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

[William Sit]

The interpretation for the sum is ok, but since x is not fixed, it cannot be
differentiated termwise (again, Martin pointed this out, but I was not paying
attention) because the sum does not have a constant number of terms.

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

-- 
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