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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
- [Axiom-developer] Bugs in combfunc.spad and partial patch, William Sit, 2004/06/10
- Re: [Axiom-developer] Bugs in combfunc.spad and partial patch, William Sit, 2004/06/10
- [Axiom-developer] Re: Bugs in combfunc.spad and partial patch, m.rubey, 2004/06/11
- [Axiom-developer] Re: Bugs in combfunc.spad and partial patch, William Sit, 2004/06/11
- Message not available
- Message not available
- Message not available
- Message not available
- [Axiom-developer] Re: EXPR POLY INT, Martin Rubey, 2004/06/15
- Re: [Axiom-developer] Re: EXPR POLY INT, root, 2004/06/15
- [Axiom-developer] Re: EXPR POLY INT, William Sit, 2004/06/15
- [Axiom-developer] Re: EXPR POLY INT, Martin Rubey, 2004/06/15
- [Axiom-developer] Re: EXPR POLY INT, Martin Rubey, 2004/06/16
- [Axiom-developer] Re: EXPR POLY INT, root, 2004/06/16
- [Axiom-developer] Re: EXPR POLY INT, William Sit, 2004/06/17
- [Axiom-developer] Re: EXPR POLY INT, William Sit, 2004/06/16
- Message not available
- [Axiom-developer] Re: EXPR POLY INT, Martin Rubey, 2004/06/18