On Mon, Oct 12, 2015 at 04:43:24AM -0500, Laurence Marks wrote:
> Rather than giving the answer to the original question, I think it is
> better to try and encourage people to solve problems themselves -- perhaps
> I am being too much of an teacher, but that is my approach.
> So, let me pose a question in response to the original question. What is
> the relationship between summing a set of values on a grid and the integral
> of a function sampled at a set of grid points. Think about this, and the
> reason why the numbers are very different will be obvious.
> If they still do not completely agree (they wont), then think about the
> accuracy of numerical integration. It is quite important in DFT codes to
> remember that numerical integrations are never exact, and this leads to
> many small limitations.


I love your approach. May I continue?

There are two main strategies in doing a numerical integration. The
first is designing the geometry of the grid by simplicity and providing
each grid point an equal weight. The alternative approach is far more
complex, but it can also be more successful. The position and weight
of grid points can be used to improve the quality of the integration.

Best regards,
             Víctor Luaña
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