From c2a50e8b7c7db53daa1c0906eaf32ab2f910f400 Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 8 Jul 2020 16:17:48 +0200 Subject: [PATCH] fix typos in num9 --- sose2020/num/uebungen/num9.pdf | Bin 268936 -> 268869 bytes sose2020/num/uebungen/num9.tex | 12 ++++++------ 2 files changed, 6 insertions(+), 6 deletions(-) diff --git a/sose2020/num/uebungen/num9.pdf b/sose2020/num/uebungen/num9.pdf index 3da293297381888d560c14679f1f0db7d0b22b34..af23f65422820671d3c5bddf01a7abf77883b450 100644 GIT binary patch delta 5674 zcmV+_7S-v9wGhR$5U?Z|12#D~mvO5BD1Y5tS(6*L5q{UN;8Th$Yhc`m9HmN*Wmg@8^1;)W5B~jt!JO%g)pMkd2*+n{t{;5& zJ)K>^S6|L3yt1>Oe8cOR6wt)L?SFdq;=w;gsMdx^xHCo(E@UA4bEJ8TqeEjBHD-=* zg}wS_bqhMv#2H~d6Nz+kRxtnJTa}+&Rqf(ofybTvyC`J2boyL{U`_Usf$- zOby}n*RLp-IT9#QOxKNm$n*gWktcOhO~ghG6w$Kz{-aC}cqi%xIR+`x5P$k8^ZPO;5NjIt{3kOYklxXNANs>d!a!Nm(b%F;+|D0k z>lqhB2nF4m^G!z$%Ul!5_$+Glic&ftHh$W08SMbtm`O(i>L-YMpB;@dksHlKjX1H) zHoC|%-FV?QO`7#8q9z~+?SI-;Q8ULRO*n6C1Ix)=c8NZ(MDq&PXK^a{i1L;$JG4~) z=xW$i4A(VMG?6H4t=v_P$&T`K=`l~Io7F-`l)bwaf+GLsYM8opPGNm1&ftyD z#mW0Z4~f!hM8z86F=1BTztUpllZ5+^GLvEwxPYB;1_nj+K${!_Z7o*sARNp3SUt(B zCFthSq7YopKVMieU#{1?3?1ANAyp?|Xgle`X3)pSCJQ@rlA`fO55t@;mK)3kb2Tq_ z`y78I(TtxMRDVzD@nCYuOeKfRhl|ROMeest4ykmBwo9QTPe>5WrhK#DcK&X`%{(g_ zm?e|9TS97Vnw4rSUoQ$6f3+Rdlmw*cBO{`->{}tIo`1U9 zuHK+`e}A{Zw_)!e-v5YCH=7)_OcSMa#|oId5vb5N%}7?jT|?3*IV*t?XZ5`pS-#P( zsb+bYZ-wdXFvmSXjH{dP%>Y0`r$L+fT$lU6r;F8fxOow{y1HJ5=jCCCH^DJ4!)nL( ztL?j2LdtdZ`)gkYfgP2raP!h5^vl(5jqNs<)qm?{HO|-D91i0U^U#*arx^kYNkMQz zrTz~I>FC5GW!AKFt+P-tDWf4;B(~jS zsLZ-E#I!NRN#ohhwsiuurvuL+M-nwPksylncUQ>ZRk>M($FJX40m^b;-oQ&X zuZpHNufTlm%gdfx#yU}OE-D9SV8OUs=zqqO!|tbrpTXW6Z|UfW@iqoa$AXxt4w+#?xV0^vh1udV2${o_`z6d>Ci^km1@Vt%djA5&ZJmo zl35i>xJIGUP(mn+p5Ihc`$vWXaJv{!aERzd#0nSziE=hU6eeL`GmXb|8vp)~#ecuF zb&t9~1nnZ0!Hiv*(*g!