From 61762f724defa8aec4ba90634c45d4d00f3dc404 Mon Sep 17 00:00:00 2001 From: flavis Date: Fri, 1 May 2020 10:28:43 +0200 Subject: [PATCH] add num1 --- sose2020/num/uebungen/nulladdition.cpp | 20 ++++ sose2020/num/uebungen/num1.pdf | Bin 0 -> 108044 bytes sose2020/num/uebungen/num1.tex | 124 +++++++++++++++++++++++++ 3 files changed, 144 insertions(+) create mode 100644 sose2020/num/uebungen/nulladdition.cpp create mode 100644 sose2020/num/uebungen/num1.pdf create mode 100644 sose2020/num/uebungen/num1.tex diff --git a/sose2020/num/uebungen/nulladdition.cpp b/sose2020/num/uebungen/nulladdition.cpp new file mode 100644 index 0000000..b2b70bd --- /dev/null +++ b/sose2020/num/uebungen/nulladdition.cpp @@ -0,0 +1,20 @@ +#include +#include + +using namespace std; + +int main() { + cout << setprecision(17); + cout << setw(10); + cout << "Enter a float > "; + float x; + cin >> x; + x = x + 1; + cout << x << endl; + + cout << "Enter a double > "; + double y; + cin >> y; + y = y + 1; + cout << y << endl; +} diff --git a/sose2020/num/uebungen/num1.pdf b/sose2020/num/uebungen/num1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..18f47623fcfd624feb21a874d801434cd19cfe1e GIT binary patch literal 108044 zcmc$`byU=A_clH-z|f5hJ#5=xhVw4kJP zDM%wdzwzif&-0u!@9&Gx_g(A7#ab@?=f3v6hgsV7PshP#gk< zI9oap%E%B3Y1-VmL%fBX;%Dcd``*|;K4PktI?+MV(Ep+-VMoUCdqbxX=;F2QlexBpJ=gL@pmLiFBq2vWJ|XXB_7_|T zy<_4GIZT~20vcm`#ob-t<`s#P(+P>jc*h$zXB{8UuI-KNR9~HLWMhMMs`9<6zE4-A zI-p4v8m>P!VX}X`U}G{>ZxuQHqI%@SyBnO5{O~kY$97RHo%-z1dBUbb5PtdXW7j_J z4fB-?OP{Kvd|z!fvVAg;FHcBnyzsP}Om_PD@}alKyxPg3pV`L@5i%iv!H*evZ_eh| zb*>6c-#Gmm8PFri9uqwZv9;Ux^V&ORk<4g*nk_=CAa%UPB?#*?j|_aLwEuMkK7K_4 zwa?4fn}Ez_$yJR&1h1(ob5IMXQ_`!sN8|JrD+Yf-uvXyG+q1(sN%AreWe@K9Nc20u zC^8Is^PSpKGO@gm!jq~jA#qsvmcYH2{fSWIsH#k@oNpYr)}|rlY=S3!bSJY;Reus? z#Bi&Zhq1nLe4#mOIUz}lWJ)84Bw@&bg5qOnb}svdWq(4LTWska%y*dEr$z$ zB+bmVQ|s?0A3O0x;{?&hs-BS&QJ(T0Kj4EtKB+Pbb4&PSvwjb^+0QQJ#j;X8?}eyd z*zv^1%~jG(q?71J+AG0Bgco$kC1fl|ISa{c zM85VMQZ3LGWaTM3B;XJTNJnw-V#RRPD^YqlK16^vgb-}%*Nq4+Q>yi{Dfxn045ai3 znj_>DS3U%OFbGVt2z(k!Dy!h)ft;~+Y#yTvOnOIsVrnp}maPXeQ_O>ZA7shh-)YCbI!I@TaprC*xQmjvHFiFIFc zo3nO^3KzJ%G9X~Niht)-Vl-zfuL^s(6$59sj-rX+NHF|GDYCpzlb6w;ck$lcjUg1dZQ^T}~ zI$hC}+SHfVQ&Fs|+m)R1s)6#>lSMoO+HK*L%2_3$ChgugKHo{y#o3e$s6iSllD=ld z7Unwamh z(iX(HFdU~ou8AF$YE~O>o819ObabgWxz{d5~m|+m*2jyEKTy4cJYr z46YjCZzeMB;c?}Hc_f;Y-zl2mo0cpPR9~+99`b3IfxJ_Uc|DV?{nqhw9$NR7dI)nT zlQ&DjfOJ-hSO1z~_gV|b!qqK$`Q+q}k0RX@BK@!