On Thu, 7 Apr 2016 18:09:10 +0100, Chris Lucas wrote:
> Sorry, you're right that I should have summarized the key
> implications (at
> least as I see them).
>
> The short version:
> (1) If you're multiplying small matrices with AMC, BlockRealMatrix
> is
> your friend. For large matrices, stick with Array2DRowRealMatrix
This is surprising, as the exact opposite of the intended purpose of
"BlockRealMatrix".
It's certainly worth ascertaining with more data points (to make a nice
plot).
> (or
> something else?).
Patch welcome. :)
> (2) Scaling isn't great  O(N^3) where N is the number of
> rows/cols in
> a square matrix)  but neither is OJAlgo's. The interlibrary
> differences
> can mostly be attributed to constant factors.
>
> The longer version:
>
> Observation 1: For Array2DRowRealMatrix, matrix multiplication
> appears to
> scale with ~ O(N^3) (i.e., the naive algorithm). In retrospect, that
> was a
> bit silly to spend time benchmarking because I've just looked at the
> source[^1] and it *is* the naive algorithm.
Not totally, there is a copy of the "current row" which otherwise makes
the computation scale really badly (and perhaps the initial reason for
the
"BlockRealMatrix" implementation).
> Naturally, better asymptotic
> performance would be nice,
What do you mean?
> but I recognize that the AMC devs likely have
> better things to do.
>
> Observation 2: When multiplying new matrices, BlockMatrix offers some
> improvement when multiplying the same matrices repeatedly
Why would someone do that?
> and for smaller
> matrices. However on one person's setup (mine) it doesn't offer
> better
> performance on large random matrices. Followup tests with N \in
> {1000,1500,2000,2500} indicate it scales worse than the naive
> algorithm
> (didn't look at growth, but factor of ~5 worse on a 2500x2500
> matrix).
In effect, that had been my conclusion some years ago. [But I hadn't
check
"small" matrices, as I was so convinced that the overhead of
"BlockRealMatrix"
was unsuitable for the smaller sizes!]
IIUC, CM (both "Array2DRowRealMatrix" and "BlockRealMatrix") only makes
data
closer in RAM with the "hope" that the CPU cache can help (at least
that's
what the doc for "BlockRealMatrix" indicates IIRC).
> Observation 3: OJAlgo doesn't scale better asymptotically from what I
> can
> tell. That's no great surprise based on
> https://github.com/optimatika/ojAlgo/wiki/Multiply.
> <https://github.com/optimatika/ojAlgo/wiki/Multiply> Scala Breeze
> w/native
> libs scales closer to O(N^2.4).
Details on how you arrive to those numbers would really be a useful
addition
to the CM documentation.
> Observation 4: Breeze with native libraries does better than one
> might
> guess based on some previous benchmarks. People still might want to
> use
> math commons, not least because those native libs might be encumbered
> with
> undesirable licenses.
Do they use multithreading?
Regards,
Gilles
>
> [^1]:
>
> https://gitwipus.apache.org/repos/asf?p=commonsmath.git;a=blob;f=src/main/java/org/apache/commons/math4/linear/Array2DRowRealMatrix.java;h=3778db56ba406beb973e1355234593bc006adb59;hb=HEAD
>
>
> On 7 April 2016 at 16:50, Gilles <gilles@harfang.homelinux.org>
> wrote:
>
>> On Thu, 7 Apr 2016 16:17:52 +0100, Chris Lucas wrote:
>>
>>> Thanks for suggesting the BlockRealMatrix. I've run a few
>>> benchmarks
>>> comparing the two, along with some other libraries.
>>>
>>
>> The email is not really suited for tables (see your message below).
>> What are the points which you want to make?
>>
>> As said before, "reports" on CM features (a.o. benchmarks) are
>> welcome but
>> should be formatted for inclusion in the userguide (for now, until
>> we might
>> create a separate document for data that is subject to change more
>> or less
>> rapidly).
>>
>> If you suspect a problem with the code, then I suggest that you file
>> a JIRA
>> report, where files can be attached (or tables can be viewed without
>> being
>> deformed thanks to the "{noformat}" tag).
>>
>> Regards,
>> Gilles
>>
>>
>> I'm using jmh 1.11.3, JDK 1.8.0_05, VM 25.5b02, with 2 warmups and
>> 10 test
>>> iterations.
