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--git a/hw2/hw2.tex b/hw2/hw2.tex new file mode 100644 index 0000000..ff3faf0 --- /dev/null +++ b/hw2/hw2.tex @@ -0,0 +1,59 @@ +\documentclass[12pt,a4paper]{article} + +\usepackage[utf8]{inputenc} +\usepackage[margin=2cm]{geometry} +\usepackage{amstext} +\usepackage{array} + +\title{Howework 2 -- Introduction to Computational Science} +\author{Claudio Maggioni} + +\begin{document} +\maketitle +\section*{Question 1} +The solutions assume that the sign bit $1$ is negative and $0$ is positive. + +\subsection*{Point a} +\begin{itemize} +\item 13 is equal to $0 1010 0000 000 1011$; +\item 42.125 is equal to $0 0101 0001 000 1101$; +\item 0.8 is equal to $0 1 1001 1001 100 0011$. 0.78 is approximated to 0.7998046875; +\end{itemize} + +\subsection*{Point b} +$1 0110 1011 100 1101$ is $(-1) * (0.25+0.125+0.03125+0.0078125+0.00320625+0.001953125) *2^5$, +which is equal to $-13.4151$. + +\subsection*{Point c} +$x_{max}$ is $0 1111 1111 111 1111$, equal to $255.9375$. Since denormalized numbers do not +belong to this representation (since the exponent $0000$ cannot be used for valid numbers other +than 0) $x_{min}$ is $0 0000 0000 000 0001$, equal to $0.0078125$. + +\section*{Question 2} +\subsection*{Point a} +$$\frac{(x + \Delta x) + (y + \Delta y) - (x + y)}{x + y} = \frac{\Delta x}{x + y} + \frac{\Delta y}{x + y} = \frac{x}{x + y}\frac{\Delta x}{x} + +\frac{y}{x + y}\frac{\Delta y}{y}$$ + +\subsection*{Point b} +$$\frac{(x + \Delta x) - (y + \Delta y) - (x - y)}{x - y} = \frac{\Delta x}{x - y} - \frac{\Delta y}{x - y} = \frac{x}{x - y}\frac{\Delta x}{x} - +\frac{y}{x - y}\frac{\Delta y}{y}$$ + +\subsection*{Point c} +$$\frac{((x + \Delta x) (y + \Delta y)) - (x y)}{x y} = \frac{y \Delta x + x \Delta y + \Delta x \Delta y}{x y} \approx +\frac{y \Delta x + x \Delta y}{x y} = \frac{\Delta x}{y} + \frac{\Delta y}{x}$$ + +\subsection*{Point d} +$$\frac{((x + \Delta x) / (y + \Delta y)) - (x / y)}{x / y} += \frac{\frac{(x + \Delta x) y}{(y + \Delta y) y} - \frac{(y + \Delta y)x}{(y + \Delta y)y}}{x / y} += \frac{y \Delta x - x \Delta y}{x (y + \Delta y)} += \frac{y \Delta x}{x(y + \Delta y)} - \frac{\Delta y}{y + \Delta y} = $$ +$$ = \left(\frac{x(y + \Delta y)}{y \Delta x}\right)^{-1} - \left(\frac{y + \Delta y}{\Delta y}\right)^{-1} += \left(\frac{x}{\Delta x} - \frac{y \Delta x}{x \Delta y}\right)^{-1} - \left(\frac{y}{\Delta y} + 1\right)^{-1} = $$ +$$ = \left(\frac{x}{\Delta x} \left(1 - \frac{\Delta y}{y}\right)\right)^{-1} - \left(\frac{y}{\Delta y} + 1\right)^{-1} +\approx \left(\frac{x}{\Delta x}\right)^{-1} - \left(\frac{y}{\Delta y}\right)^{-1} += \frac{\Delta x}{x} - \frac{\Delta y}{y} +$$ + +\subsection*{Point e} +Division and multiplication may suffer from cancellation. +\end{document}