0 causes the solution to diverse Sometimes get trapped in local maxima and minima .

14. THANK YOU..!!!!!!