>>>
>>> The suffixes denote what's being tested:
>>> MC: AMC 3.6.1 using Array2DRowRealMatrix
>>> SB: Scala Breeze 0.12 with native libraries
>>> OJ: OJAlgo 39.0. Some other benchmarks suggest OJAlgo is an
>>> especially
>>> speedy purejava library.
>>> BMC: MC using a BlockRealMatrix.
>>>
>>> I'm using the same matrix creation/multiplication/inverse code as I
>>> mentioned in my previous note. When testing BlockReadMatrices, I'm
>>> using
>>> fixed random matrices and annotating my class with
>>> @State(Scope.Benchmark).
>>> For the curious, my rationale for building matrices on the spot is
>>> that
>>> I'm
>>> already caching results pretty heavily and rarely perform the same
>>> operation on the same matrix twice  I'm most curious about
>>> performance
>>> in
>>> the absence of caching benefits.
>>>
>>> Test 1a: Creating and multiplying two random/uniform (i.e., all
>>> entries
>>> drawn from math.random()) 100x100 matrices:
>>> [info] MatrixBenchmarks.buildMultTestMC100 thrpt 10
>>> 836.579
>>> ± 7.120 ops/s
>>> [info] MatrixBenchmarks.buildMultTestSB100 thrpt 10
>>> 1649.089
>>> ± 170.649 ops/s
>>> [info] MatrixBenchmarks.buildMultTestOJ100 thrpt 10 1485.014 ±
>>> 44.158
>>> ops/s
>>> [info] MatrixBenchmarks.multBMC100 thrpt 10 1051.055 ± 2.290
>>> ops/s
>>>
>>> Test 1b: Creating and multiplying two random/uniform 500x500
>>> matrices:
>>> [info] MatrixBenchmarks.buildMultTestMC500 thrpt 10
>>> 8.997
>>> ± 0.064 ops/s
>>> [info] MatrixBenchmarks.buildMultTestSB500 thrpt 10
>>> 80.558
>>> ± 0.627 ops/s
>>> [info] MatrixBenchmarks.buildMultTestOJ500 thrpt 10
>>> 34.626
>>> ± 2.505 ops/s
>>> [info] MatrixBenchmarks.multBMC500 thrpt 10 9.132 ± 0.059
>>> ops/s
>>> [info] MatrixBenchmarks.multCacheBMC500 thrpt 10 13.630 ± 0.107
>>> ops/s
>>> [no matrix creation]
>>>
>>> 
>>>
>>> Test 2a: Creating a single 500x500 matrix (to get a sense of
>>> whether the
>>> mult itself is driving differences:
>>> [info] MatrixBenchmarks.buildMC500 thrpt 10
>>> 155.026
>>> ± 2.456 ops/s
>>> [info] MatrixBenchmarks.buildSB500 thrpt 10
>>> 197.257
>>> ± 4.619 ops/s
>>> [info] MatrixBenchmarks.buildOJ500 thrpt 10
>>> 176.036
>>> ± 2.121 ops/s
>>>
>>> Test 2b: Creating a single 1000x1000 matrix (to show it scales
>>> w/N):
>>> [info] MatrixBenchmarks.buildMC1000 thrpt 10 36.112 ± 2.775
>>> ops/s
>>> [info] MatrixBenchmarks.buildSB1000 thrpt 10 45.557 ± 2.893
>>> ops/s
>>> [info] MatrixBenchmarks.buildOJ1000 thrpt 10 47.438 ± 1.370
>>> ops/s
>>> [info] MatrixBenchmarks.buildBMC1000 thrpt 10 37.779 ± 0.871
>>> ops/s
>>>
>>> 
>>>
>>> Test 3a: Inverting a random 100x100 matrix:
>>> [info] MatrixBenchmarks.invMC100 thrpt 10 1033.011 ± 9.928
>>> ops/s
>>> [info] MatrixBenchmarks.invSB100 thrpt 10 1664.833 ± 15.170
>>> ops/s
>>> [info] MatrixBenchmarks.invOJ100 thrpt 10 1174.044 ± 52.083
>>> ops/s
>>> [info] MatrixBenchmarks.invBMC100 thrpt 10 1043.858 ± 22.144
>>> ops/s
>>>
>>> Test 3b: Inverting a random 500x500 matrix:
>>> [info] MatrixBenchmarks.invMC500 thrpt 10 9.089
>>> ±
>>> 0.284 ops/s
>>> [info] MatrixBenchmarks.invSB500 thrpt 10 43.857
>>> ±
>>> 1.083 ops/s
>>> [info] MatrixBenchmarks.invOJ500 thrpt 10 23.444
>>> ±
>>> 1.484 ops/s
>>> [info] MatrixBenchmarks.invBMC500 thrpt 10 9.156 ± 0.052
>>> ops/s
>>> [info] MatrixBenchmarks.invCacheBMC500 thrpt 10 9.627 ± 0.084
>>> ops/s
>>> [no matrix creation]
>>>
>>> Test 3c:
>>> [info] MatrixBenchmarks.invMC1000 thrpt 10 0.704 ± 0.040
>>> ops/s
>>> [info] MatrixBenchmarks.invSB1000 thrpt 10 8.611 ±
>>> 0.557
>>> ops/s
>>> [info] MatrixBenchmarks.invOJ1000 thrpt 10 3.539 ± 0.229
>>> ops/s
>>> [info] MatrixBenchmarks.invBMC1000 thrpt 10 0.691 ± 0.095
>>> ops/s
>>>
>>> Also, isn't matrix multiplication at least O(N^2.37), rather than
>>> O(N^2)?