(+y@Oj9G}K_V!av|Mag9-*-?fL zXI+K8Z;@@Rbbk8sc~p%kh@M!yzk?Eh&zE2LoxAyComo3qw6AO3Rrgw6)Or`fE!yZFI$v!ryY-zdDsQ`sZQFCKFON;6J;~)$tyCQE z=>^TPYV+>wc>CMKY`w{FJ_E0^;#*HH?45w|Z7Z@X@MGOxF2w4jk}8pl`S|89_jzzY z7%gdXpD=$xE^10AHE5^-7foHqXd{x!qFMKB4JI_h`Q|yy@g|*y2X-|8LS(D29*_6S zL3g39YuAh^r9_bRq;CpTGkD>GkuOU#_)TbbCOK8bR;vVJIaEq1xV~gAbTX$gUg!FP zaAh>*XL)v~&5H9~r+Q&zb75yZlRdnz?LwJ{aR#$$&hl>E+l;3Dfcg2o`mrr@Y43ii zDSZoRlm~9bu}Wc`?DK4uYr$(KTLzWLv4AQ9CRVF*E7ohubWfa{?ne8C(b?0d@uouP zBWFJ9ZSj`-@+IE3T9{Yau7Qi?1M6wKSmnkh>2bGxO+t0=RbT7LP?^Twq_ubycst#{ zVz%^f=iP)iP%I5K|MeEjj@j2?RG;!$mC?kvv1ePh;e*QqD44RW-K?-w0_~(}Ck05Q zJjvi5T&M96P7BI7)q&*14CoyvPkx4Ao?pgh9E-!@3%I<`yQJuA4_hG6~2`?qVULA_NIAIrOTS)%O!^|h*k!D(*HX#Mp zhiH2rV%ObouEq|_MmE&O^zxBxrR^3E)xv5ZsW9e}XW3u#4C;Y2FITVRbAMklb{ZzlWhU9`4BeU+GUJQSPy-JA4&vw7}wIA1j~wOlc!iY znkd+X%XY(Xb?i-b^HnJ7Sa%0_?)Dh!)2A|a>ESrz$9&_k#^F|b#z2W)w?I)PNBrs1rnz^Xl5Upvgte6cY1jnx5oYciHOjI5Q(ZA zwGO~l162Fa;V#?Vy4mXI&iBlX$>_h)VOtK1Kd3c&M>usb3cs>KbzkY&bICTx0O2nv z|3O3KzCt)SU#s8w`?J)52>2Ve2dkrc`4=yg-Js6po_PfA)UP!aBFRm@;fWnlxy65% zEB1bM`FKb?717}66yS|7D@pD*c~eh^r%#M_74=wQ@y ziI2MSfpXlS_cJq0;Z3Bv^c~+`*PE#g(+AkRLXuK5?$w`&dkH&T1~+zB@gs+xXG<<+YwDhX$S~lZ@)pm9$zPr>U3;BV`-1 z!jXl~w#D&_r6b<<8;uheT5pgEIC$~k>`YNq%C_B0G%8`V(vcmnBDb@we>Ry*nG4og zYM08UVRdfUa;Bv1iIE--e{^}2%Y;3#I9BdPtApLAl=ojta(eQy^6I0c4vavl4O&VF z{pXtT#^QxCkJ`Q@aNf!C6ph&XpxV=sG%6F*<|`V3>qj5a?n#+cv?X)taoeuFTUkd^ znruv)w@k$02UW+8Bx9Mtw*74l{_<0iO!dbpX$o2vc>WT+9erM)xN!hoi=)+7$6y

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$x_1 = (0.3)_{10} \in \R$ + in der normierten Fließkommadarstellung + $\mathbb{F}(2,11,2)$ ergibt + \[ + (0.10011001100 \times 2^{1})_{2} + .\] + Zurück in $\mathbb{F}(10,r,1)$ ergibt + \[ + (0.2998046875 \times 10^{0})_{10} + .\] Das heißt für $r = 1$, $r = 2$ und $r = 3$ ergibt natürliche Rundung + jeweils $(0.3)_{10}$. Größere $r$ ergeben andere Ergebnisse. + \item Die größte positive Zahl in $\mathbb{F}(4, 6, 2)$ ergibt sich im Dezimalsystem dargestellt + als: + \[ + 3 \cdot (4^{-1} + 4^{-2} + 4^{-3} + 4^{-4} + 4^{-5} + 4^{-6}) \cdot 4^{3\cdot 4^{0} + 3\cdot4^{1}} + = 0.1073 \cdot 10^{10} + .\] Die kleinste negative Zahl in $\mathbb{F}(3, 7, 1)$ ergibt sich im Dezimalsystem dargestellt + als: + \[ + - 2 \cdot (3^{-1} + 3^{-2} + \ldots + 3^{-7}) \cdot 3^{2 \cdot 3^{0}} \approx -0.8996 \cdot 10^{1} + .\] Damit folgt für den maximalen Abstand zweier Zahlen aus $\mathbb{F}(4,6,2)$ und + $\mathbb{F}(3,7,1)$: + \[ + \max_{x_2, x_3} | x_2 - x_3 | \approx + 0.1073 \cdot 10^{10} + 0.8996 \cdot 10^{1} \approx 0.1073 \cdot 10^{10} + .