`gM1{PR0G-Pd1vF<#x*6BX6F;-Oe)anLXJxW!6CpZC@@uz zF_nW$YTFDHASkd)T+WmP7OK(8LM|($6DL~Zd2?ln19Yw=C5-FQ`P*dvWqykuM~ex# zHMeYzTOBjbV2Mhfj~lGLGeo!MVt7}?Qgc=fV?HT#?sMTrUncx7aerh zF>vTA;KF2i2_b@(4N)%+*AU2D zcvS~pWrGR>_?I?mA3318e1BXCqWy`1bxUKHl$*IFio0RMg7f!=#ozUpbBDsTQke@> zWRr$i$E7uf(5UWZ#TiWZvV5Iku>eh!_7qLqP*TbPcEHUMe}IS+tKWP7#^T_#li5$%APM-v=@Cq;Ks|wbnpMK!S$`9Qn>$6+iUg5; zIaWC4PeaTwIim3d1msx8b}!zXA6>(*|uX zR@=%LVi~0rsrusDu7aIP@BIC)3~8hKsL1ZJ@mrZy$sN&czE&RTwvOO_8YyZk66PQ4 z-ck!_r0IfZIT&kxoqy|rQ+rBXUY=YK@Ho9b0~V8ej8o_)54pgDE^aD&IF!`TiAg>s zaIG?rHpx6X?zPe1bQ2sQyJ3BH!#7xM-`;GYIclV1Rb~X0B_q6`FY02T3<(Vi^|@!;j_x+^s)=U zB6geXLM~`bHq(x;NJuj=4n(JKk+kR_F9n!}gF3p~Bx*XlRjh^7gF97>nR_~(l?#`R zY4Am7(I@>;4}a}O04Np|W_@>1`S(5~gqt>+4$p_p1>Vre+2)Hfn8Hl28#_D*x=Ps; zpo-W#5dcS)@eTp-!|-a^UQ{- z1|i;p1b_a4@wr<2JsaR?c>ZE}0EU=csDj)5?iQfT4yDMDeh9rh{X$x@R1$X!mDw5b*&E^se`ON%pNyz74y??z~UsksvV#2$h%9Xe4c<-?f-URG(9UcHmf&?6EMx{T^GMZ}U6L=_-YEJmWh=CqqPFy6t=w!PeXNqT;bbm76 zGGDu_mxt`j1J6^#WT+8&VO2>k3INy;(w5p*C9fFzWTAmJb-^likO?tjoK?` z0arl@<}u<^jn8w4in^0>3RgM2p(xAc*Zuh?j3c?oE`6{gQPIO0XV=-IHteJdm5PKE zTY3`OC*FI?U_B0u(SJPhJOpAIbkgULL-L@q8D6p``8m(0IJ7#4WYrl+=YJ{bo;-jP z0J;#-6RVG*PNsTmlYb&(`E$U{z-gB!Qb>7tEkL)`=t$%u~yZMq9Ibj3oOT~w_-wI~YfdF9wVqPjbOT_=Rl=ivN z_Wu!w#^P~qXZU{z(&Wx@J3ie%{9#1lUq}MFl5`a0azm&;Gauoq1TazQtKgYsgw_@A zd_W-T)Qu{jRR^^J0AT9P=9;^>baT9{jD+-I-ozgR6;Z3By@H>aWPiT&xWMCN+@2FG z^6ZWgmYDTP>i=t0O40PG#Ydy(j+lYKLZtYpf9^=08zXC9YzfwVB`G5KhF z`29k$`OW(0&CNA=gZFPjuS8+LaK(8@76mv3gW%5Kz5rA0-|ltt*vZeq*5ZG+!cy^~ zaH7%Vj~9hua2@(>?tdK+YN?z*SrpF94{+7jf8w={$rr*vSf1tuez{~G)$G-H5`{yWydy6S&p20M`RhocRsuKq4*q=#SqOlz+` z)^a~JR)2Hj2xb|ITXY6#MQ-m;7Z(N0QaIuD0QZ^kTKem{xqsryJGEngR4?GCxIpmw z&R+?o`_+sn#N ze%o97SZLJPYFM0BdD0bf5U%h2b?vW9f34ee3>^j%f39A3heWDmSo(7QGhBVYPI_4> zZw4gx4%ch{rhnJFrF{OaUk6Y9^@+be!)jtV|I4I#3`BY^$!Mb~_7Q_T`PS3>)L)z)d`AHaMK~ub$ zdbseKx2sktUNK3aC?am))T8k5qyHwT_8YI}p>43En17{a9Hi)tNy}~;vs(eX3nRt+ zPm4-b3z+E+`**!