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Re: Paul Abbott Chebyshev Article
*To*: mathgroup at smc.vnet.net
*Subject*: [mg79798] Re: Paul Abbott Chebyshev Article
*From*: chuck009 <dmilioto at comcast.com>
*Date*: Sat, 4 Aug 2007 06:03:41 -0400 (EDT)
I worked this problem using just regular power series:
a = -1;
b = 1;
n = 20;
xs = N[Table[x, {x, -1, 1, (b - a)/n}], 6];
xs[[11]] = 0.00001; (* set the zero point to 0.0001 so 0^0 not used *)
cs = Thread[Subscript[c, Range[0, n]]];
lhs = cs . Table[xs^i, {i, 0, n}];
rhs = 1 + (1/Pi)*cs . Table[NIntegrate[Evaluate[
t^i/((xs - t)^2 + 1)], {t, -1, 1}], {i, 0, n}];
sol = Solve[lhs == rhs, cs]
f[x_] = Sum[Subscript[c, i]*x^i, {i, 0, n}] /. First[sol]
Plot[f[x], {x, -1, 1}]
The results are comparable to using Chebyshev polynomials. Although I used 21 equations in 21 unknowns.
Things I learned in this thread:
1. The appearance of a notebook in the front end is different than what the notebook looks like on disk. On disks, its a text file with Cell commands.
2. Never use Traditional Form in a working cell. Use Standard or Input form. Traditional form is probably best used for documentation and publications.
3. Listable constructs such as Cos[{1,2,3,4}] is a new mathematical concept for me. Most functions in Mathematica have Listable attributes.
4. The code written by Paul Abbott is some of the most sophisticated code I have ever studied. I'll never be able to write code sufficiently sophisticated to get published in the Mathematica Journal.
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