>>>
>>> On 6 April 2016 at 12:55, Chris Lucas <clucas2@inf.ed.ac.uk> wrote:
>>>
>>> Thanks for the quick reply!
>>>>
>>>> I've pasted a small selfcontained example at the bottom. It
>>>> creates the
>>>> matrices in advance, but nothing meaningful changes if they're
>>>> created
>>>> on a
>>>> peroperation basis.
>>>>
>>>> Results for 50 multiplications of [size]x[size] matrices:
>>>> Size: 10, total time: 0.012 seconds, time per: 0.000 seconds
>>>> Size: 100, total time: 0.062 seconds, time per: 0.001 seconds
>>>> Size: 300, total time: 3.050 seconds, time per: 0.061 seconds
>>>> Size: 500, total time: 15.186 seconds, time per: 0.304 seconds
>>>> Size: 600, total time: 17.532 seconds, time per: 0.351 seconds
>>>>
>>>> For comparison:
>>>>
>>>> Results for 50 additions of [size]x[size] matrices (which should
>>>> be
>>>> faster, be the extent of the difference is nonetheless striking to
>>>> me):
>>>> Size: 10, total time: 0.011 seconds, time per: 0.000 seconds
>>>> Size: 100, total time: 0.012 seconds, time per: 0.000 seconds
>>>> Size: 300, total time: 0.020 seconds, time per: 0.000 seconds
>>>> Size: 500, total time: 0.032 seconds, time per: 0.001 seconds
>>>> Size: 600, total time: 0.050 seconds, time per: 0.001 seconds
>>>>
>>>> Results for 50 inversions of a [size]x[size] matrix, which I'd
>>>> expect to
>>>> be slower than multiplication for larger matrices:
>>>> Size: 10, total time: 0.014 seconds, time per: 0.000 seconds
>>>> Size: 100, total time: 0.074 seconds, time per: 0.001 seconds
>>>> Size: 300, total time: 1.005 seconds, time per: 0.020 seconds
>>>> Size: 500, total time: 5.024 seconds, time per: 0.100 seconds
>>>> Size: 600, total time: 9.297 seconds, time per: 0.186 seconds
>>>>
>>>> I hope this is useful, and if I'm doing something wrong that's
>>>> leading to
>>>> this performance gap, I'd love to know.
