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zUj9RQYm9LnXXYA$z9vuqZ0M6A$;LJ*7U3DmB)B-9$$i2Z{xQocHg`4j=x+MbD{+#Q zH+|mi?nxQzN4jq?@uHos@_SDhk-Heb!a8j}9L+3QgxoW7k{>|9iZRIl0gW*JH#G7; zPvYu|CZJe+K-Y(j!UjEsN&a0+s;Gchs>vx+jXi;8ftGPAJ=iSQBp-@E+I zB5h)8=4?*D$o{`{3IzWF2iDHNFoj;!`WFicn;6*{oBRh`{4Y_2`v&e5CmNsvLdd>X zjE(KXoO$ybPJl28L6W2inm({9VIz2g_9+rK;-mxh+l<~#4i&(ws6P%U1WG>Ckbtv@ zT=bA|w1}Sc_*`2|MtCH|{?L^Tg*PHPW8t3<@yP#gEEg+ozv!6a(cN>RYtD)8rUKW% zDIDjw-Zk3DyY}Dqe(Tj*g*PLvc2p;Xs@qksW_!7I?W`xSQm-oUb8Gk8=&!x?{$}l<{$}} + +\title{Midterm -- Introduction to Computational Science} +\author{Claudio Maggioni} + +\begin{document} +\maketitle +\section*{Question 1} +\subsection*{Point a)} +$$7.125_{10} = (1 + 2^{-1} + 2^{-2} + 2^{-5}) * {2^2}_{10} = 0 | 1100 1000 0000 | 110_F$$ +$$0.8_{10} = (1 + 2^{-1} + 2^{-4} + 2^{-5}+ 2^{-8} + 2^{-9} + 2^{-12}) * {2^{-1}}_{10} \approx 0 | 100110011001|011_F$$ +$$0.046875_{10} = (2^{-2} + 2^{-3}) * {2^{-3}}_{10} = 0 | 0011 0000 0000|000_F$$ + +For the last conversion, we assume that the denormalized mode of this floating point representation implicitly includes an exponent of $-3$ (this makes the first bit in the mantissa of a denormalized number weigh $2^{-3}$). This behaviour is akin to the denormalized implementation of IEEE 754 floating point numbers. + +\subsection*{Point b)} +$$1|011010111000|110_F = -(1+2^{-2} +2^{-3} +2^{-5} +2^{-7} +2^{-8} + 2^{-9}) * {2^{2}}_{10} \approx +-5.6796875$$ +$$1|101010101010|010_F = -(1+2^{-1} +2^{-3} +2^{-5} +2^{-7} +2^{-9} + 2^{-11}) * {2^{-2}}_{10} \approx +-0.4166259766$$ + +\subsection*{Point c)} +$$1|0000 0000 0000|001_F = 2^{-3}_{10} = 0.125_{10}$$ + +\subsection*{Point d)} +$$1|1111 1111 1111|111_F = (1 + 2^{-1}+ 2^{-2} + 2^{-3} +2^{-4} + 2^{-5} +2^{-6} + 2^{-7} + 2^{-8} + 2^{-9}+ 2^{-10}+ 2^{-11}+ 2^{-12}) * {2^3}_{10} = $$ +$$ = 15.998046875_{10}$$ + +\subsection*{Point e)} +With 12 independent binary choices (bits to flip), there are $2^{12}$ different denormalized numbers in this encoding. + +\subsection*{Point f)} +With 12 independent binary choices (bits to flip) and 3 extra bits for the exponent, there are $2^{15}-1$ different denormalized numbers in this encoding. We subtract 1 in order to be consistent with the assumption in point \textit{a)}, since $0.125_{10}$ would be representable both as $1|0000 0000 0000|001_F$ and +as $1|1000 0000 0000|000_F$ + +\section*{Question 2} + +\subsection*{Point a)} +$$ \sqrt[3]{1 + x} - 1 = (\sqrt[3]{1 + x} - 1) \cdot + \frac{ \sqrt[3]{1 + x} + 1}{ \sqrt[3]{1 + x} + 