The Toad
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 Jan4 comment Speed up replacement using Rule Dispatch Associtation I'm not surprised. It's not unconceivable that Transpose[DD1].DD1 is more complex (in terms of leaf count and computations involved) than just DD1. What I said in my previous two comments was that you should go back to the drawing board and see if you can use mathematics to simplify the problem before throwing it to Mathematica. Since we don't know the origins of your problem or how you arrive at kappa, we cannot help you there (nor is this necessarily the site for that). Jan4 comment Speed up replacement using Rule Dispatch Associtation Note that the rank of your matrix $X$ is bounded by the number of columns which is much less than the number of rows. Does your formulation permit an easy calculation of $X^\top X$ which is only $30\times 30$ (or $100\times 100$ in the bigger one)? If so, then you can try and compute the rank of that, as it should be the same as the rank of $X$. Assuming that all elements of $X\in \mathbb{R}^{n\times m}, n \gg m$ take roughly the same time to compute and the time taken to compute an element of $X$ is the same as $X^\top X$, then you've gone from having to compute $O(n m)$ elements to $O(m^2)$. Jan4 comment Speed up replacement using Rule Dispatch Associtation I think this is a dead-end in terms of having a general answer. Suggestions and tricks that might help speed this up will likely depend on how the kappa expression is obtained, about which you don't say anything. We're effectively being asked to analyze and optimize a blackbox expression. You should perhaps consider stepping away from Mathematica for a bit and turning to mathematics to try and simplify the problem. Jan4 comment Rapid density estimation and rendering of decision regions One thing you could do when using Manipulate with compute-intensive demonstrations is to use ControlActive with the variable being changed. That way, the evaluation doesn't try to kick in midway during the slider adjustment and the plot is redrawn only when the slider is released. It doesn't address the inherent slowness of the computation, but does give a semblance of interactivity and overall less choppiness. Jan3 comment Can a System symbol spontaneously leak into a Global namespace? To give an example, see this discussion between me, Leonid and Rojo. In the end, the mistake was mine — I hadn't fully isolated all my private symbols and a few of them happened to get exposed only in some very specific order of internal function calls. The point I'm trying to make is that this bug (albeit minor) crept into a smallish but non-trivial package that had extensive unit-tests and was identified only a year later. It is very likely that there are hundreds if not thousands of these in the Mathematica code base. Jan3 comment Can a System symbol spontaneously leak into a Global namespace? Re: A – I have seen behavior like this, yes. Mathematica is a complex piece of software and some hard to detect bugs like this can often come up even in internal code. Re: B — As you say, it's hard to produce a minimal example for bugs like this (understandable), but it also makes it hard for us to try and troubleshoot. The question then becomes: Is this affecting your work? If so, how? That would give us a concrete path to proceed. If not, then I would suggest ignoring it or silencing it. Dec29 reviewed Leave Open RegionPlot fails when given function with internally redefined variables Dec28 awarded Nice Answer Dec22 revised Interpretation of CopulaDistribution added 2 characters in body; edited tags Dec22 reviewed Close Interpretation of CopulaDistribution Dec21 reviewed Close solving singular matrix Dec21 reviewed Close Why is Likelihood / LogLikelihood not working? Dec21 reviewed Close import from Excel adds a point to each number …How to get rid of it? Dec21 comment import from Excel adds a point to each number …How to get rid of it? "but lacks the resources to fix." — have you tried telling WRI that you're doing Tweetable data science of the computational world using the Internet of Things in Excel? Maybe you'll get it fixed faster that way ;) Dec21 reviewed Close MathematicaPolarPlot Question Dec21 comment How can I use ContourPlot to creat streamline for a free surface domian: proper arrow orientation and correct domain? If you include a space after @ then the user will not be pinged. You should ping like so @Ixy Dec21 reviewed No Action Needed Determine the Number of Lines in a Text File Dec19 awarded Enlightened Dec19 awarded Nice Answer Dec15 awarded Announcer