Question

6.5 Employ the Newton-Raphson method to determine a real root for 4x20.5 using initial guesses of (a) 4.52 f(x) 15.5x Pick the best numerical technique, justify your choice and then use that technique to determine the root. Note that it is known that for positive initial guesses, all techniques except fixed-point iteration will eventually converge. Perform iterations until the approximate relative error falls below 2 %. If you use a bracket- ing method, use initial guesses of x 0 and x, = 2. If you use the Newton-Raphson or the modified secant method, use an ini- tial guess of x guesses of x-1 0.7. If you use the secant method, use initial = 0 and x 2.

Answer 1

f()= -155-4x2t0.53 fltn)= -l5.5xm-4x,2+0.5x f'(x)= 5.5-8x+ 1-5x f(n) = 5.5-g znt l.5 Zn Neoton Raphson methoc ing f(xn) f(xn) Put n- o f(z0) initial guesses Taking o4.52 4.52-f(4.52) fl4.52) 2240.54(452 fl4.52) -+5.5x4.52- f4.52) 688 f'(4-52) = 5.5-8xu52 +154.522 0.0144 X14.52 (-1-688). -807.I5 Foo144) we use initialgue.s of i=0.7 Xo-O-7 o.7- f lo. 7) Again, 07-+55 Xo 7-4x0-7210.5x073 5.5 8x07+ 15x0.72 0.7- 1.0615 o.9716 o.635 2 x-f (x1) (2),P I+55al-0.9716)-4x(09T16405X(0976 -0-9716 55-8x1-0-9716)+1:5x{-0.9716 2-o.9716- D-58) -0.2514 14.69 13 = f'(12) -0.2514-+5-5 x (- 02514)-4x(-0-25/4)2 +05x(-025143 55-8x(-025147415 x (-0 2514)2 0.2514-2.64) --0 2514t0.3469 7.6 2 0.0955