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MathGroup Archive 2007

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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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