>>>>
>>>> 
>>>> import org.apache.commons.math3.linear.LUDecomposition;
>>>> import org.apache.commons.math3.linear.Array2DRowRealMatrix;
>>>> import org.apache.commons.math3.linear.RealMatrix;
>>>>
>>>>
>>>> public class AMCMatrices {
>>>>
>>>> public static void main(String[] args) {
>>>> miniTest(0);
>>>> }
>>>>
>>>> public static void miniTest(int tType) {
>>>> int samples = 50;
>>>>
>>>> int sizes[] = { 10, 100, 300, 500, 600 };
>>>>
>>>> for (int sI = 0; sI < sizes.length; sI++) {
>>>> int mSize = sizes[sI];
>>>>
>>>> org.apache.commons.math3.linear.RealMatrix m0 =
>>>> buildM(mSize,
>>>> mSize);
>>>> RealMatrix m1 = buildM(mSize, mSize);
>>>>
>>>> long start = System.nanoTime();
>>>> for (int n = 0; n < samples; n++) {
>>>> switch (tType) {
>>>> case 0:
>>>> m0.multiply(m1);
>>>> break;
>>>> case 1:
>>>> m0.add(m1);
>>>> break;
>>>> case 2:
>>>> new LUDecomposition(m0).getSolver().getInverse();
>>>> break;
>>>> }
>>>>
>>>> }
>>>> long end = System.nanoTime();
>>>>
>>>> double dt = ((double) (end  start)) / 1E9;
>>>> System.out.println(String.format(
>>>> "Size: %d, total time: %3.3f seconds, time per: %3.3f
>>>> seconds",
>>>> mSize, dt, dt / samples));
>>>> }
>>>> }
>>>>
>>>> public static Array2DRowRealMatrix buildM(int r, int c) {
>>>> double[][] matVals = new double[r][c];
>>>> for (int i = 0; i < r; i++) {
>>>> for (int j = 0; j < c; j++) {
>>>> matVals[i][j] = Math.random();
>>>> }
>>>> }
>>>> return new Array2DRowRealMatrix(matVals);
>>>> }
>>>> }
>>>>
>>>> 
>>>>
>>>> On 5 April 2016 at 19:36, Gilles <gilles@harfang.homelinux.org>
>>>> wrote:
>>>>
>>>> Hi.
>>>>>
>>>>> On Tue, 5 Apr 2016 15:43:04 +0100, Chris Lucas wrote:
>>>>>
>>>>> I recently ran a benchmark comparing the performance math commons
>>>>>> 3.6.1's
>>>>>> linear algebra library to the that of scala Breeze (
>>>>>> https://github.com/scalanlp/breeze).
>>>>>>
>>>>>> I looked at det, inverse, Cholesky factorization, addition, and
>>>>>> multiplication, including matrices with 10, 100, 500, and 1000
>>>>>> elements,
>>>>>> with symmetric, nonsymmetric, and nonsquare cases where
>>>>>> applicable.
>>>>>>
>>>>>>
>>>>> It would be interesting to add this to the CM documentation:
>>>>> https://issues.apache.org/jira/browse/MATH1194
>>>>>
>>>>> In general, I was pleasantly surprised: math commons performed
>>>>> about as
>>>>>
>>>>>> well as Breeze, despite the latter relying on native libraries.
>>>>>> There
>>>>>> was
>>>>>> one exception, however:
>>>>>>
>>>>>> m0.multiply(m1)
>>>>>>
>>>>>> where m0 and m1 are both Array2DRowRealMatrix instances. It
>>>>>> scaled very
>>>>>> poorly in math commons, being much slower than nominally more
>>>>>> expensive
>>>>>> operations like inv and the Breeze implementation. Does anyone
>>>>>> have a
>>>>>> thought as to what's going on?
>>>>>>
>>>>>>
>>>>> Could your provide more information such as a plot of performance
>>>>> vs
>>>>> size?
>>>>> A selfcontained and minimal code to run would be nice too.
>>>>> See the CM microbenchmarking tool here:
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> https://github.com/apache/commonsmath/blob/master/src/test/java/org/apache/commons/math4/PerfTestUtils.java
>>>>> And an example of how to use it:
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> https://github.com/apache/commonsmath/blob/master/src/userguide/java/org/apache/commons/math4/userguide/FastMathTestPerformance.java
>>>>>
>>>>> In case it's useful, one representative test
>>>>>
>>>>>> involves multiplying two instances of
>>>>>>
>>>>>> new Array2DRowRealMatrix(matVals)
>>>>>>
>>>>>> where matVals is 1000x1000 entries of math.random and the second
>>>>>> instance
>>>>>> is created as part of the loop. This part of the benchmark is
>>>>>> not
>>>>>> specific
>>>>>> to the expensive multiplication step, and takes very little time
>>>>>> relative
>>>>>> to the multiplication itself. I'm using System.nanotime for the
>>>>>> timing,
>>>>>> and
>>>>>> take the average time over several consecutive iterations, on a
>>>>>> 3.5 GHz
>>>>>> Intel Core i7, Oracle JRE (build 1.8.0_05b13).
>>>>>>
>>>>>>
>>>>> You might want to confirm the behaviour using JMH (becoming the
>>>>> Java
>>>>> standard
>>>>> for benchmarking):
>>>>> http://openjdk.java.net/projects/codetools/jmh/
>>>>>
>>>>>
>>>>> Best regards,
>>>>> Gilles
>>>>>
>>>>>
>>>>> Thanks,
>>>>>>
>>>>>> Chris
>>>>>>

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