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Re: Manually nested Tables faster than builtin nesting?

Bill Rowe wrote:
> On 10/27/10 at 5:13 AM, sschmitt at (Sebastian
> Schmitt) wrote:
>> I find that manually nested Tables are faster (factor 2 in my
>> example) compared to the builtin nesting. I probably do a stupid
>> thinko. Could you please have a look?
>> In[52]:= imax = jmax = 5000;
>> In[53]:= (tabletable = Table[Table[i*j, {i, imax}], {j, jmax}];) //
>> Timing // First
>> Out[53]= 1.84
>> In[54]:= (table = Table[i*j, {i, imax}, {j, jmax}];) // Timing //
>> First
>> Out[54]= 14.55
> I get similar timing results. My speculation is as follows:
> When you use the built-in nesting, you explicitly do 25 million
> multiplies. No possibility of vector processing.
> But when you use explicit nesting, you are multiplying a packed
> array by a scalar. My guess is Mathematica employs some vector
> processing here and gains a speed up over individual multiplies.
> If my speculation is right, then there should be some operations
> which will not gain such a dramatic speed-up.

This is heading in the right direction. Actually the entire issue 
involves use, or not, of packed arrays. No vectorization of 
multiplication is done here. If it were, there would be an additional 
speed gain.

imax = jmax = 2000;

First the basic nested version.

In[2]:= First[Timing[tnest1 = Table[Table[i*j, {i, imax}], {j, jmax}];]]
Out[2]= 0.239963

Now we vectorize and it gets noticeably faster.

In[3]:= First[Timing[tnest2 = Table[j *
   Table[i, {i, imax}], {j, jmax}];]]
Out[3]= 0.099985

In[4]:= tnest1===tnest2
Out[4]= True

Now without explicit nesting.

In[5]:= First[Timing[tnonest1 = Table[j*i, {i, imax}, {j, jmax}];]]
Out[5]= 1.46478

In[6]:= tnonest1===tnest2
Out[6]= True

Why so much slower? It has to do with packing. Neither is packed, but 
the first ones have packed rows.


In[10]:= And @@ Map[PackedArrayQ,tnest2]
Out[10]= True

In[11]:= And @@ Map[PackedArrayQ,tnest1]
Out[11]= True

In[12]:= And @@ Map[PackedArrayQ,tnonest1]
Out[12]= False

One might well wonder why the packing is not done in this case. It has 
to do with evaluation semantics. Table sees the second iterator upper 
bound as a variable (because it is Held). Indeed, it could be changed in 
process of handling the first iterator. Hence Table cannot set up a 
packed array.

We can force preevaluation of that iterator bound as below. This 
recovers the expected speed.

In[13]:= First[Timing[tnonest2 = With[{jmax=jmax},
   Table[j*i, {i, imax}, {j, jmax}]];]]
Out[13]= 0.138979

We can even pack the entire thing, simply by forcing numerical 
evaluation of both inner and outer iterator bounds.

In[15]:= First[Timing[tnonest3 = With[{imax=imax,jmax=jmax},
   Table[j*i, {i, imax}, {j, jmax}]];]]
Out[15]= 0.13898

In[16]:= PackedArrayQ[tnonest3]
Out[16]= True

In[17]:= tnonest3===tnonest2===tnonest1
Out[17]= True

One will notice the construction of tnest2 was slightly faster. This is 
presumably an indication of the speed gain from vectorization of the 

Daniel Lichtblau
Wolfram Research

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