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Recently I tried to do what I thought was a fairly small (relative to the 6 GB of RAM that I have on my machine) row reduction calculation on a matrix representing an undetermined linear system and was very surprised by how much trouble I had with it. The 2590 X 12155 matrix, A (available for download via Mediafire), takes a little over 17 MB to store, has rational entries, and is very sparse (~3% of the entries non-zero). The data structure I used to store A is Mathematica's rule-based representation for sparse arrays on a row-by-row basis (if this is unclear please see the example matrix from my last post).

Curiously, I find that using RowReduce to find a row echelon form for A leaves a huge memory footprint well beyond what Mathematica reports with MemoryInUse. I was able to observe this effect without leaving Mathematica in an (to me miraculous) allegedly platform-independent way by using the answer that @WReach gave here in response to a question about memory monitoring. I began with

In[1] := $HistoryLength = 0;

to turn off caching and then set up a pair of commands which give respectively my machine's and Mathematica's estimates of how much memory is currently in use using the ideas of @WReach:

In[2] := Needs["JLink`"]
         InstallJava[];
         LoadJavaClass["java.lang.management.ManagementFactory"];

In[5] := temp = JavaBlock[{#,java`lang`management`ManagementFactory
         `getOperatingSystemMXBean[]
         @#[]} & /@ {getFreePhysicalMemorySize, 
         getTotalPhysicalMemorySize}]; memstart1 = 
         temp[[2, 2]] - temp[[1, 2]]

Out[5] := 3096420352

In[6] := memstart2 = MemoryInUse[]

Out[6] := 18286664

In[7] := temp = JavaBlock[{#, 
         java`lang`management`ManagementFactory`getOperatingSystemMXBean[]
         @#[]} & /@ {getFreePhysicalMemorySize, 
         getTotalPhysicalMemorySize}]; memnow1 = 
         temp[[2, 2]] - temp[[1, 2]] - memstart1

Out[7] := 94208

In[8] := memnow2 = MemoryInUse[] - memstart2

Out[8] := 1808

I've found that copying and running the preceding two lines of code wherever I want a memory sanity-check seems to give a fairly consistent way to compare how much memory Mathematica has actually used since the start of the session and how much memory Mathematica thinks it has used since the start of the session (the numbers 94208 and 1808 from the initialization above should be thought of as being effectively zero). Although it is probably obvious, I have of course assumed that the amount of RAM used by all other programs is constant throughout the session (which should be approximately correct if they are just running idly in the background).

Now I read in the file containing the matrix and assign it to A:

In[9] :=  f = OpenRead["rrdemo"];
          A = Read[f];
          Close[f];

I also go ahead and define an array that will be used momentarily (recall that 12155 is the number of columns in A)

In[12] := W = Array[w, 12155];

Actually, although it's a bit off-topic, it's worth pointing out that there already seems to be some missing memory associated with the process of reading in the file "rrdemo" (in a separate session I checked that there is no missing memory associated with the definition of W):

In[13] := temp = JavaBlock[{#, 
          java`lang`management`ManagementFactory`getOperatingSystemMXBean[]
          @#[]} & /@ {getFreePhysicalMemorySize, 
          getTotalPhysicalMemorySize}]; memnow1 = 
          temp[[2, 2]] - temp[[1, 2]] - memstart1

Out[13] := 425725952

In[14] := memnow2 = MemoryInUse[] - memstart2

Out[14] := 268374384

This is actually nothing at all compared to what happens with RowReduce. To obtain a row echelon form for my matrix I use RowReduce in a straightforward way on a sparse array. To show that the problem is not with my generation of the sparse array, I first check the memory footprint of that step of the calculation:

In[15] := AbsoluteTiming[C1 = CoefficientArrays[Plus @@@ (#[[2]] w @@ 
          #[[1]] & /@ # & /@ A), W][[2]];]

Out[15] := {5.447995, Null}

In[16] := temp = JavaBlock[{#,java`lang`management`ManagementFactory
          `getOperatingSystemMXBean[]@#[]} & /@ {getFreePhysicalMemorySize, 
          getTotalPhysicalMemorySize}]; memnow1 = 
          temp[[2, 2]] - temp[[1, 2]] - memstart1

Out[16] := 539504640

In[17] := memnow2 = MemoryInUse[] - memstart2

Out[17] := 385560664

In[18] := ByteCount[C1]

Out[18] := 100604400

The performance of Mathematica here seems reasonable. The actual and reported memory usage go up from what they were before in proportion to the ByteCount of C1.

Now look what happens if we RowReduce C1:

In[19] := AbsoluteTiming[C2 = Drop[ArrayRules[#], -1] & /@ 
          RowReduce[C1, Method -> "OneStepRowReduction"];]

Out[19] := {402.172945, Null}

In[20] := temp = JavaBlock[{#, 
          java`lang`management`ManagementFactory`getOperatingSystemMXBean[]
          @#[]} & /@ {getFreePhysicalMemorySize, 
          getTotalPhysicalMemorySize}]; memnow1 = 
          temp[[2, 2]] - temp[[1, 2]] - memstart1

Out[20] := 2461675520

In[21] := memnow2 = MemoryInUse[] - memstart2

Out[21] := 731499200

In[22] := ByteCount[C2]

Out[22] := 280479264

Something seems to be seriously wrong here. The ByteCount of C2 is only ~ 280 Mb and this is close to the increase in the total memory usage reported by Mathematica. But the actual memory usage is ~3.3 times what Mathematica thinks it is! I'm forced to conclude that Mathematica fails to garbage collect more than 1.5 Gb of memory at the end of the calculation.

