Mathematical Foundations for Signal Processing, Communications, and Networking

Exercise 5.8.10. On a computer using a decimal floating-point system with ϵ_mach = 10⁻⁸ and an exponent range of [−16, 15], what is the result of the following computations?

1.  10⁴ + 10⁻⁵

2.  1 − 10⁻⁴

3.  10¹⁰/10⁹

4.  10⁻¹⁰/10⁹

5.  10¹⁵ × 10⁻¹⁶

6.  10¹⁵ × 10¹⁶

Exercise 5.8.11. Which of the following two formulas to compute the midpoint of an interval [a, b] is preferable in floating-point arithmetic?

$m=\frac{a+b}{2},\hspace{1em}m=a+\frac{b-a}{2}$

Is it possible to compute m that lies outside the interval with either one of these formulas?

Exercise 5.8.12. The standard deviation of a set of real numbers can be computed in two mathematically equivalent ways

$\sigma ={\left[\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}\right]}^{1/2},\hspace{1em}\sigma ={\left[\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{x}_i^2-{\overline{x}}^2}\right]}^{1/2}$

where $\overline{x}={\displaystyle {\sum }_{i=1}^n{x}_i}$ is the mean. Which one ...