MaplePrimes - answers and comments on Question, Padé Approximants

I tried it

lemelinm — Wed, 16 Apr 2008 18:39:33 Z

I have tried your suggestion. Of course, this was my mistake about collecting x^k.

But even then, I still unable to reproduce the answer of Maple. PLease help me solve this one as an example and if as a bonus you have a simple application, I will be very thankfull.

mario.lemelin@cgocable.ca

Pade

Robert Israel — Wed, 16 Apr 2008 23:35:20 Z

You should be using the coefficients of (x-4)^n in fx*qx - px, not the coefficients of
x^n. You could use a change of variables t = x-4, or series(fx*qx-px, x=4, ...).

Thanks!

lemelinm — Thu, 17 Apr 2008 00:55:47 Z

I have learn a lot in the way you have done it. Since I want to show it to students, I will try to do it without "taylor(qx*fx - px, x=4, 9)". Do you know (or someone else) a simple application of the use of the Padé Approximant?

Thanks!

problem with chebpade?

jakubi — Sat, 19 Apr 2008 20:25:06 Z

I was looking for a simple example of a function defined by a numerical integral and calculated for it a 'pade' and a 'chebpade' approximant:

with(numapprox):
with(orthopoly):
J:=Int(sin(u^3)*tan(u),u=0..x);
f:=x->evalf(Int(sin(u^3)*tan(u),u=0..x)):

J := Int(sin(u^3)*tan(u),u = 0 .. x); This is a toy example and here 'f' evaluates fast:

f(.1);
                                         -5
                          0.2004776617 10

I get these approximants:

p:=pade(f(x), x, [5,5]);

            5
           x
-------------------------
  /          2    23   4\
5 |1 - 5/21 x  - ---- x |
  \              1323   /

chp:=chebpade(f(x), x, [5,5]);

                                     3                 5
-0.0001075954404 x + 0.002076036171 x  + 0.2184405434 x
--------------------------------------------------------
                                2                 4
    1.148539583 - 0.3990232808 x  + 0.1359254864 x

Now, their evaluation at the same point gives:

evalf(eval(p,x=.1));
                                         -5
                          0.2004776764 10

evalf(eval(chp,x=.1));
                                          -5
                          -0.5678239432 10

So, something seems to be wrong with this 'chebpade' approximant in this region:

plot([f(x),p,chp],x=0..0.2,color=[red,green,blue]);

What?

chebpade

gkokovidis — Sun, 20 Apr 2008 07:07:10 Z

I tried your example above on my laptop at home, where I am running Maple 9.52. The line with chebpade gave me the following error:

chp:=chebpade(f(x), x, [5,5]);
Error, (in numapprox:-chebpade) singularity in or near interval

I'm wondering if this is the cause of the difference that you are observing between the two approximations.

Regards,
Georgios Kokovidis

Dräger Medical

rounding problems?

Axel Vogt — Sun, 20 Apr 2008 12:52:54 Z

using 14 Digits in Maple 11 i have no problems, however chebpade has larger approximation error in the plotted range

I think one should specify a range, chpp:=chebpade(f(x), x=0 .. 1/3, [5,5]) works quite well