Problem 1

What are the approximate absolute and relative errors in approximating π� by each of the following quantities?

1. 33
2. 3.143.14
3. 22/722/7

You can use either single or double precision for your computations. Please state your choice.

Problem 2

In either single or double precision, is the machine epsilon the smallest number ε� that can be stored on the computer, such that 1+ε≠11+�≠1? Justify your answer.

Problem 3

Write a program to compute the absolute and relative errors in Stirling’s approximation

for n=1,2,…,10�=1,2,…,10. Does the absolute error grow or shrink as n� increases? Does the relative error grow or shrink as n� increases? Is the result affected when using double precision instead of single precision?

Problem 4

Let x∈Rn�∈�� be an n�-dimensional vector. Show that ∥x∥2‖�‖2 and ∥x∥∞‖�‖∞ are equivalent.

Problem 5

Consider the image blurring example discussed in class, and suppose we denote the matrix of grayscale pixel values as I�. Modify the Python script blur.py to use the following operation instead:

Compute the blurred image after 2020 iterations of this modified scheme.