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