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In the book of "Implementing Spectral Methods for Partial Differential Equations" by David A. Kopriva it is written on page no. 34 (ch. # 1): For problems with Legendre weighted integrals, the abscissas and weights for the Gauss-Lobatto rule are calculated as: The abscissas (nodes) are

$x_j$ = +1, -1, Zeros of $L'_N(x)$

the weights are

$w_j = \frac{2}{N(N+1)}\frac{1}{[L_N(x_j)]^2}$ where j = 0, 1,...,N

by using these formulas kindly help me calculating nodes and weights for N = 16? how to calculate Zeros of $L'_N(x)$?

Abbeha
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  • i need Legendre gauss lobatto nodes and weights, the above link gives Gauss Legendre nodes and weights only. Lobatto includes end points and zeros of $L'_N(x)$ – Abbeha Dec 14 '16 at 05:03
  • http://keisan.casio.com/exec/system/1280801905 – Jonathan Hahn Dec 14 '16 at 05:04
  • Thanks i think it will work but there is a small difference in the above mentioned formulas and the formulas mentioned in link. can anyone suggest how to calculate it manually for confirmation. – Abbeha Dec 14 '16 at 05:15
  • [This input was provided by @TobiasK.S.Ritschel, who can't comment yet.]The book "Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers" by David A. Kopriva (2009) which you refer to actually presents algorithms for how to compute the Legendre and Chebyshev Gauss and Gauss Lobatto nodes (see Algorithm 23, 25, 26, and 27 on p. 64-68 in Chapter 3). The purpose of the book is exactly to provide algorithms which allow the reader to implement spectral methods.

    Here's a link: https://link.springer.com/book/10.1007/978-90-481-2261-5

     – InanimateBeing Sep 06 '22 at 17:17

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The Legendre Polynomials have a recursive formula: $(n+1)P_{n+1}(x)=(2n+1)xP_{n}(x)-nP_{n-1}(x)$ (Here $P_n(x) = L_n(x)$ in your notation above). You can also use an explicit formula to find them: $P_{n}(x) ={\frac {1}{2^{n}}}\sum _{k=0}^{n}{n \choose k}^{2}(x-1)^{n-k}(x+1)^{k}$. Mathematica will find the polynomial with the command LegendreP[n,x]. The polynomial you are looking for is $$P_{16}(x) = (1/32768)(6435 - 875160 x^2 + 19399380 x^4 - 162954792 x^6 + 669278610 x^8 - 1487285800 x^{10} + 1825305300 x^{12} - 1163381400 x^{14} + 300540195 x^{16}).$$ The derivative is $$P'_{16}(x) = (1/32768)(-1750320 x + 77597520 x^3 - 977728752 x^5 + 5354228880 x^7 - 14872858000 x^9 + 21903663600 x^{11} - 16287339600 x^{13} + 4808643120 x^{15})$$ which is a 15th order polynomial with 15 real roots. I have no hope of find the roots of this polynomial by hand, but a computer can find approximations (with Mathematica, try Solve[D[LegendreP[16,x],x] == 0.0]).

You'll get 15 roots: $[-0.973132, -0.91088, -0.815696, -0.691029, -0.541385, -0.372174, -0.189512, 0, 0.189512, 0.372174, 0.541385, 0.691029, 0.815696, 0.91088, 0.973132]$

Add $-1$ to the beginning of this list, and $1$ to the end of it, and you have your nodes $x_i$. Using $P_{16}(x)$, you can now find your weights. For $x_0 = -1$ and $x_{16} = 1$, you'll use the weights $$w_0 = w_{16} = \frac{2}{n(n+1)} = \frac{2}{(16)(17)}.$$ For $x_1 = -0.973132$, for example, you'll use the weight $$w_1 = \frac{2}{n(n+1)P_n(x_1)} = \frac{2}{(16)(17)P_n(-0.973132)} = -0.0181744.$$

Then, to do the Gauss-Lobatto integration rule for $\int_{-1}^{1} f(x)dx$ you compute $\sum_{i=0}^n w_i f(x_i)$.

The casio website computes the nodes and weights for you, but their formula starts at $n=1$ instead of $n=0$, so it is off by one of yours, and you'll want to compute for $n=17$ instead on that website.

Jonathan Hahn
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