Roots Finding: Open Methods

Problem 6.4.

Determine the lowest positive root of f(x) = 7sin(x)e^-x - 1: (a) Graphically. (b) Using the Newton-Raphson method (three iterations, x₀ = 0.3). (c) Using the secant method (three iterations, x_-1 = 0.5 and x₀ = 0.4). (d) Using the modified secant method (five iterations, x₀ = 0.3, δ = 0.01).

Solution:

(a) Plot the graph using MATLAB. The commands are shown as below:

>> x = linspace(-1,2);

>> fx = 7*sin(x).*exp(-x)-1;

>> plot(x,fx),grid,xlabel('x'),ylabel('f(x)')

The graph generated is shown as above. As shown by the graph, there are two roots. The lowest positive root is between 0 and 0.5. By maximizing the graph, the root is estimated as 0.17. 

(b) Firstly, compute the first derivative of the function. (Recall Product Rule)

which can be substituted along with the original function to give

Starting with the initial guess of x₀ = 0.3, the iterative equation can be applied to compute:

Hence, after three iterations, the root estimate is 0.17018 with an approximate error of 0.453%.

(c) The iterative equation using secant method is 

Begin the iteration with x_-1 = 0.5 and x₀ = 0.4 yields

First iteration:

x_-1 = 0.5     f(0.5) = 1.0355

x₀ = 0.4      f(0.4) = 0.8272

x₁ = 0.0028 

The calculation can be continued to yield

Thus, after three iterations, the root estimate is 0.17899 with an approximate error of 21.9%.

(d) The iterative equation using modified secant method is 

First iteration:

x₀ = 0.3                                    f(0.3) = 0.5325

x₀ + δx₀ = 0.3030                  f(0.3030) = 0.5427

x₁ = 0.1437

The calculation can be continued to yield

Hence, after five iterations, the root estimate is 0.17018 with a zero percent approximate relative error.

 Additional: The real root of the equation can be found using fzero function in MATLAB.

>> format long

>> fx =@(x) 7*sin(x).*exp(-x)-1;

>> xr = fzero(fx,0.3)

xr =

   0.170179993753835

Discussion:

In conclusion, the Newton-Raphson method and modified secant method both give comparably accurate result with the true root value. In addition, modified secant method is able to attain the efficiency of Newton-Raphson method without having to compute derivatives.