Timeline for Is there an example where the error of Gauss-Laguerre quadrature does not vanish?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 13, 2018 at 9:27 | answer | added | user111 | timeline score: 2 | |
| May 8, 2014 at 5:23 | answer | added | chinese communist | timeline score: 1 | |
| Mar 24, 2014 at 21:13 | comment | added | gondolier | @DouglasZare I see now. Thanksa lot. But is there any way to check a specific function such as $f(x)=x \log x$ with infinite derivative converges or not? The error estimate I found seems to be useless. | |
| Mar 24, 2014 at 21:11 | comment | added | gondolier | @FedericoPoloni, The Runge function seems to be an example for equidistant case instead of Gauss quadrature. | |
| Mar 24, 2014 at 16:10 | comment | added | Douglas Zare | I didn't suggest that $f$ should depend on $n$. I meant that you can easily construct $f$ by placing infinitely many bumps, one at the largest root of the $n$th polynomial, so that the $n$th degree Gauss-Laguerre quadrature estimate is off by $1$, for example. Since the largest roots have no limit point you can put a smooth bump there to adjust the $n$th degree estimate arbitrarily without affecting lower estimates. | |
| Mar 24, 2014 at 7:57 | comment | added | Federico Poloni | The Runge function $x\mapsto \frac{1}{1+25x^2}$ is the standard example of hard-to-approximate function for the Gauss-Legendre quadrature in the interval $[-1,1]$ (which is exact for integrating polynomials of degree up to $2n-1$). My first attempt would be trying a change of variable of some form to take that interval into $[0,\infty]$ and Legendre into Laguerre, but I am not an expert in quadrature myself. | |
| Mar 24, 2014 at 7:24 | comment | added | gondolier | Douglas, I meant for a fixed function $f$ and let $n$ goes to infinity. So $f$ cannot depend on $n$. | |
| Mar 24, 2014 at 6:37 | comment | added | Douglas Zare | You need some extra condition to rule out placing bumps at the largest roots of the Laguerre polynomials. | |
| Mar 24, 2014 at 4:53 | history | asked | gondolier | CC BY-SA 3.0 |