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Re: Re: Transforming a polynomial into a trigonometric format tia sal2

  • To: mathgroup at smc.vnet.net
  • Subject: [mg88346] Re: [mg88299] Re: Transforming a polynomial into a trigonometric format tia sal2
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 2 May 2008 03:44:31 -0400 (EDT)
  • References: <fv42j9$5ui$1@smc.vnet.net> <200805010719.DAA26032@smc.vnet.net>

I think what you want is this function:


Chebyshev[f_, x_] :=
  With[{n = Exponent[f, x]},
   Sum[a[i]*Cos[i ArcCos[x]], {i, 0, n}] /.
    SolveAlways[f == Sum[a[i]*ChebyshevT[i, x], {i, 0, n}], x][[1]]]

So now, for example:

Chebyshev[21 - 10*x + x^2, x]
-10*x + (1/2)*Cos[2*ArcCos[x]] + 43/2

TrigExpand[%]
  x^2 - 10*x + 21

Chebyshev[21 - 10*x + x^2 + 3*x^3, x]
  -((31*x)/4) + (1/2)*Cos[2*ArcCos[x]] + (3/4)*Cos[3*ArcCos[x]] + 43/2


and so on.

Andrzej Kozlowski


On 1 May 2008, at 16:19, ratullochjk2 at gmail.com wrote:

> I'm sorry if I didn't explain myself better I hope this clarifies it
> better
>
> when I test for x =3 x= 7 for equation:
> 21 - 10 x + x^2 I get zero for both answers
>
> I used another math program using the ChebyshevT command and I
> got this
> 1/2 Cos[2 ArcCos[x]] + 43/2 - 10 x I tested with x=3 and x=7 and I
> also got zero for both
>
> but when I do a TrigReduce
>
> In[148] := 21 - 10 x + x^2 /. x -> Cos[y] // TrigReduce
>
> I get
>
> Out[149]:= 1/2 (43 - 20 Cos[x] + Cos[2 x])
> x=3 gives me 31.88
> x=7 gives me 14.0293
>
> I would like to use the ChebyshevT in mathematica 6 because I prefer
> that software but I'm not sure
> how to get the 1/2 Cos[2 ArcCos[x]] + 43/2 - 10 x answer in
> mathematica 6
>
> Am I doing something wrong with the TrigReduce function or leaving a
> part out why are the answers not even close?
>
> tia sal2
>
> On Apr 27, 10:41 pm, ratulloch... at gmail.com wrote:
>> Transforming a polynomial into a trigonometric format tiasal2
>>
>> Greetings All
>>
>> I'm using mathematica 6 and I have a polynomial and would like to
>> convert it into
>> a Trigonometric format.  Is this possible?
>>
>> Example:
>> I have a polynomial
>> 0.00154991- 4.01371 x + 1.81197 x^2 + 8.00183 x^3 - 9.3462 x^4
>>
>> How can I transform this into a trigonometric format
>> Example:
>> 0.00596679 Cos[6.98132 x] + 0.00358397 Cos[7.21403 x] +
>> 2.25013 Sin[0.232711 x] - 4.51511 Sin[0.465421 x]
>>
>> Note: these aren't correct answers I just wanted to include and
>> example
>>
>> tiasa
>



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