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zWNQn)MKJy`yd`Z+Z3*voM0eJ@qzIul<$qdss0jb_qdOxFSbH46cXI;^;K{-UPevvx HFZTZc&f8sL literal 0 HcmV?d00001 diff --git a/S2/CWR/CWRVL1.typ b/S2/CWR/CWRVL1.typ new file mode 100644 index 0000000..806b672 --- /dev/null +++ b/S2/CWR/CWRVL1.typ @@ -0,0 +1,107 @@ += Einleitung + +Dozent: marcus.muillr\@uni-goettingen.de + +== Pruefungsvorleitstung + +- 4 Testat-Aufgaben jeweils eine Woche +- git repo $-> $ Tutor +- Pass/Fail 1 Verbesserung pro Testat moeglich + +== Pruefung + +Projekt + Report eine Woche Zeit + +ca. 10 Seiten + +1. Periode 4-11 August +2. Periode 6-13 Oktober + +Programmiersprache C. Die Programme muessen lauffaehig im CIP Pool sein. + +Graphische Auswertung in Python. + +== Literatur + +Numerical Recipies Cambridge University Press + +== Ziele + +$"Probleme"-> "Algorithmen"-> "Programme"-> "Auswertung"$ + += Numerische Integration + +Das Problem ist ein einfaches Integral auszurechnen + +$ + I = integral_(a)^(b) f(x) d x. +$ + +Dafuer kann die *Mittelpunktsregel* verwendet werden + +$ + I = lim_(Delta x -> 0) sum_(i = 0)^(N) Delta x f(x_(i)) \ + x_0 = a, x_N = b\ + Delta x = (b-a) / (N) \ + x_i = a + i Delta x\ + "Mittelpunkt": x_(i) + (Delta x) / (2) l\ + I approx sum_(i)^(N) Delta x f(x_i + (Delta x) / (2) ) \. +$ + + +Oder die *Trapez-Regel* + +$ + f_("app") = f(x_i) + (f(x_(i+1) - f(x_i) ) / (Delta x) (x - x_i)\ + I_1 = integral_(x_(i+1) )^(x_i) f(x) d x approx integral_(x_(i+1) )^(x_i) f_("app")(x) d x = Delta x (f(x_(i+1) )+ f(x_(i))) / (2 ) . +$ + +== Simpson regel + +Quadratische Naeherung der Funktion auf dem intervall + +$ + I_(i) = integral_(x_(i+1) )^(x_(i) ) d x f(x) approx integral_(x_(i+1) )^(x_i) d x f_("app") (x) = (Delta x) / (6) [f(x_(i)) + 4 f(x_(i)) + f(x_(i)) ]. +$ + +== Fehlerabschaetzung + +Berechnung der Ordnungen der Fehler und Abschaetzung des Fehlers. + +#highlight[TODO: Literatur lesen und die Kapitel ausbessern] + += Berechnung von Nullstellen + ++ Intervallschachtelung + Pruefen von Intervallen, welche durch die Bedingung $f(a)f(b) < 0$ eine Nullstelle enthalten muessen. + Fuer den Algorithmus waelt man dann fuer die neue Intervallgrenze den Mittelpunkt zwischen $a$ und $b$, je nachdem ob die Bedingung fuer eine Nullstelle wieder erfuellt ist faehrt man dann mit dem einen oder dem anderen Intervall fort ++ Approximation durch eine lineare Funktion ($hat(f) = f_("app") $) + $ + hat(f)(x) = f(a) + (f(b) - f(a)) / (b-a) (hat(x)-a) = 0\ + hat(x) = a- (b-a) / (f(b) - f(a)) f(a) + $ + + Mit der Iterationsvorschrift + $ + x_(n+1) = x_(n-1) (f(x_(n)) - f(x_(n+1)) ) / (x_(n) -x_(n-1) ) f(x_(n-1) ), + $ + wobei die Abbruchbedingung $abs(f(x_(n))) < epsilon$ ist. ++ Newton-Raphson ist ein iteratives Verfahren. + + #highlight[TODO: understand and implement this] + += Auswahl von Algorithmen + ++ Rechenzeit/Effizienz ++ Robustheit/Stabilitaet ++ Genauigkeit + +== Bewertung der Algorithmen + ++ Intervallschachtelung\ + Robustheit ++\ + Effizient + +== Uebung + +$f(z) = z^3 -1 = 0, quad z in CC$ mit Newton.