1} = \frac{\sqrt[3]{(1 + x)^2} - 1}{\sqrt[3]{1 + x} + 1} = +\frac{\sqrt[3]{(1 + x)^2} - 1}{\sqrt[3]{1 + x} + 1} \cdot \frac{\sqrt[3]{(1 + x)^2} + 1}{\sqrt[3]{(1 + x)^2} + 1} = $$ +$$\frac{(1 + x)\sqrt[3]{1 + x} - 1}{(\sqrt[3]{1 + x} + 1) \cdot (\sqrt[3]{(1 + x)^2} + 1)} $$ +\subsection*{Point b)} +$$ \frac{1 - cos(x)}{sin(x)} = \frac{sin^2(x)cos^2(x) - cos(x)}{sin(x)} \cdot \frac{sin(x)}{cos(x)} \cdot \frac{cos(x)}{sin(x)} = (sin^2(x)cos(x) - 1)\cdot\frac{cos(x)}{sin(x)}$$ +\subsection*{Point c)} +$$ \frac{1}{1-\sqrt{x^2-1}} = \frac{x}{x^2-\sqrt{x^4-x^2}}$$ +\subsection*{Point d)} +$$ x^3\cdot\left(\frac{x}{x^2-1}-\frac{1}{x}\right) = x^3\cdot\left(\frac{x^2-x^2+1}{x^3-x}\right) = +\frac{x^2}{x^2-1}$$ +\subsection*{Point e)} +$$ \frac{1}{x} - \frac{1}{x+1} = \frac{x + 1 - x}{x^2 + x} = \frac{1}{x^2 + x}$$ + +\section*{Question 3} +\subsection*{Point a)} +First we point out that: +$$ \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \stackrel{k := -h}{=} \lim_{k \to 0} \frac{f(x - (-k)) - f(x)}{-k} = $$ +$$ = \lim_{k \to 0} \frac{f(x) - f(x + k)}{k} = - \lim_{h \to 0} \frac{f(x) - f(x - h)}{h} $$ + +Then, we find an equivalent way to represent $f'(x)$: +$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = 2 \cdot \lim_{h \to 0} \frac{f(x + h) - f(x)}{2h} =$$ +$$ = \lim_{h \to 0} \frac{f(x + h) - f(x)}{2h} + \left(- \lim_{h \to 0} \frac{f(x) - f(x - h)}{2h}\right) = \lim_{h \to 0} \frac{f(x + h) - f(x - h)}{2h} $$ + +Then we consider the \textit{epsilon-delta} definition of limits for the last limit: +$$ \forall \epsilon > 0 \exists \delta > 0 | \forall h > 0, \text{if } 0 < h < \delta \Rightarrow$$ +$$\left| \frac{f(x + h) - f(x - h)}{2h} - f'(x)\right| = \left|f'(x) - \frac{f(x + h) - f(x - h)}{2h}\right| < \epsilon$$ + +I GIVE UP :( + +\section*{Question 4} +\subsection*{Point a)} +\[ +A_1 = + \begin{bmatrix} + 1 & 1 & 1 & 1 & 1 \\ + 2 & 4 & 4 & 4 & 4 \\ + 3 & 7 & 10 & 10 & 10 \\ + 4 & 10 & 16 & 20 & 20 \\ + 5 & 13 & 22 & 30 & 35 \\ + \end{bmatrix}, + b = + \begin{bmatrix} + 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ + \end{bmatrix} +\] +\[ +l_1 = + \begin{bmatrix} + 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ + \end{bmatrix}, +u_1 = + \begin{bmatrix} + 1 & 1 & 1 & 1 & 1 \\ + \end{bmatrix}, + A_2 + \begin{bmatrix} + 0 & 0 & 0 & 0 & 0 \\ + 0 & 2 & 2 & 2 & 2 \\ + 0 & 4 & 7 & 7 & 7 \\ + 0 & 6 & 12 & 16 & 16 \\ + 0 & 8 & 17 & 25 & 30 \\ + \end{bmatrix} +\] +\[ +l_2 = + \begin{bmatrix} + 0 \\ 1 \\ 2 \\ 3 \\ 4 \\ + \end{bmatrix}, +u_2 = + \begin{bmatrix} + 