Read More
Date: 14-12-2021
![]()
Date: 8-12-2021
![]()
Date: 7-12-2021
![]() |
A Gaussian quadrature-like formula for numerical estimation of integrals. It uses weighting function in the interval
and forces all the weights to be equal. The general formula is
![]() |
(1) |
where the abscissas are found by taking terms up to
in the Maclaurin series of
![]() |
(2) |
and then defining
![]() |
(3) |
The roots of then give the abscissas. The first few values are
![]() |
![]() |
![]() |
(4) |
![]() |
![]() |
![]() |
(5) |
![]() |
![]() |
![]() |
(6) |
![]() |
![]() |
![]() |
(7) |
![]() |
![]() |
![]() |
(8) |
![]() |
![]() |
![]() |
(9) |
![]() |
![]() |
![]() |
(10) |
![]() |
![]() |
![]() |
(11) |
![]() |
![]() |
![]() |
(12) |
![]() |
![]() |
![]() |
(13) |
(OEIS A002680 and A101270).
Because the roots are all real for and
only (Hildebrand 1956), these are the only permissible orders for Chebyshev quadrature. The error term is
![]() |
(14) |
where
![]() |
(15) |
The first few values of are 2/3, 8/45, 1/15, 32/945, 13/756, and 16/1575 (Hildebrand 1956). Beyer (1987) gives abscissas up to
and Hildebrand (1956) up to
.
![]() |
![]() |
2 | ![]() |
3 | 0 |
![]() |
|
4 | ![]() |
![]() |
|
5 | 0 |
![]() |
|
![]() |
|
6 | ![]() |
![]() |
|
![]() |
|
7 | 0 |
![]() |
|
![]() |
|
![]() |
|
9 | 0 |
![]() |
|
![]() |
|
![]() |
|
![]() |
The abscissas and weights can be computed analytically for small .
![]() |
![]() |
2 | ![]() |
3 | 0 |
![]() |
|
4 | ![]() |
![]() |
|
5 | 0 |
![]() |
|
![]() |
REFERENCES:
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 466, 1987.
Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 345-351, 1956.
Salzer, H. E. "Tables for Facilitating the Use of Chebyshev's Quadrature Formula." J. Math. Phys. 26, 191-194, 1947.
Sloane, N. J. A. Sequences A002680/M2261 and A101270 in "The On-Line Encyclopedia of Integer Sequences."
|
|
منها نحت القوام.. ازدياد إقبال الرجال على عمليات التجميل
|
|
|
|
|
دراسة: الذكاء الاصطناعي يتفوق على البشر في مراقبة القلب
|
|
|
|
|
هيئة الصحة والتعليم الطبي في العتبة الحسينية تحقق تقدما بارزا في تدريب الكوادر الطبية في العراق
|
|
|