Lab 7: Numerical Analysis

Total Points: 30

Part 1. Basic Root-Finding Problems (8 pts)

Use both the bisection method and fixed-point iteration to find an approximation to \(\sqrt[3]{25}\) that is accurate to within \(10^{-7}\).

1. For the bisection method, use \(f(x)=x^3-25\) with the search interval \([1,3]\).

2. For the fixed-point iteration, use \(g(x)=\frac{2x^3+25}{3x^2}\) to identify the fixed point \(g(x)=x\) with some initial point (says, \(p_0=1\) or \(p_0=2\)).

# Your code goes here

Part 2. Non-quadratic Convergence of Newton’s Method (10 pts)

Recall that Newton’s method does not exhibit quadratic convergence when one of the following two cases occur:

• Case 1: The derivative of the function \(f\) is zero at the root \(p\) with \(f(p)=0\).

• Case 2: The second-order derivative of the function \(f\) does not exist at the root \(p\) with \(f(p)=0\).

We already demonstrate the exact linear convergence of Newton’s method under Case 1 in the Lecture 8 slides. Now, we will explore Case 2 by implementing Newton’s method on \(f(x)=x+x^{\frac{4}{3}}\). In addition, output the limiting point \(p^*\) of Newton’s method under the initialization \(p_0=3\) and the tolerance level \(\epsilon=10^{-13}\). Moreover, apply the Aitken’s \(\Delta^2\) method to the sequence produced by Newton’s method. Finally, plot the logarithm of the error \(|p_n-p^*|\) against the number of iterations for both the Newton’s method and Aitken’s \(\Delta^2\) method in the same plot. (Hint: You can adopt the code in Lecture 8 slides, but the ylim and legend location for the plot need adjusting.)

library(latex2exp)
# Your code goes here

Part 3. Numerical Differentiation and Integration (6+6 pts)

1. Compute the first-order derivative of \(f(x)=\log(x) + cos(x)-\sqrt{x}\) at \(x=2\) using the forward-difference, three-point endpoint formula, and five-point midpoint formula with \(h=0.005\). Then, also compute its second-order derivative at \(x=2\) using the second-order derivative midpoint formula. Output all these values. Finally, also output \(f'(2)\) and \(f''(2)\) that are computed by hand with R as your calculator.

# Your code goes here

2. Compute the integral \(\int_0^{\pi}\exp(2\cos(x)) dx\) using the trapezoidal rule, Simpson’s rule, composite Simpson’s rule with \(n=60\), and composite Trapezoidal rule with \(n=30\). Output all these values. In addition, output the integral value using the build-in function integrate().

# Your code goes here