This is very similar to the problem I'm stuck on:
*1) Problem:*
-I have two functions defined by linear inequalities
-I would like to add these functions
-*they are piecewise*
*
*
My case is signal processing. Like you, these inequalities
-define bounded regions equally expressible as simple geometric envelopes
and
-I would like to integrate over them as well.
but that requires adding them first.
Piecewise is missing ordinary manipulation methods. I could compute a
Fourier series approximation of two triangles, and add
those approximations. (And so could you) But I cannot add them directly.
Hilarious.
*2) General considerations*
(echoing your comments. Let me know if we disagree, or if this does not
relate to your problem.)
The functions are simple enough to be computed by hand. I have a large
number of them.
Yes, can write my own program to sum and integrate them in SAGE. I can
also open notepad.exe and do it in C++. I would like to use SAGE *instead,*
not
*as a programming environment* in which to do them. The tasks are too
simple to merit the overhead of interfacing SAGE. If I have to hand-define
integration for regions enclosed by lines, in space, I'll do it in GL or
DirectX where I can then immediately leverage geometric generalization,
topological reordering and display, 3D navigation/vector methods.
*3) Offer*
*I'd love to help revamp the Piecewise class,* but I'm new enough to
programming that I'll need a buddy.
Suppose that happened.
*4) *Would such a project assist you, Mr. Irving?
Say, if we proposed reworking Piecewise. Goal: provide a flexible container
for accessing continuous computation over regions defined by a finite
number of continuous boundaries?
Your example of trapping the region of interest in a polyhedron is
charming. I'm a bit bewildered this is not possible at present. You
describe the most fundamental definition of an integral. We've had this
one for 2500 years. Which leads me to...
*5)Plea*
Personally, the last thing I want is for the SAGE developers to believe
that the way to make a coherent environment is to extend further and
acquire more packages. SAGE is an OCD madhouse of methods. To have a
coherent environment, SAGE needs a coherent environment. Right now, I
could *use* SAGE (python) to write such an application. SAGE is supposed
to *be* that application. I want reliable generalizations that allow me to
draw from the methods already present. I want to define and manipulate
arbitrary mathematical problems without knowing the syntax of the python
library after calling it by hand.
I know development takes time. But another package simply will not solve
the problem. Suppose we then want to *visualize* our data structures. Or *add
points and lines to them, directly*. *Define boundaries to assist
integration, by drawing them?*
I do this every day in a CAD environment. The tools to define connection
graphs for manifolds in a natural way have *nothing* to do with syntax.
They are *topological* tools, for topological problems. It would be an
extraordinary thing to have such tools as a mathematician as well.
Let me advance the following position: We could *decide* that adding new
packages is by default NOT the answer to any problem involving elementary
mathematics that SAGE can already handle. Where, and when it proves too
difficult to build a solution into the existing methods and libraries, *then
*seek to acquire new ones.
{as usual, please know if these questions or comments do not properly
relate. And feel free to contact me directly if they do}
On Saturday, June 16, 2012 6:54:15 AM UTC-7, Alastair Irving wrote:
>
> Hi
>
> I want to numerically evaluate the integral of a function f(x,y) over a
> region defined by linear inequalities, for example
> 1/8<=y<=x<=1/3
> x+y<=1/3.
> I can do this with a repeated call to numerical_integral because I can
> re-write the constraints as y<=min(x,1/3-x). However, this solution
> isn't very satisfying in general. If I have more constraints and/or
> more variables then working out the limits on the repeated integrals by
> hand is going to be rather unpleasant.
>
> Is there any way I can get sage to compute the integral directly from
> the inequalities. For example, if I use a Polyhedron object to store
> the region of interest is there any way of getting functions from it
> which describe the limits of integration? Ideally, if P is a Polyhedron
> it would be nice to have something like P.integral(f) which computes the
> integral of f over P.
>
> Many thanks
>
> Alastair Irving
>
>
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