QUIZ 1

1. What are the advantages and disadvantages of numerical analysis?
   • possible to solve problems that may not be solvable by hand
   • possible to solve problems (that you may have solved before) in a different way
   • only need four operations (add, substract, multiply, division) and Comparasion
   • analytical answer is not the true (exact) answer that it is always an approximation
   • accuracy and precision concepts are important
2. Describe truncation and round-off errors. Give example.
   • Truncation Error: i.e., approximate $e^x$ by the cubic power
     
     $$P_3(x)=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!};      e^x=P_3(x)+\sum_{n=4}^\infty \frac{x^n}{n!}$$
     
     
     approximating $e^x$ with the cubic gives an inexact answer. The error is due to truncating the series.
   • Round-off Error: All computing devices represents numbers, except for integers and some fractions, with some imprecision
     Floating-point numbers of fixed word length; the true values are usually not expressed exactly by such representations
3. Describe a hypothetical numbering system with six bit representation?
   • Say we have six bit representation (not single, double)
     • $1 bit\rightarrow sign$
     • $3(+1) bits\rightarrow mantissa$
     • $2 bits\rightarrow exponent$
     
     For positive range $\frac{9}{32}\longleftrightarrow \frac{15}{4}$
     For negative range $\frac{-15}{4}\longleftrightarrow \frac{-9}{32}$; even discontinuity at point zero since it is not in the ranges.
     Very simple computer arithmetic system $\Rightarrow$ the gaps between stored values are very apparent. Many values can not be stored exactly.