COMPUTER ORIENTED NUMARICAL & STATIC METHODS x1 0.5
0.377583 3.479426
75
f '(x0)=3+sin(0.5) =3.479426 }
= 0.608519 Step-2: x1= 0.608519
x2 x1
f ( x1 ) f ( x1 )
x2 0.608519
f (0.608519) f (0.608519)
= 0.607102 Step-3: x2=0.607102
x3 x 2
f ( x2 ) f ( x2 )
x3 0.607102
f (0.607102) f (0.607102)
x3 =0.607102 Hence , the root of the given eqn is 0.607102 Example: 2 Using Newton-Raphson method, establish the formula +N/ xn) to calculate the square root of N. hence find the
xn+1=1(xn
5
2
correct to 4 decimals.
Solution : (i)To derive the formula: If x = Let
N , then x2-N=0 is the equation to be solved. 2
f (x)=x -N. (1)
Then
f'(x)=2x.
(2)
By Newton Raphson rule, th
(for n iteration)
xn 1 xn
f ( xn ) f ( xn )
, n=0,1,2,3,….
using (1) & (2) in the above equation,
xn N 2 xn 2
xn 1 xn
xn N 2 xn 2
xn+1= ½ (xn +N/ xn)
, n=0,1,2,3,….
1 2
N xn xn , n=0,1,2,3,….
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