6.5

Employ the newton-Raphson method to determine a real root for f(x) = -2 +6x - 4x^2 +.5x^3 using initial guesses of (a) 4.2 and (b) 4.43. Discuss and use graphical and analytical methods to explain any peculiarities in your results.


Matlab Code:

x = 0:.001:5;

f = @(x) -2 +6 * x - 4 * x .^2 + 0.5 * x .^3;

hold on

plot (x,f(x))

%(a) Newton-Raphson method (three iterations, x_0 = 4.2).

df = @(x) 6 - 2 * 4 * x + 3 * 0.5 * x .^2; 

ax = 4.2;

for n = 1:1:23

ax_n = ax - f(ax) ./ df(ax);

ax = ax_n;

end

plot (ax_n, x, 'k')

error_approximatea = (ax_n - ax) ./ ax_n;

%(b) Newton-Raphson method (three iterations, x_0 = 4.43).

bx = 4.43;

for n = 1:1:23

bx_n = bx - f(bx) ./ df(bx);

bx = bx_n;

end

plot (bx_n, x, 'r')

error_approximateb = (bx_n - bx) ./ bx_n;

Red is the approximation to the real root 4.43
Black is the approximation to the real root 4.2
Blue is f(x) = -2 +6x - 4x^2 +.5x^3



EDU>> ax_n


ax_n =


    0.4746


EDU>> bx_n


bx_n =


    0.4740


EDU>> error_approximatea


error_approximatea =


     0


EDU>> error_approximateb


error_approximateb =


     0