Re: FYI: optimizing certain matrix arithmetic

[Jaroslav Hajek]

On Tue, Sep 29, 2009 at 1:34 PM, Michael Creel <address@hidden> wrote:
> On Tue, Sep 29, 2009 at 12:47 PM, Jaroslav Hajek <address@hidden> wrote:
>> On Tue, Sep 29, 2009 at 12:26 PM, Michael Creel <address@hidden> wrote:
>>> On Tue, Sep 29, 2009 at 10:44 AM, Jaroslav Hajek <address@hidden> wrote:
>>>> On Tue, Sep 29, 2009 at 10:38 AM, Michael Creel <address@hidden> wrote:
>>>>> Hi all,
>>>>>
>>>>> On an Apple Macbook Pro running Ubuntu Jaunty amd64, using the benchmark
>>>>>
>>>>> %%%%%%%%%%%%%%%%%%%%%%
>>>>> n = 500;
>>>>> R = triu (rand (n));
>>>>> u = rand (n, 1);
>>>>>
>>>>> tic; for i = 1:1000; R \ u; endfor; toc
>>>>> tic; for i = 1:1000; u' / R; endfor; toc
>>>>> tic; for i = 1:1000; R' \ u; endfor; toc
>>>>>
>>>>> R = tril (rand (n));
>>>>> u = rand (n, 1);
>>>>>
>>>>> tic; for i = 1:1000; R \ u; endfor; toc
>>>>> tic; for i = 1:1000; u' / R; endfor; toc
>>>>> tic; for i = 1:1000; R' \ u; endfor; toc
>>>>>
>>>>> u = u + I*rand (n, 1);
>>>>> tic; for i = 1:1000; R \ u; endfor; toc
>>>>> tic; for i = 1:1000; R' \ u; endfor; toc
>>>>>
>>>>>
>>>>> n = 800;
>>>>> a = rand (n);
>>>>> b = rand (n) + i*rand (n);
>>>>> tic; a * b; toc
>>>>> tic; b * a; toc
>>>>> tic; a' * b; toc
>>>>> tic; b * a'; toc
>>>>> tic; a \ b; toc
>>>>> tic; b / a; toc
>>>>> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
>>>>>
>>>>> Octave3.0.1 that comes with Ubuntu Jaunty amd64, I get
>>>>>
>>>>>
>>>>> octave:4> bench
>>>>> Elapsed time is 0.20216 seconds.
>>>>> Elapsed time is 1.93894 seconds.
>>>>> Elapsed time is 2.33824 seconds.
>>>>> Elapsed time is 0.188448 seconds.
>>>>> Elapsed time is 1.95657 seconds.
>>>>> Elapsed time is 2.43552 seconds.
>>>>> Elapsed time is 4.08299 seconds.
>>>>> Elapsed time is 7.84752 seconds.
>>>>> Elapsed time is 0.213021 seconds.
>>>>> Elapsed time is 0.21117 seconds.
>>>>> Elapsed time is 0.218387 seconds.
>>>>> Elapsed time is 0.217174 seconds.
>>>>> Elapsed time is 0.452714 seconds.
>>>>> Elapsed time is 0.391383 seconds.
>>>>> octave:5>
>>>>>
>>>>>
>>>>>
>>>>> Matlab 2008b gives
>>>>>>> bench
>>>>> Elapsed time is 0.289161 seconds.
>>>>> Elapsed time is 0.566446 seconds.
>>>>> Elapsed time is 0.562623 seconds.
>>>>> Elapsed time is 0.253456 seconds.
>>>>> Elapsed time is 0.574304 seconds.
>>>>> Elapsed time is 0.570281 seconds.
>>>>> Elapsed time is 0.253070 seconds.
>>>>> Elapsed time is 0.572601 seconds.
>>>>> Elapsed time is 0.102086 seconds.
>>>>> Elapsed time is 0.102677 seconds.
>>>>> Elapsed time is 0.103080 seconds.
>>>>> Elapsed time is 0.103759 seconds.
>>>>> Elapsed time is 0.165608 seconds.
>>>>> Elapsed time is 0.181704 seconds.
>>>>>>>
>>>>>
>>>>>
>>>>> Octave 3.2.3+ from today, self compiled, gives
>>>>> octave:1> bench
>>>>> Elapsed time is 0.208794 seconds.
>>>>> Elapsed time is 0.189178 seconds.
>>>>> Elapsed time is 0.186724 seconds.
>>>>> Elapsed time is 0.188649 seconds.
>>>>> Elapsed time is 0.192915 seconds.
>>>>> Elapsed time is 0.19166 seconds.
>>>>> Elapsed time is 0.186277 seconds.
>>>>> Elapsed time is 0.19102 seconds.
>>>>> Elapsed time is 0.212707 seconds.
>>>>> Elapsed time is 0.211013 seconds.
