Re: subscribe and ask questions
- To: mathgroup at smc.vnet.net
- Subject: [mg44830] Re: subscribe and ask questions
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Wed, 3 Dec 2003 04:24:10 -0500 (EST)
- Organization: The University of Western Australia
- References: <bq59vq$jr2$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <bq59vq$jr2$1 at smc.vnet.net>,
Harry Sheng <harryying at yahoo.com> wrote:
> In [Abramowitz and Stegun, 1964] Table 25.9, I try to find the zeros of
> hermite polynomial. For x_i and w_i, there are (-1) (0) (-2) before the
> values. Can you tell me what they are?
They are the exponent. (-1) means multiply the following value by
10^(-1).
You can use Mathematica to compute the x_i as follows. Here I compute
x_1 and x_2 for n=2;
x[2, 1] = x /. FindRoot[LaguerreL[2, x], {x, 0.5}]
0.585786437626905
x[2, 2] = x /. FindRoot[LaguerreL[2, x], {x, 3}]
3.414213562373095
To compute w_i use the formula 25.4.45 (however, there is a discrepancy
between this formula and the results tabulated in 25.9. I have dropped
the (n!)^2 factor):
w[n_, i_] := x[n, i]/((n + 1)^2 LaguerreL[n + 1, x[n, i]]^2)
Here I compute w_1 and w_2 for n=2;
w[2, 1]
0.8535533905932734
w[2, 2]
0.14644660940672627
> If I would use w_i to solve the integral, such as int(exp(-t^2)) from
> -infinity to infinity, how can I use the weight w_i?
In Mathematica you would not, in general, use Laguerre integration.
Instead you would either compute the integral in closed-form,
Integrate[Exp[-t^2],{t,-Infinity, Infinity}]
or numerically (usually when you cannot get a closed-form result),
NIntegrate[Exp[-t^2],{t,-Infinity, Infinity}]
letting Mathematica decide what numerical integration (or quadrature)
method to use.
Cheers,
Paul
--
Paul Abbott Phone: +61 8 9380 2734
School of Physics, M013 Fax: +61 8 9380 1014
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