diff --git a/hw1/assignment1.m b/hw1/assignment1.m index ce6de6a..cd28416 100644 --- a/hw1/assignment1.m +++ b/hw1/assignment1.m @@ -45,7 +45,7 @@ A = [1 -2 0 ; -2 1 -2; 0 -2 1] Z = zeros(9,9) B = ones(9,9) * 3 C = (eye(9) - 1) * -1 -D = diag([[1:5],[4:-1:1]]) +D = diag([1:5,4:-1:1]) E = repmat(transpose(1:9), 1, 5) %% Problem 3 @@ -87,7 +87,8 @@ function xmax = getxmax x = y; y = y * 2; end - xmax = typecast(bitor(typecast(x, 'uint64'), 0x000FFFFFFFFFFFFF), 'double'); + xmax = typecast(bitor(typecast(x, 'uint64'), 0x000FFFFFFFFFFFFF), ... + 'double'); end % geteps does not differ from eps, as getxmax does not differ from realmax. % However, getxmin returns the nearest positive floating point value to 0 diff --git a/hw1/hw1.aux b/hw1/hw1.aux new file mode 100644 index 0000000..f23e546 --- /dev/null +++ b/hw1/hw1.aux @@ -0,0 +1 @@ +\relax diff --git a/hw1/hw1.log b/hw1/hw1.log new file mode 100644 index 0000000..7ac493d --- /dev/null +++ b/hw1/hw1.log @@ -0,0 +1,165 @@ +This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) (preloaded format=pdflatex 2019.1.23) 3 MAR 2020 21:53 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**hw1.tex +(./hw1.tex +LaTeX2e <2018-04-01> patch level 2 +Babel <3.18> and hyphenation patterns for 84 language(s) loaded. +(/usr/local/texlive/2018/texmf-dist/tex/latex/base/article.cls +Document Class: article 2014/09/29 v1.4h Standard LaTeX document class +(/usr/local/texlive/2018/texmf-dist/tex/latex/base/size12.clo +File: size12.clo 2014/09/29 v1.4h Standard LaTeX file (size option) +) +\c@part=\count80 +\c@section=\count81 +\c@subsection=\count82 +\c@subsubsection=\count83 +\c@paragraph=\count84 +\c@subparagraph=\count85 +\c@figure=\count86 +\c@table=\count87 +\abovecaptionskip=\skip41 +\belowcaptionskip=\skip42 +\bibindent=\dimen102 +) +(/usr/local/texlive/2018/texmf-dist/tex/latex/base/inputenc.sty +Package: inputenc 2018/04/06 v1.3b Input encoding file +\inpenc@prehook=\toks14 +\inpenc@posthook=\toks15 +) +(/usr/local/texlive/2018/texmf-dist/tex/latex/geometry/geometry.sty +Package: geometry 2018/03/24 v5.7 Page Geometry + +(/usr/local/texlive/2018/texmf-dist/tex/latex/graphics/keyval.sty +Package: keyval 2014/10/28 v1.15 key=value parser (DPC) +\KV@toks@=\toks16 +) +(/usr/local/texlive/2018/texmf-dist/tex/generic/oberdiek/ifpdf.sty +Package: ifpdf 2017/03/15 v3.2 Provides the ifpdf switch +) +(/usr/local/texlive/2018/texmf-dist/tex/generic/oberdiek/ifvtex.sty +Package: ifvtex 2016/05/16 v1.6 Detect VTeX and its facilities (HO) +Package ifvtex Info: VTeX not detected. +) +(/usr/local/texlive/2018/texmf-dist/tex/generic/ifxetex/ifxetex.sty +Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional +) +\Gm@cnth=\count88 +\Gm@cntv=\count89 +\c@Gm@tempcnt=\count90 +\Gm@bindingoffset=\dimen103 +\Gm@wd@mp=\dimen104 +\Gm@odd@mp=\dimen105 +\Gm@even@mp=\dimen106 +\Gm@layoutwidth=\dimen107 +\Gm@layoutheight=\dimen108 +\Gm@layouthoffset=\dimen109 +\Gm@layoutvoffset=\dimen110 +\Gm@dimlist=\toks17 +) +(/usr/local/texlive/2018/texmf-dist/tex/latex/amsmath/amstext.sty +Package: amstext 2000/06/29 v2.01 AMS text + +(/usr/local/texlive/2018/texmf-dist/tex/latex/amsmath/amsgen.sty +File: amsgen.sty 1999/11/30 v2.0 