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Re: [Axiom-developer] Re: Bugs in combfunc.spad and partial patch


From: William Sit
Subject: Re: [Axiom-developer] Re: Bugs in combfunc.spad and partial patch
Date: Mon, 14 Jun 2004 10:55:11 -0400

The patch supplied below by me would not work in all cases, as pointed out by
Martin that

  D(sum(f(i),i=0..x),x) 

would give a wrong answer. Worse, I believe now the interpretation is wrong as
well (at least in certain cases). So sorry. 

William
----------

William Sit wrote:

> --------------------------- Patch for dvdsum in combfunc.spad ---------------
 
> The meaning of differenting a sum with variable bounds can be understood this
> way, by treating the RESULT as a function of the variable. For example, 
> assuming
> x takes integer values,
> 
>   x +-> sum(f(i,x), i=x^2..x^3)
> 
> defines a function of x as f(x^2,x)+ f((x^2+1),x) + ... + f(x^3,x)
> whose domain can be generalized to the real numbers or even complex numbers.
> Then we may ask what is its derivative and we can actually write it down with 
> a
> summation. More generally, we want to compute
> 
>   D(sum(f(i,x),i=a(x)..b(x)),x)
> 
> which should give
> 
>   sum(D(f(a(x) + i,x),x), i=0..(b(x) - a(x)))
> 
> So if you accept this interpretation, here is my new dvdsum (which evaluates 
> in
> all cases!) Note: before calling dvdsum, the system already checks that the
> lower and upper limits are functions of x, so no checking is needed here.
> 
>   dvdsum(l,x)==
>     x = retract(y := third l)@SE => 0
>     k := retract(d := second l)@K
>     f := first l
>     g := third rest l
>     h := third rest rest l
>     opdsum [differentiate(eval(f,k,g+d),x),d,y,0,h-g]
> 
> Test this:
> 
>   f := operator 'f
>   D(sum(f(i,x), i=a..b),x)
> 
> I tried other differentiation and they seem fine.
> 
>   D(sum(x^i, i=0..n),x)
>   D(sum(f(i,x), i=x^2..x^3)
>   D(sum(x/i, i=x^2..x^3)
> 
> The patch leaves the summation unevaluated if it cannot be done (this is
> automatic due to Bronstein's original implementation).

-- 
William Sit
Department of Mathematics..............Email: address@hidden
City College of New York..........................Tel: 212-650-5179
Convent Ave at West 138th Street..................Fax: 212-862-0004
New York, NY 10031............Axiom, A Scientific Computation Sytem
USA..........................http://www.nongnu.org/axiom/index.html




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