Re: [eigen] matrix::linspace

[Benoit Jacob <jacob.benoit.1@xxxxxxxxx>]

2010/1/23 Hauke Heibel <hauke.heibel@xxxxxxxxxxxxxx>:
> Hi there,
>
> I am working on some functionality which would allow something like this:
>
>  VectorXd a = VectorXd::linspace(low,high,num_steps)
>
> which should be corresponding to:
>
>  const double step_size = (high-low)/(num_steps-1);
>  VectorXd a(num_steps);
>  for (int i=0; i<num_steps; ++i)
>   a(i) = low+i*step_size;
>
> In Eigen, this is best implemented based on NullaryExpressions, i.e.
> Matrices, where each element can be computed based on some functor.

I agree!

> In
> theory, this should not only be nice from a usability point of view
> but I also hoped of gaining from vectorization. It tuns out, that this
> is not that easy - it maybe due to MSVC but maybe also due to my lack
> of SSE skills. For the most simple example
>
>  for (int i=0; i<num_steps; ++i) a(i) = i;
>
> I managed to implement the functor's method ei_linspace_op::packetOp
> such that we do actually gain from vectorization as (note that I added
> integer parameters to it):
>
>  // for doubles only
>  ei_linspace_op::packetOp(int i, int = 0)
>  {
>    return ei_padd(ei_pset1<Scalar>(i),ei_pset<Scalar>(0,1));
>  }
>
> For me the first surprise was that this performs way better than just
> returning ei_pset<Scalar>(i,i+1).

Notice that reading from memory is costly, and ei_pset1 is costly
because it needs a SHUFPD instruction, so naively I would think that
the best that can happen is that you make once and for all a packet
(0,1) and a packet (1,1) and at each step you only add the latter into
the former. Perhaps your code is fast because it allowed the compiler
to do that optimization?

> Next, I looked into the case:
>
>  for (int i=0; i<num_steps; ++i) a(i) = low+i
>
> and here, I totally failed. Returning
>
>  return ei_pset<Scalar>(m_low+i,m_low+i+1);
>
> resulted in no statistically meaningful gain (~0.06%) speedup. The
> same holds for
>
>  return ei_padd(ei_pset1<Scalar>(i),ei_padd(ei_pset1(m_low),ei_pset<Scalar>(0,1)));
>
> which is even sometimes slower and permuting the operators did not help either.
>
> So, I am sort of puzzled about what I could improve. Any ideas? Any
> SSE operation, I am not aware of for creating (a, a+1, a+2, ...)

Same remark as above. I would initially store the packet (low,low+1)
in a SSE register, store (1,1) into another SSE register, and only do
a ei_padd at each iteration. Naively I would think that that is much
faster?

> Finally, note that ei_pset is not yet in PacketMath. Over here, I
> added forward delarations into GenericPacketMath and the actual SSE
> implementation in the corresponding PacketMath header (no Altivec
> yet). They GenericPacketMath declarations (as the name says) have no
> implementations since I don't see any meaningful way of providing them
> nor do I think they are useful.

Indeed I think they're only used in the matrix product code, which
doesn't use your new ei_pset function.

> The second change in the lib was
> mentioned above - I added indices to the packetOp functions.

Oh right, you need to fit your stuff with our packetOp infrastructure.
Hm, not easy indeed, as I was assuming sequential, not random access.

>
> If everything regarding the vectorization fails, we could still
> provide the functionality (with a specialization for the trivial case
> where we gain from vectorizing) because at least, I think it is nifty.

We might fail to vectorize random access efficiently. But 95% of use
cases will be sequential access. This could then be a use case for a
ReturnByValue function. When nesting this into a larger expression,
this would have the drawback of introducing a temporary, and the
advantage of enabling full vectorization instead of preventing
vectorization of the larger expression.

Benoit