generic functions +\@emptytoks=\toks18 +\ex@=\dimen111 +)) +(/usr/local/texlive/2018/texmf-dist/tex/latex/tools/array.sty +Package: array 2018/04/07 v2.4g Tabular extension package (FMi) +\col@sep=\dimen112 +\ar@mcellbox=\box26 +\extrarowheight=\dimen113 +\NC@list=\toks19 +\extratabsurround=\skip43 +\backup@length=\skip44 +\ar@cellbox=\box27 +) (./hw1.aux) +\openout1 = `hw1.aux'. + +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 13. +LaTeX Font Info: ... okay on input line 13. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 13. +LaTeX Font Info: ... okay on input line 13. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 13. +LaTeX Font Info: ... okay on input line 13. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 13. +LaTeX Font Info: ... okay on input line 13. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 13. +LaTeX Font Info: ... okay on input line 13. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 13. +LaTeX Font Info: ... okay on input line 13. + +*geometry* driver: auto-detecting +*geometry* detected driver: pdftex +*geometry* verbose mode - 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2^{-4}) * 2^7$ +\subsection*{Point e)} +The best approximation for 30.1 is 30, or $1.1110_2 * 2^4$, so $d_0=1$, $d_1=1$, $d_2=1$, $d_3=1$, $d_4=0$, and $e=4$. +This representation is not exact. The number 30.1 cannot be represented exactly in this format or any format with $e=2$. + +\section*{Question 6} +\subsection*{Point a)} +$$f(x) = \frac{1}{x} - \frac{1}{x + 1} = \frac{x + 1}{x (x + 1)} - \frac{x}{x (x + 1)} += \frac{x + 1 - x}{x (x + 1)} = \frac{1}{x (x + 1)} = g(x) $$ +\subsection*{Point b)} +$$ f(10_f) = \frac{1}{1.00 * 10^1} - \frac{1}{(1.00 * 10^1) + 1} = +(1.00 * 10^{-1}) - \frac{1}{1.10 * 10^1} \approx$$ +$$\approx (1.00 * 10^{-1}) - (0.91 * 10^{-1}) = 9.00 * 10^{-3} = 0.009 $$ +$$ g(10_f) = \frac{1}{(1.00 * 10^1) ((1.00 * 10^1) + 1)} = \frac{1}{1.00 * 10^1 * 1.10 * 10^1} \approx +(9.09 * 10^{-3}) = 0.00909$$ + +$$ f(100_f) = \frac{1}{1.00 * 10^2} - \frac{1}{(1.00 * 10^2) + 1} = +(1.00 * 10^{-2}) - \frac{1}{1.01 * 10^2} \approx$$ +$$\approx (1.00 * 10^{-2}) - (0.99 * 10^{-2}) = 1.00 * 10^{-4} = 0.0001 $$ +$$ g(100_f) = \frac{1}{(1.00 * 10^2) ((1.00 * 10^2) + 1)} = \frac{1}{1.00 * 10^2 * 1.01 * 10^2} \approx +(0.99 * 10^{-4}) = 0.000099$$ +$$ f(1000_f) = \frac{1}{1.00 * 10^3} - \frac{1}{(1.00 * 10^3) + 1} \approx +(1.00 * 10^{-3}) - \frac{1}{1.00 * 10^3} =$$ +$$= (1.00 * 10^{-3}) - (1.00 * 10^{-3}) = 0 $$ +$$ g(1000_f) = \frac{1}{(1.00 * 10^3) ((1.00 * 10^3) + 1)} \approx \frac{1}{1.00 * 10^3 * 1.00 * 10^3} = +(1.00 * 10^{-6}) = 0.000001$$ +\subsection*{Point c)} +\begin{tabular}{L | L | L | L} + \text{Value} & \text{Correct} & \text{Abs. Error} & \text{Rel. error} \\ + f(10_f) & 0.\overline{90} & 9.\overline{09} * 10^{-5} & 8.2645 * 10^{-7} \\ + g(10_f) & 0.\overline{90} & 9.\overline{09} * 10^{-7} & 8.26 * 10^{-9} \\ + f(100_f) & 9.\overline{9009} * 10^{-5} & 9.901 * 10^{-7} & 10^{-2} \\ + g(100_f) & 9.\overline{9009} * 10^{-5} & 9.9 * 10^{-9} & 10^{-4} \\ + f(1000_f) & 9.99 * 10^{-7} & 9.99 * 10^{-7} & 1 \\ + f(1000_f) & 9.99 * 10^{-7} & 10^{-9} & 10^{-3} \\ +\end{tabular} +\end{document}