Question

Consider the function, f(x) = - x⁴ - 2x³ - 8x² - 5x

Use parabolic interpolation (x₀ = -2, x₁ = -1, x₂= 1, iterations = 4). Select new points sequentially as in the secant method.

Answer 1

First iteration

X2 = X0 - f(x0)((x1-x0)/(f(x1)-f(x0)))

Now X0 = -1 and X1 = 1

so F(X1) = - (1)⁴ - 2(1)³ - 8(1)² - 5(1) = -1-2-8-5 = -16

F(X0) = - (-1)⁴ - 2(-1)³ - 8(-1)² - 5(-1) = -1 +2-8+5 = -2

X2 = -1 - (-2)(((1+1)/(-16-(-2)) = -1 + 2(2/-14) = -1.2857

x2 = -1.2857

Second Iteration

x2 = -1.2857 , f(x2) =-5.277

x1 = 1, f(x1) =-16

X3 = X1 - f(x1)((x2-x1)/f(x2)-f(x1))

if we plug the value :

X3 =-2.410

Third Iteration

X3 =-2.410, f(X3) =-40.1933

x2 = -1.2857 , f(x2) =-5.277

X4 = X2 - f(x2)((x3-x2)/f(x3)-f(x2))

if we plug the value :

X4 =-1.1156

Fourth Iteration

X4 =-1.1156, F(X4) = -3.1513

X3 =-2.410, f(X3) =-40.1933

X5 = X3 - f(x3)((x4-x3)/f(x4)-f(x3))

if we plug the value :

X5 =-1.0055

So X5 = -1.0055 after fourth iteraitons