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

• To: mathgroup at smc.vnet.net
• Subject: [mg88337] Re: [mg88299] Re: Transforming a polynomial into a trigonometric format tia sal2
• From: "Szabolcs HorvÃt" <szhorvat at gmail.com>
• Date: Fri, 2 May 2008 03:42:55 -0400 (EDT)
• References: <fv42j9$5ui$1@smc.vnet.net> <200805010719.DAA26032@smc.vnet.net>

Hi,

Actually the TrigReduce approach I posted does the same thing, one
just has to substitute ArcCos[x] for the y variable.  Here's a
complete function:

chebyshev2[f_, x_] :=
 Module[{y}, Expand@TrigReduce[f /. x -> Cos[y]] /. y -> ArcCos[x]]

(Applying Expand is not really necessary.  In this case it ensures
that chebyshev[] will give the result in the same form as
Chebyshev[].)

Szabolcs

On Thu, May 1, 2008 at 11:30 PM, Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
> It is slightly better to localize the symbol a:
>
>  Chebyshev[f_, x_] :=
>   Module[{a, 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]]]
>
>  Andrzej Kozlowski
>
>
>
>  On 1 May 2008, at 22:19, Andrzej Kozlowski wrote:
>
>
> > 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
> > > >
> > >
> > >
> >
> >
>
>

• References:
  • Re: Transforming a polynomial into a trigonometric format tia sal2
    • From: ratullochjk2@gmail.com