0 & 2 & 2 & 2 & 2 \\ + \end{bmatrix}, + A_3 + \begin{bmatrix} + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 3 & 3 & 3 \\ + 0 & 0 & 6 & 10 & 10 \\ + 0 & 0 & 9 & 17 & 22 \\ + \end{bmatrix} +\] +\[ +l_3 = + \begin{bmatrix} + 0 \\ 0 \\ 1 \\ 2 \\ 3 \\ + \end{bmatrix}, +u_3 = + \begin{bmatrix} + 0 & 0 & 3 & 3 & 3 \\ + \end{bmatrix}, + A_4 + \begin{bmatrix} + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 4 & 4 \\ + 0 & 0 & 0 & 8 & 13 \\ + \end{bmatrix} +\] +\[ +l_4 = + \begin{bmatrix} + 0 \\ 0 \\ 0 \\ 1 \\ 2 \\ + \end{bmatrix}, +u_4 = + \begin{bmatrix} + 0 & 0 & 0 & 4 & 4 \\ + \end{bmatrix}, + A_5 + \begin{bmatrix} + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 5 \\ + \end{bmatrix} +\] +\[ +\l_5 = + \begin{bmatrix} + 0 \\ 0 \\ 0 \\ 0 \\ 1 \\ + \end{bmatrix}, +u_5 = + \begin{bmatrix} + 0 & 0 & 0 & 0 & 5 \\ + \end{bmatrix}, + L = + \begin{bmatrix} + 1 & 0 & 0 & 0 & 0 \\ + 2 & 1 & 0 & 0 & 0 \\ + 3 & 2 & 1 & 0 & 0 \\ + 4 & 3 & 2 & 1 & 0 \\ + 5 & 4 & 3 & 2 & 1 \\ + \end{bmatrix}, + U = + \begin{bmatrix} + 1 & 1 & 1 & 1 & 1 \\ + 0 & 2 & 2 & 2 & 2 \\ + 0 & 0 & 3 & 3 & 3 \\ + 0 & 0 & 0 & 4 & 4 \\ + 0 & 0 & 0 & 0 & 5 \\ + \end{bmatrix} + \] +\subsection*{Point b)} +\[ +\l_1 = + \begin{bmatrix} + 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ + \end{bmatrix}, +e^{T}_1 = + \begin{bmatrix} + 1 & 0 & 0 & 0 & 0 \\ + \end{bmatrix}, + L_1 = + \begin{bmatrix} + 0 & 0 & 0 & 0 & 0 \\ + -2 & 1 & 0 & 0 & 0 \\ + -3 & 0 & 1 & 0 & 0 \\ + -4 & 0 & 0 & 1 & 0 \\ + -5 & 0 & 0 & 0 & 1 \\ + \end{bmatrix} + \] + \[ +\l_2 = + \begin{bmatrix} + 0 \\ 1 \\ 2 \\ 3 \\ 4 \\ + \end{bmatrix}, +e^{T}_2 = + \begin{bmatrix} + 0 & 1 & 0 & 0 & 0 \\ + \end{bmatrix}, + L_2 = + \begin{bmatrix} + 1 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & -2 & 1 & 0 & 0 \\ + 0 & -3 & 0 & 1 & 0 \\ + 0 & -4 & 0 & 0 & 1 \\ + \end{bmatrix} + \] + \[ +\l_3 = + \begin{bmatrix} + 0 \\ 0 \\ 1 \\ 2 \\ 3 \\ + \end{bmatrix}, +e^{T}_3 = + \begin{bmatrix} + 0 & 0 & 1 & 0 & 0 \\ + \end{bmatrix}, + L_3 = + \begin{bmatrix} + 1 & 0 & 0 & 0 & 0 \\ + 0 & 1 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & -2 & 1 & 0 \\ + 0 & 0 & -3 & 0 & 1 \\ + \end{bmatrix} + \] + \[ +\l_4 = + \begin{bmatrix} + 0 \\ 0 \\ 0 \\ 1 \\ 2 \\ + \end{bmatrix}, +e^{T}_4 = + \begin{bmatrix} + 0 & 0 & 0 & 1 & 0 \\ + \end{bmatrix}, + L_4 = + \begin{bmatrix} + 1 & 0 & 0 & 0 & 0 \\ + 0 & 1 & 0 & 0 & 0 \\ + 0 & 0 & 1 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & -2 & 1 \\ + \end{bmatrix} +\] +\subsection{Point c)} + \end{document}