On Birkhoff quadrature formulas. (English) Zbl 0599.41049
Es seien \(1=x_{1n}>x_{2n}>...>x_{n-1,n}>x_{n,n}=-1\) die Nullstellen des Polynoms \(\Pi_ n(x):=(1-x^ 2)P'_{n-1}(x),\) wobei \(P_{n-1}\) das Legendre-Polynom vom Grad n-1 ist. In Vereinfachung des Beweises in einer früheren Arbeit [Trans. Am. Math. Soc. 274, 797-808 (1982; Zbl 0525.41002)] beweist der Verf., daß die Quadraturformel
\[
\int^{+1}_{-1}f(x)dx=\frac{3}{n(2n-1)}[f(1)+f(-1)]+\frac{2(2n- 3)}{n(n-2)(2n-1)}\sum^{n-1}_{k=2}\frac{f(x_{kn})}{(P_{n- 1}(x_{kn}))^ 2}+
\]
\[ \frac{1}{n(n-1)(n-2)(2n-1)}\sum^{n- 1}_{k=2}\frac{(1-x^ 2_{kn})f''(x_{kn})}{(P_{n-1}(x_{kn}))^ 2} \] für alle Polynome vom Grad \(\leq 2n-1\) exakt ist.
\[ \frac{1}{n(n-1)(n-2)(2n-1)}\sum^{n- 1}_{k=2}\frac{(1-x^ 2_{kn})f''(x_{kn})}{(P_{n-1}(x_{kn}))^ 2} \] für alle Polynome vom Grad \(\leq 2n-1\) exakt ist.
Reviewer: K.-H.Hoffmann
MSC:
41A55 | Approximate quadratures |
41A05 | Interpolation in approximation theory |
65D30 | Numerical integration |
65D05 | Numerical interpolation |
Citations:
Zbl 0525.41002References:
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