MaplePrimes - answers and comments on Question, series approximation

fdiff ?

Axel Vogt — Tue, 06 Dec 2011 01:15:46 Z

At least for the example values one can use fdiff (for cross checks)

'eval(H, [a=convert(v, rational),b=convert(w, rational)])';
fdiff(%, y=1e-9);
fdiff(%%, y=0);

    -621.372 .... (100 Digits)

Is it true that lHospital can not be used for H ?

For your actual task: I can not follow your code (too technical for
me), but may be you can use (for your last 'example')

multiseries((exp(y)-1-y)/y^2, y, 8, 'exact_order');

re fdiff

icegood — Tue, 06 Dec 2011 01:36:58 Z

unfortunately fdiff is not case for me because i need also zero limits of higher orders.

For lhopital: actually should work, because series exists and has finite limit, but it's not case for this task anyway because of slowness of limit itself.

multiseries... could be useful, will try!

got it: 4-th derivative will do

Axel Vogt — Tue, 06 Dec 2011 01:51:10 Z

A:=numer(H); B:=denom(H): # no assumptions on a,b are used

diff(A,y$4): 
A4:=eval(%, y=0);

diff(B,y$4):
eval(%, y=0): 
B4:=simplify(%);

A4/B4;

      4/a^2*(b+1)*2^(1/2)/LerchPhi(b/(b+1),1,a)


'eval(%, [a=v,b=w])';
evalf(%);

       - 3.19 ...

Edited: even if not that exact one can guess needed order by
multiseries(B, y=0, 2, 'exact_order'): series(%,y): 
convert(%, polynom):
ldegree(%,y); 
                                  4

The same for A, thus H = A/B = constant* (y^4 + .. )/(y^4 + ...)

ldegree : ya-ya

icegood — Tue, 06 Dec 2011 02:18:06 Z

ldegree :)))) One more stone. 
Instead of it now i prefer my own 'j' calculation. 

It was wrong because coefficients, that calculated, wasn't simplified, 
so wrong value returned. 
And from multiseries to polynom. intermediate 'series' call doesn't help. O-term remains. 
It's not so easy :(
Diff(...,y)+eval(...,y=0) could be case!

DH | y=0

Axel Vogt — Wed, 07 Dec 2011 01:31:24 Z

If it is still of interest (not sure, whether by that you only want to show some
of the weakness of Maple (15)):

diff(H, y=0) in y = 0 equals AA8/BB8, where

  AA8 := -1720320*2^(1/2)*a^7*b*((a-1/2)*LerchPhi(b/(b+1),1,a)-1-b)*(b+1)^2
  BB8 := -322560*LerchPhi(b/(b+1),1,a)^2*2^(1/2)*(b+1)*b*a^8

The approach is similar to the one sketched above.

For some test I take your u,v,w as rationals and Hvw the test with a=v,b=w:

  u,v,w:=op( convert([u,v,w], rational) );
  Hvw:='eval(H, [a=v,b=w])';

From limit( H,y=0) = A4/B4 one has the value 3.19 ... for a=v, b=w and then
form limit(DH,y=0) = AA8/BB8 one gets the value -48.27 ... for that test.

Now 'check' the according tangent against the function (your Digits = 100)

  'Hvw'; plot([%, -48*y + 3.19], y=-0.01..0.01, color=[red,blue]);
  

It is worth to mention, that fdiff fails (for me), it gives ~ - 621 as above
said and increasing (work) precision even gives the wrong sign and magnitude:

  oldDigits:=Digits: Digits:=400:
    fdiff(Hvw, y=0, workprec=1.5);
  Digits:=oldDigits:

is roughly + 300 (even if staying with Digits = 100)

ahhh :(

icegood — Tue, 06 Dec 2011 02:07:04 Z

i recalled, i tried multiseries before, but many functions are not supported and i see that for H even...

Error, (in MultiSeries:-multiseries) unable to compute series.

And for smaller:

s:=multiseries(LerchPhi(exp(-2*y*a)*b/(b+1), 1, a), y, 4, 'exact_order');
s := LerchPhi(b/(b+1), 1, a)+(-8*b^4*a/((b+1)^4*(a+4))-2*b*a/((b+1)*(a+1))-6*b^3*a/((b+1)^3*(a+3))-4*b^2*a/((b+1)^2*(a+2)))*y+(32*b^4*a^2/((b+1)^4*(a+4))+18*b^3*a^2/((b+1)^3*(a+3))+8*b^2*a^2/((b+1)^2*(a+2))+2*a^2*b/((b+1)*(a+1)))*y^2+(-36*b^3*a^3/((b+1)^3*(a+3))-(32/3)*b^2*a^3/((b+1)^2*(a+2))-(256/3)*b^4*a^3/((b+1)^4*(a+4))-(4/3)*a^3*b/((b+1)*(a+1)))*y^3+O(y^4)

is OK but how to convert it to polynom without that O(y^4) trail?
eval(s, O=proc() 0; end proc); - nothing. Again long and windy road.