Gaussian Quadrature Weights and Abscissae

This page is a tabulation of weights and abscissae for use in performing Legendre-Gauss quadrature integral approximation, which tries to solve the following function

by picking approximate values for n, w_i and x_i. While only defined for the interval [-1,1], this is actually a universal function, because we can convert the limits of integration for any interval [a,b] to the Legendre-Gauss interval [-1,1]:

The summation function is called the Legendre-Gauss quadrature rule because the abscissae x_i in the Gauss quadrature function for [-1,1] are defined as the roots of the Legendre polynomial for n:

with the weights w_i coming from the following function:

(The procedure is explained in more detail here, with a very accessible video lecture on the theory behind the Gauss quadrature rule here)

The tables provided here give the values for x_i and w_i for n=2 through n=64, with an internal decimal precision of 256, limited to 16 decimals due to floating point number limits in PHP. These tables were made using the numbers that are generated by the following series of Mathematica instructions:

    symboliclegendre[n_, x_] := Solve[LegendreP[n, x] == 0];
    legendreprime[n_, a_] := D[LegendreP[n, x], x] /. x -> a;
    weights[n_, x_] := 2/((1 - x^2) legendreprime[n, x]^2);

    (*how many terms should be generated*)
    h = 64;

    (* what numerical precision is desired? *)
    precision = 256;

    str = OpenWrite["lgvalues.txt"];
    Write[str, "abcissae"];
    Do[Write[str]; Write[str, n]; Write[str];
      nlist = symboliclegendre[n, x];
      xnlist = x /. nlist;
      Do[Write[str, N[Part[xnlist, i], precision]], {i, Length[xnlist]}];, {n, 2, h}];
    Write[str, "weights"];
    Do[Write[str]; Write[str, n]; Write[str];
      slist := symboliclegendre[n, x];
      xslist = x /. slist;
      Do[Write[str, N[weights[n, Part[xslist, i]], precision]], {i, Length[xslist]}];, {n, 2, h}];
    Close[str];

If you need higher precision than is presented on this web page you can either download the abscissae and weights files below, or you can run this program in Mathematica yourself, with higher precision and/or higher h values.

• abscissae (607KB, easily converted to not-PHP)
• weights (615KB, easily converted to not-PHP)

Weights and Abscissae Tables for n=2 to n=64