# Nodes and Weights of Gauss-Kronrod Calculator

## Calculates the nodes and weights of the Gauss-Kronrod quadrature.

 $\normal Gauss-Kronrod\ quadrature\\[10] {\large\int_{\small -1}^{\hspace{25}\small 1}}f(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}f(x_i)\\{\large\int_a^{\hspace{25}b}}f(x)dx\simeq{\large\frac{b-a}{2}\sum_{\small i=1}^{n}}w_{i}f({\large\frac{b-a}{2}}x_i+{\large\frac{b+a}{2}})\\\vspace{20}$
 display Kronrod xi wi + Gauss wi order n 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99  n=3,5,7,..   (odd) 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit
 $\normal Gaussian\ quadrature\\\hspace{20} {\large\int_{\small a}^{\hspace{25}\small b}}w(x)f(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}f(x_i)\\Gauss-Kronrod\ quadrature\\\hspace{30} interval(a,b):\hspace{20} [-1,\ 1]\\\hspace{30} w(x):\hspace{80} 1\\\hspace{30} polynomialsl:\hspace{10} P_n (x)\\$
Nodes and Weights of Gauss-Kronrod
 [1-8] /8 Disp-Num5103050100200
[1]  2018/09/21 19:54   Male / 60 years old level or over / A teacher / A researcher / Useful /
Purpose of use
Research
Bug report
The known symmetries of w[i]=w[n+1-i] and x[i]=-x[n+1-i] are not preserved. It is possible that this is a display bug rather than a calculation bug, since for the tested case (n=51, with 50digit output), the longer of the paired values rounded to the shorter. This impacts nodes, and both Kronrod and Gauss weights, although it is most visually obvious in the case of Gauss weights (as noted by a previous respondent). In the tested case, Gauss nodes and weights were correctly rounded to the last reported figure, as verified by comparison to much higher precision values calculated in Maple.
[2]  2018/07/07 00:14   Male / 20 years old level / An engineer / Very /
Purpose of use
Master''s dissertation research
[3]  2018/03/02 22:53   Male / 30 years old level / High-school/ University/ Grad student / - /
Bug report
Function which "Calculates the nodes and weights of the Gauss-Kronrod quadrature" with order n=81: Some of the Gauss weights w_i seem to be wrong (from too few digits to larger deviations). Deviations can also be seen from results of the function for Gauss Legendre quadratue nodes and weights on this website.
[4]  2017/10/13 05:22   Male / 40 years old level / A teacher / A researcher / Very /
Purpose of use
Scientific research.
[5]  2015/05/23 20:48   Male / 40 years old level / High-school/ University/ Grad student / Very /
Purpose of use
integration of sommerfield function
Comment/Request
increase the value of node to 1023!
I don't care about the time of computation....
[6]  2014/05/10 00:16   Male / 30 years old level / High-school/ University/ Grad student / Very /
Purpose of use
research
Comment/Request
-
[7]  2014/01/30 11:10   Male / 60 years old level or over / Self-employed people / Very /
Purpose of use
Integration of a function in a research project on air pollution.
[8]  2011/09/21 13:53   Male / 20 years old level / A teacher / A researcher / Very /
Purpose of use
Make a program in a calculator.
Comment/Request
Very Good.
Thanks.

Sending completion

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