Estimate a solution to the following nonlinear equation usin

Estimate a solution to the following nonlinear equation using two iterations of the method of false position. Choose your initial guess(es) from the following values and explain your choice: 0.25, 1.0, 2.5.

Solution

f(x) = xln(x) - sqrt(x)

f(.25) = .25*ln(.25) - sqrt(.25) = -0.847

f(1) = -1

f(2.5) = 2.5*ln(2.5) - sqrt(2.5) = 0.709

now from the above three results we can see that

f(.25)*f(1) = will be a positive number

f(.25)*f(2.5) = will be a negative number

f(1)*f(2.5) = will be a negative number

we want the lower bound value Xl and the upper bound value Xu such that

f(Xl)*f(Xu) < 0

so we can choose Xl = .25 and Xu = 2.5

or Xl = 1 and Xu = 2.5

lets go with

The next predicted/improved root Xr for the equation xln(x) - sqrt(x) can be computed as the midpoint between Xl and Xu as :

1st iteration :

the estimated root Xr is :

Xr = [Xu*f(Xl) - Xl*f(Xu)]/[f(Xl) - f(Xu)]

Xr = [2.5*f(1) - 1*f(2.5)]/[f(1)-f(2.5)]

Xr = [2.5*(-1) - 1*.709]/[(-1)- .709] = 1.877

Xr = 1.877

f(Xr = 1.877) = x*ln(x)-sqrtx = 1.877*ln(1.877) - sqrt(1.877) = - .188

now we\'ll check

f(Xl)f(Xr) = -1*1.877 = -1.877 < 0

=> the root lies between Xl and Xr

and for the next iteration

=> Xl = -1 and Xu = Xr = 1.877

2nd iteration :

The estimate of the root is Xr

Xr = [Xu*f(Xl) - Xl*f(Xu)]/[f(Xl) - f(Xu)]

Xr = [1.877*f(1) - 1*f(1.877]/[f(1)-f(1.877)]

Xr = [(1.877*-1) - (1*(-.188))][-1 - (-.188)] = .848

=> Xr = .848

hence the next estimatd root is

Xr = .848