Homework 1

Note

You are free to discuss these questions with your classmates and on our class slack, but you must write your own solutions, including your own source code.

All code should be uploaded to Brightspace along with any analytic derivations, notes, etc.

1. Consider the function:

   \[f(x) = \frac{1}{\sqrt{x^3 + 1} - 1}\]

   defined for \(x > 0\). When \(x\) is small, roundoff error from the subtraction will dominate the answer. We want to see if we can get a better result.

   • Show that by multiplying and dividing \(f(x)\) by \(\sqrt{x^3 + 1} + 1\), you get an analytically equivalent expression without any subtractions. Call this new expression \(g(x)\).

   • Using the fact that \(\sqrt{x^3 + 1} \approx 1 + x^3/2\) when \(|x| \ll 1\), show that you get:

     \[f(x) \approx \frac{2}{x^3}\]

   • Write a program to evaluate \(f(x)\), \(g(x)\), and the approximation to \(f(x)\) for \(|x| \ll 1\) for the values of \(x\): 1.e-4, 1.e-5, and 1.e-6.

     What do you observe?

2. In class, we derived a finite-difference approximation to the first-derivative that was second-order accurate:

   \[f^\prime |_i = \frac{f_{i+1} - f_{i-1}}{2\Delta x}\]

   This uses one point to the left and right of point \(i\). Assume that the spacing between the points, \(\Delta x\), is constant.

   Now, instead, let’s construct a one-sided difference for the first-derivative that is second-order accurate. To do this, consider the points \(f_i\), \(f_{i+1}\), and \(f_{i+2}\), and Taylor-expand them about \(x_i\) and solve for \(df/dx |_i\).

   Demonstrate that this converges second-order accurate by computing the derivative of \(f(x) = \sin(x)\) at \(x = 1\) for several values of \(\Delta x\).