\] + \item Die Aussage ist falsch. Gegenbeispiel: $\mathbb{F}(2, 2, 1)$. Die Menge der darstellbaren + Zahlen ist hier, analog zur Vorlesung: + \[ + \mathbb{F}(2,2,1) = \left\{- \frac{3}{2}, -1, -\frac{3}{4}, -\frac{1}{2}, -\frac{3}{8}, + -\frac{1}{4}, 0, \frac{1}{4}, \frac{3}{8}, \frac{1}{2}, \frac{3}{4}, 1, \frac{3}{2}\right\} + .\] Hier ist sofort zu sehen, dass beispielsweise für $x_4 = 1$ gilt: + \[ + \left| 1 - \frac{3}{4}\right| = \frac{1}{4} \neq \frac{1}{2} = \left| \frac{3}{2} - 1 \right| + .\] + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Die ersten zehn Folgenglieder in Dezimaldarstellung (nicht normiert): + \begin{table}[h!] + \centering + \label{tab:label} + \begin{tabular}{ccc} + $n$ & Aufrunden & gerades Runden \\ + $0$ & $2.46$ & $2.46$ \\ + $1$ & $2.47$ & $2.46$ \\ + $2$ & $2.48$ & $2.46$ \\ + $3$ & $2.49$ & $2.46$ \\ + $4$ & $2.50$ & $2.46$ \\ + $5$ & $2.51$ & $2.46$ \\ + $6$ & $2.52$ & $2.46$ \\ + $7$ & $2.53$ & $2.46$ \\ + $8$ & $2.54$ & $2.46$ \\ + $9$ & $2.55$ & $2.46$ \\ + $10$ & $2.56$ & $2.46$ \\ + \end{tabular} + \end{table} + + Beim Aufrunden ist deutlich zu sehen, dass, da nach jedem Rechenschritt die 4. Stelle + auf $5$ steht, jedes mal aufgerundet wird. Da zwei Rundungen pro Folgenglied stattfinden, + führt das zu einer Erhöhung um $2 \cdot 0.005 = 0.01$ pro Folgenglied. + + Beim geraden Runden wird zwar nach der ersten Operation aufgerundet, nach der zweiten jedoch nicht. + Deshalb bleibt das Ergebnis exakt. +\end{aufgabe} + +\begin{aufgabe} + Programm zum Test der Nulladdition: + \begin{lstlisting}[language=C++, title=nulladdition.cpp, captionpos=b] +#include +#include + +using namespace std; + +int main() { + // set output width and precision + cout << setprecision(17); + cout << setw(10); + + // float + cout << "Enter a float > "; + float x; + cin >> x; + x = x + 1; + cout << x << endl; + + // double + cout << "Enter a double > "; + double y; + cin >> y; + y = y + 1; + cout << y << endl; +}\end{lstlisting} + \begin{enumerate}[a)] + \item Bei Verwendung von \lstinline{float} muss $x$ als $5\cdot 10^{-8}$ gewählt werden, bei + Verwendung von \lstinline{double} muss $x$ als $10^{-16}$ gewählt werden, damit + exakt $1$ zurückgegeben wird. + \item Das ist bei weitem nicht die kleinste positive Zahl, die \lstinline{float} bzw. + \lstinline{double} darstellen können, da im Exponenten noch weit mehr Stellen zur Verfügung + stehen, um sehr viel kleinere positive Zahlen darstellen zu können. + + Allerdings kann dies hier nicht genutzt werden, da durch die Addition mit $1$ gerundet werden + muss. + \end{enumerate} +\end{aufgabe} + +\end{document}