*zyN+4Y+xGj{hO$5Gz}4^A;zQ*(~w~r8nvpLhNg|p&FCRcuk?d$ zGS&H>t+x z4*uxo5Di2KhARPvD+0DF1Ao{mI59Rb zI5jsbFfu1FFefPrFHLV`L}7GgASgsSGB7eQFgGwbF*YzbH8;1Z0|ZzTQ7|?|I5Ig# zGeR{%GB!mxLpL=-I50vsFfuVPMMN+-Gd>_ZFg8UvGC4*wLN!4$Hbpo?H#I^yFhVvk zGBGekL@+orK3xhgOl59obZ8(qG&YyvqyZ^^CDdt9PEi;K@bjGes+4vuqO?a!p`^vG zQbLvwW-!B;VK8I)V3=VT%QwvUU~Hqv7P4ilkjSn=i>)&Dwd_-tkpKVm0#b&jb6%&a?v?$+>U4lq| z8W}^n6CdJ9IuVb}+GsX) zBksgQWcOAS^0tFLTfp88VBcD>KNTFXC#@j=9XOZ^4y^}=GeLm|DF;W6fTOd(v4P|p zIGztqRD;4YaPl}fwFaD?0?s&+3UKy+5-4ISehW&Tf^+rYd?mPW6O{5E7xO^b4RDD& zUtR#p7lJD>;HoES2G^d0>o38L7I3o`+`0m8?*SEyLFF`1l?Lt%2X`~TJ*Mh3)1j`! 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Also keine Operationen nötig. \item Auswertung von $L_i^{(n)}(\xi) = \prod_{j=0,j\neq i}^{n} \frac{(\xi-x_j)}{x_i - x_j} $: - $n+1$ Faktoren mit $3$ Operationen plus $n$ Multiplikationen für das Produkt. Gesamt: - $3(n+1) + n = 4n+4+n=5n+4=5(n+1)-1$. + $n$ Faktoren mit $3$ Operationen plus $n-1$ Multiplikationen für das Produkt. Gesamt: + $3n + n-1 = 4n-1$. - Auswertung von $p(\xi)$: $n+1$ Summanden mit $5(n+1)-1$ Operationen plus Multiplikation - mit $y_i$, ergibt $(n+1)\cdot (5(n+1)-1+1) = 5(n+1)^2$. Mit zusätzlich $n$ Additionen - für die Auswertung der Summe ergibt sich insgesamt $5(n+1)^2 + n = \mathcal{O}(n^2)$. + Auswertung von $p(\xi)$: $n+1$ Summanden mit $4n-1$ Operationen plus Multiplikation + mit $y_i$, ergibt $(n+1)\cdot (4n-1+1) = 4n^2$. Mit zusätzlich $n$ Additionen + für die Auswertung der Summe ergibt sich insgesamt $4n^2 + n = \mathcal{O}(n^2)$. \end{itemize} \item \begin{itemize} \item Berechnung der Koeffizienten erfordert die Lösung eines LGS mit unterer Dreicksmatrix. - Dies erfordert $\mathcal{O}^2$ Operationen. + Dies erfordert $\mathcal{O}(n^2)$ Operationen. \item Auswertung von $N_i(\xi) = \prod_{j=0}^{i-1} (\xi - x_j) $ erfordert $1$ Addition pro Faktor und insgesamt $i-1$ Multiplikationen für das Produkt, also insgesamt: $2i - 1$.