# Nodes and Weights of Gauss-Legendre Calculator

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

 $\normal{\large\int_{\small -1}^{\hspace{25}\small 1}}f(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}f(x_i)\\\ nodes\hspace{30} x_i:\hspace{10} P_n(x_i)=0\\\ weights\hspace{15} w_i={\large\frac{2(1-x_i^2)}{[nP_{n-1}(x_i)]^2}}={\large \frac{2}{[P_n^1(x_i)]^2}\\$
 order n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 n=2,3,4,..,100 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit
 $\normal Gaussian\ quadrature\\\hspace{20} {\large\int_{\small a}^{\hspace{25}\small b}}w(x)f(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}f(x_i)\\Gauss-Legendre\ quadrature\\\hspace{30} interval(a,b):\hspace{20} [-1,\ 1]\\\hspace{30} w(x):\hspace{80} 1\\\hspace{30} polynomialsl:\hspace{10} P_n (x)\\$
Nodes and Weights of Gauss-Legendre
 [1-9] /9 Disp-Num5103050100200
[1]  2018/10/17 17:57   Male / Under 20 years old / High-school/ University/ Grad student / Useful /
Purpose of use
Coding
[2]  2018/09/30 16:58   Female / 30 years old level / High-school/ University/ Grad student / A little /
Purpose of use
can you calculate me gauss points and weights for n = 2000
Comment/Request
can you calculate me gauss points and weights for n = 2000
[3]  2017/05/23 22:01   Male / 60 years old level or over / A retired people / Very /
Purpose of use
transform 2-D singular integrals to singularity-canceled form, use Cartesian product of Gauss-Legendre rules for high precision integration for solving integral equations
Comment/Request
1) Sign is incorrect in last equality, eq. (3) above. Weights would be negative!
2) Programmed the algorithm summarized above and found, for 34 sig. figs. at least, Newton's method more efficient than Halley's method! Not sure why, and perhaps not always true. .
from Keisan
We modified the last equality, eq.(3).
[4]  2015/04/20 22:32   Male / 30 years old level / High-school/ University/ Grad student / Useful /
Purpose of use
Calculate spherical harmonic decomposition harmonics of 3D Gaussian
[5]  2014/04/24 06:44   Male / 50 years old level / Others / Very /
Comment/Request
Very, Very, good work. Thank you.
[6]  2014/03/29 00:37   Female / 20 years old level / High-school/ University/ Grad student / A little /
Purpose of use
Helping ferrence for finding errors in my own Fortran written Gauss-Legendre code.
[7]  2013/12/26 00:06   Male / 20 years old level / High-school/ University/ Grad student / A little /
Purpose of use
Use in programmation with Fortran for a Master's degree.
[8]  2011/10/07 06:24   Male / 20 years old level / A teacher / A researcher / Very /
Purpose of use
Comment/Request
Prety impressed, perfect job.
[9]  2011/05/30 07:05   Male / 60 years old level / A teacher / A researcher / Very /
Purpose of use
Solving an initial value problem for a symmetric hyperbolic system of PDE-s
Comment/Request

Sending completion

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