Legendre economization

Posted on 2008-11-09 11:04 By figment (1)

Categories:

How do I ...? with Maple (Student)

Hi,

I have this taylor series expansion:

> taylor(exp(-x)*sin(x), x, 11);

Now I need to do a legendre economization. So I want to write x, x², x³ etc (from the taylor series) as linear combinations of the Legendre polynomials. My idea was to simply write something like:

>solve({aP(0,x)+bP(1,x)+cP(2,x) = x²}, {a,b,c});

This is supposed to give coefficients for the legendre expansion of x². This doesn't work because it gives solutions in terms of x, which is useless to me. Is there any way?

---
The analagous in Chebyshev is to write:
x²=1/2(T0 + T2)
---

I need to find the same, but in terms of legendre polynomials.

Thanks.

OrthogonalSeries

Comment on 2008-11-09 12:57 from Axel Vogt ( Maple Rating 4

994)

It is a bit hidden, You can do that for example with the package 'OrthogonalSeries':

Define your polynom from Taylor series:

  taylor(exp(-x)*sin(x), x, 11): 
  myP:=convert(%,polynom);

Convert it to the Legendre base and check it:

  OrthogonalSeries:-ChangeBasis(myP,LegendreP(v, x)):
  OrthogonalSeries:-ConvertToSum(%):
  convert(%,polynom):
  'myP=(%)';
  is(simplify(%));

                                 true

The same for Tschebyscheff:

  OrthogonalSeries:-ChangeBasis(myP,ChebyshevT(n,x)):
  OrthogonalSeries:-ConvertToSum(%):
  convert(%,polynom):
  'myP=(%)';
  is(simplify(%));
                                 true

If it is homework, then give credit to mapleprimes ...

economization?

Comment on 2008-11-09 15:03 from jakubi ( Maple Rating 4

943)

was not aware of a thing named "Legendre economization". Does it mean that the truncated Taylor series, expressed in terms of these polynomials, is computationally more economical?