In summary, I have a couple of related questions. Does anybody have any insights into the anomalous behavior of RowReduce that I just illustrated? Am I doing something wrong? More generally, does anybody have any idea as to whether or not Mathematica has special algorithms for symbolic sparse linear algebra? Memory issues aside, it makes me suspicious that RowReduce takes so long to process C1 (I have a 2.5 GHz processor) and that it returns a result that is no longer a sparse array. At this point I'm wondering whether Mathematica is even the right tool for the job (symbolic linear algebra on large, very sparse undetermined linear systems). Any advice from more experienced users would be greatly appreciated. If it makes any difference, I'm using 64 bit Linux and am running Mathematica v.8.0.4.

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1  
If you believe you have a memory leak, please report it to wolfram technical support by email: wolfram.com/support/contact –  Searke Jun 12 '12 at 19:56
1  
Well, I think that, at a minimum, it would be nice to check whether my issue can reproduced on other platforms before I file a bug report. I'm also interested in hearing from the various experts on this site. I have gotten the impression from reading various posts on this site that memory leaks in Mathematica are very rare occurrences. Since I am (very) new to serious Mathematica programming it is entirely possible that I am doing something wrong. –  Rob2181 Jun 12 '12 at 20:58
    
Could somebody with a Mac please check if this issue is reproducible with a Mac OS? I have access to a Windows box but it might be quite some time before I have access to a Mac. –  Rob2181 Jun 13 '12 at 9:35
    
I just tried it on my Windows machine and the performance was even worse than on a Linux box (no real surprise there). I wasn't even able to run RowReduce on C1 twice in a row. –  Rob2181 Jun 14 '12 at 15:59
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1 Answer 1

up vote 9 down vote accepted

There might be a memory leak. More likely is that memory was used at intermediate steps and, while (I hope) freed by Mathematica, was not returned to the OS.

As for why massive memory will be consumed at all, I got into a debug version of the Mathematica kernel and had a look. By row 850 or so, in the forward elimination step, I see numbers going around in the range of 1300 bytes. As there are quite a few of these it is no surprise that memory usage grows. My guess is there is some amount of fragmentation as ever larger integers get generated in the elimination.

I guess one way to get a better idea of whether memory is made available for further use would be to clear caches and repeat the row reduction. Or do it with a different matrix of similar size, if you happen to have one available. Then check to see if the max memory consumption went up again by a significant amount. If so, that might indicate reuse was either not happening at all, or else not happening in the way one would expect.

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Thanks again for getting back to me so quickly. I would be happy to experiment some more but I'm not entirely sure I understood your suggestion. When you say clear caches what do you mean exactly? I thought caching was turned off from the beginning of my Mathematica session since I set $HistoryLength to zero. –  Rob2181 Jun 12 '12 at 22:45
    
I meant use ClearSystemCache[] (just as I typed it there). Then rerun or run another RowReduce[...] –  Daniel Lichtblau Jun 12 '12 at 23:26
    
Interesting. I just read up a bit on ClearSystemCache and related commands. So, if I understand correctly, there are really two types of caching in Mathematica. Roughly speaking there is an "internal" cache and an "external" cache ($HistoryLength = 0 only turns off the latter of the two). If I execute the command SetSystemOptions[ "CacheOptions" -> {"CacheKeyMaxBytes" -> 0, "CacheResultMaxBytes" -> 0}] at the beginning of my notebook as well will this completely disable internal caching or is there something more that needs to be done beyond that? –  Rob2181 Jun 12 '12 at 23:50
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I went ahead and ran the test that you suggested over night. The results were extremely interesting. I assume that the command in my last comment successfully turned internal caching off because running RowReduce on C1 took ~ 378 seconds. The memory utilized but not detected by Mathematica does go up dramatically after the second run of RowReduce (actual memory usage 3117580288, reported memory usage 731498328) if one runs the whole notebook in one go. However, when I measured the memory usage again this morning I found that the actual and reported numbers coincided –  Rob2181 Jun 13 '12 at 9:08
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(actual usage 740745216, reported usage 731498336)! I think this means that your analysis was right on target. The memory allocated for the job is released by Mathematica once the job finishes running but there seems to be a very significant lag between when the memory is deallocated and when it is returned to the OS. Unfortunately, this behavior seems to seriously hamper Mathematica's ability to do heavy calculations in real time; for example, evaluating AbsoluteTiming[C1 = CoefficientArrays[Plus @@@ (#[[2]] w @@ #[[1]] & /@ # & /@ A), W][[2]];] 3 times in a row would crash my computer. –  Rob2181 Jun 13 '12 at 9:14
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