>>>>> Elapsed time is 0.210491 seconds.
>>>>> Elapsed time is 0.210447 seconds.
>>>>> Elapsed time is 0.431791 seconds.
>>>>> Elapsed time is 0.367412 seconds.
>>>>> octave:2>
>>>>>
>>>>> Congratulations!
>>>>> Michael
>>>>>
>>>>
>>>> It's interesting you didn't get any speed-up in the second part of the
>>>> benchmark, compared to 3.0.1...
>>>> What BLAS and LAPACK are you using? What's your compiler configuration?
>>>> Also, what exactly is your tip? The "3.2.3+" is a bit unclear, did you
>>>> mean "3.3.50+", i.e. the development version?
>>>>
>>>> thanks
>>>>
>>>> --
>>>> RNDr. Jaroslav Hajek
>>>> computing expert & GNU Octave developer
>>>> Aeronautical Research and Test Institute (VZLU)
>>>> Prague, Czech Republic
>>>> url: www.highegg.matfyz.cz
>>>>
>>>
>>> Oops, sorry, it's 3.3.50+, updated this morning.
>>>
>>> I make using
>>> make -j2 CFLAGS="-O3 -march=native -funroll-loops" FFLAGS="-O3
>>> -march=native -funroll-loops" XTRA_CFLAGS="-O3 -march=native
>>> -funroll-loops" XTRA_CXXFLAGS="-O3 -march=native -funroll-loops"
>>>
>>
>> In general, if you're with a newer gcc on a 64-bit architecture, I
>> advise you against -funroll-loops. For me, it usually gets some +1% of
>> additional speed of some operations, at the cost of increasing the
>> binaries' size by more than 50%. Seems like a bad tradeoff.
>>
>>> ./configure reports
>>>  BLAS libraries:       -llapack -lcblas -lf77blas -latlas
>>>
>>> so I assume that Octave is using Atlas (the atlas dev package that
>>> comes with Kubuntu Jaunty amd64).
>>>
>>> Michael
>>>
>>
>> Apparently, yes. Hmm. It's really weird you got almost exactly the same 
>> figures.
>> If you apply the attached patch, rebuild and re-run the benchmark,
>> what do you get?
>>
>> --
>> RNDr. Jaroslav Hajek
>> computing expert & GNU Octave developer
>> Aeronautical Research and Test Institute (VZLU)
>> Prague, Czech Republic
>> url: www.highegg.matfyz.cz
>>
>
> With that patch applied, I get
> octave:1> bench
> Elapsed time is 0.194493 seconds.
> Elapsed time is 0.192309 seconds.
> Elapsed time is 0.189026 seconds.
> Elapsed time is 0.188679 seconds.
> Elapsed time is 0.195958 seconds.
> Elapsed time is 0.193521 seconds.
> Elapsed time is 0.187596 seconds.
> Elapsed time is 0.193254 seconds.
> Elapsed time is 0.215135 seconds.
> Elapsed time is 0.213705 seconds.
> Elapsed time is 0.21341 seconds.
> Elapsed time is 0.212501 seconds.
> Elapsed time is 0.363992 seconds.
> Elapsed time is 0.368094 seconds.
>
> so there is an improvement in the second to last number.
>
> Cheers, M.
>

OK, it's funny. I now understand where the problem is. Just change the line

b = rand (n) + i*rand (n);

to

b = rand (n) + I*rand (n);

(note the big I). At this point, i is still defined from the previous
loops as a real numeric value (!)
And run the benchmarks again. I think this affects Matlab, too.
In any case, it is apparent that your Matlab is linked to something
faster than ATLAS; probably Intel's MKL.

regards

-- 
RNDr. Jaroslav Hajek
computing expert & GNU Octave developer
Aeronautical Research and Test Institute (VZLU)
Prague, Czech Republic
url: www.highegg.matfyz.cz