Approximate Calculation of Integrals by V. I. Krylov

By V. I. Krylov

A systematic advent to the primary rules and result of the modern idea of approximate integration, this quantity techniques its topic from the perspective of practical research. additionally, it bargains an invaluable reference for sensible computations. Its fundamental concentration lies within the challenge of approximate integration of services of a unmarried variable, instead of the more challenging challenge of approximate integration of features of multiple variable.
The three-part therapy starts with suggestions and theorems encountered within the thought of quadrature. the second one half is dedicated to the matter of calculation of convinced integrals. This part considers 3 simple themes: the idea of the development of mechanical quadrature formulation for sufficiently soft integrand features, the matter of accelerating the precision of quadratures, and the convergence of the quadrature technique. the ultimate half explores tools for the calculation of indefinite integrals, and the textual content concludes with necessary appendixes.

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F1 h a x1 f2 h f3 h x2 h x3 b Abb. 7. , xn−1/2 = b − h/2 ist. 8 dargestellt. Dabei haben wir die alte Bezeichnung wieder beibehalten. Die Teilintervalle f 1/2 a h f 3/2 x1 h f 5/2 x2 h b=x3 Abb. 8. MacLaurin-Formeln sind durch [xi−1 , xi ] f¨ ur i = 1, ... , n gegeben. Zum Exaktheitsgrad all dieser Formeln l¨ asst sich feststellen: Eine Newton-Cotes-Formel mit k ¨ aquidistanten St¨ utzstellen hat • den Exaktheitsgrad k , falls k ungerade • den Exaktheitsgrad k − 1, falls k gerade ist, das heißt, Polynome bis zum k -ten bzw.

Ist die gew¨ unschte Genauigkeit nicht erreicht, kommt eine neue Zeile dazu bis |Tj ,J − Tj +1,J | < ε, so dass Tj +1,j +1 als N¨ aherungswert akzeptiert wird. Das Verfahren sucht sich so bei vorgegebener Genauigkeit die Ordnung selbst. 2. Wenn man zu feineren Unterteilungen u ¨bergeht, steigt der Rechenaufwand des Romberg-Verfahrens relativ stark an, wie die Zusammenstellung der St¨ utzordinaten zeigt: Schrittw. O) h5 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ +8 = 17 (10. NN ): T i, n, m, N , p Integer: ¨ Ubernehme (a, b, N ) T1,1 = (b − a) f (a) + f (b) 2 n=2 Solange n ≤ N h= b−a 2n−1 q =0 i =1 Solange i ≤ 2n−1 − 1 q = q + f (a + ih) i =i +2 Tn,1 = Tn−1,1 + hq 2 p=4 m=1 Solange m ≤ n Tn,m = Tn,m−1 + Tn,m−1 − Tn−1,m−1 p−1 p = 4p m =m +1 n =n +1 Gib TN ,N zur¨ uck.

6 (Simpson-Regel, mitMaple-Worksheet). 5) vom letzten Abschnitt auf: 3 (x 3 − 2x ) dx . 25 3 und stimmt genau mit dem exakten Wert u urlich ¨berein. Ist dies Zufall? Nat¨ nicht! 27) zeigt, dass dort die vierte Ableitung des Integranden steht. Diese ist bei dem Polynom dritten Grades Null und somit liefert die N¨ aherungsformel den exakten Wert. 6 diskutiert, liefert die Simpson-Regel f¨ ur jedes Polynom vom Grade kleiner oder gleich drei den exakten Wert. Dies kann man als ein Maß der G¨ ute der Quadraturformel festhalten: Die Simpson-Regel integriert Polynome bis Grad 3 exakt.

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