17.1.1.1. Round-off Error

Try these first

1. 1.

   What is round-off error?

   Hint

   Think about how numbers might change when approximated.

2. 2.

   Give an example of round-off error.

   Hint

   What common constant can be rounded?

3. 3.

   What causes round-off error?

   • Finite precision in numbers
   • Infinite series
   • Data overflows

   Hint

   Think about how computers handle numbers.

4. 4.

   True or False: Round-off errors can accumulate over multiple computations.

   • True
   • False

   Hint

   What's the effect of repetitive approximations?

5. 5.

   Consider a simple numerical integration using Euler's method with a step size of h = 0.1. If the initial value is x = 1, compute the next value after one iteration. Now assume rounding to the nearest hundredth at each step. Analyze how the result is affected by round-off error.

   Hint

   Focus on the results of Euler's method and how errors may accumulate.

6. 6.

   If you take π and represent it with three different levels of precision (1 decimal, 2 decimals, and 5 decimals), how do you expect the accuracy of numerical computations to change? Illustrate how using more digits would minimize round-off error.

   Hint

   Try evaluating a mathematical operation using different representations of π.