Hi Stavro,
Thanks for the detailed response. I was wondering whether the
difficutlies you mention
can be overcome in the following manner.
A new Maxima integration function, num_integration, internally calls a
user defined Maxima function, say integrandf, and passes the values of
the integration variables, x1, x2, together with the values of other
parameters, p1,p2,..., needed to compute the integrand, as follows,
........
I1 : apply(integrandf,[x1,x2,p1,p2,...]);
.........
The user would call num_integration with something like,
I : num_integration(integrandf,list_of_limits,p1,p2,....);
Any suggestions would be appreciated.
Regards,
Constantine.
On 1/12/12, Stavros Macrakis <macrakis at alum.mit.edu> wrote:
> As Alexander says, you don't need the temporary variable etc.
>
> Also, if you explicitly quote everything, you are less likely to have bad
> surprises. Something like this:
>
> (%i27) (cnt:0,quad_qags('(quad_qags('(cnt:cnt+1,if x>y then 1 else
> 0),x,0,1)[1]),y,0,1));
> (%o27) [0.49999763949292, 3.4177098928722671E-9, 735, 0]
> (%i28) cnt;
> (%o28) 621243
>
> Note the number of function evaluations for a very simple case! Note also
> that the result is not very accurate, and that the error estimate is way
> off. I imagine there are better routines for this kind of integration.
> Obviously in this case it is easy to transform to a single integral (even
> analytical), but presumably you want to handle more complicated integrands.
>
> -s
>
> On Thu, Jan 12, 2012 at 05:38, Alexander Klimov <alserkli at inbox.ru> wrote:
>
>> Hi.
>>
>> On Thu, 12 Jan 2012, Constantine Frangos wrote:
>> > (1) Its not clear why the function definition I am using does not
>> > work. According to the Maxima 5.24 manual, paragraph 20.4, the
>> > integrand can be the name of a Maxima function, presumably defined by
>> > the user in a file, for example, integrand.mac
>>
>> Your function does not always return a number. Consider your original
>> definition:
>>
>> (%i1) ind(x,y) := block([s],if is(x*x + y*y <= 1) then (s : 1) else (s :
>> 0),return(s))$
>>
>> Let us see how it works
>>
>> (%i2) trace(ind);
>> (%o2) [ind]
>>
>> If one gives numerical arguments, then your function returns the
>> number which is assigned to "s"
>>
>> (%i3) ind(0,1);
>> 1 Enter ind [0, 1]
>> 1 Exit ind 1
>> (%o3) 1
>>
>> but with symbols as arguments nothing is assigned to "s" and thus the
>> function returns just a symbol "s"
>>
>> (%i4) ind(x1,x2);
>> 1 Enter ind [x1, x2]
>> 1 Exit ind s
>> (%o4) s
>>
>> and this is exactly what happens when you use it in integration
>>
>> (%i5) I:quad_qags(quad_qags(ind(x1,x2),x1,-1,1)[1],x2,-1,1);
>> 1 Enter ind [x1, x2]
>> 1 Exit ind s
>> COERCE-FLOAT-FUN: no such Lisp or Maxima function: s
>> -- an error. To debug this try: debugmode(true);
>>
>> > (2) Are there perhaps other Maxima numerical integration functions
>> > that may perform better ?
>>
>> As I said, it is better to
>>
>> > > use a simple integral:
>> > >
>> > > (%i4) quad_qags(2*sqrt(1-y^2),y,-1,1);
>> > > (%o4) [3.141592653589797, 2.000470900043183e-9, 399, 0]
>>
>> the problem with indicators is that while they may be suitable for
>> symbolic manipulations, they are bad functions for Gaussian quadrature
>> rule which is only good if the function is similar to a polynomial.
>>
>> --
>> Regards,
>> ASK
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>>
>