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times faster at multiplying 128x128 matrices. It's time is pretty much the same as that block-multiplication for 256x256 and 512x512 matrices. For 1024x1024 and larger matrices (when the entire problem set no longer fits into processor's L2 cache), Strassen's algorithm is 3 times faster than block-multiplication and 12 times as fast as the classical three nested-loop implementation.
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While a lot of older textbooks will say that "additions don't matter, but multiplications do". That is usually not relevant anymore. Multiplying numbers is slower if you are working with multi-word representations of numbers (sometimes called 'big numbers'), but if you are just multiplying "float"s
1504:
The section entitled "Numerical
Analysis" should be "Performance Analysis". Numerical Analysis could include an analysis of any of: numerical stability, error accumulation, bit accuracy (vs. precision), or the correctness of the algorithm (though that may be a stretch). It does mention the first
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A popular misconception is that
Strassen's algorithm outperforms the classic multiplication approach only for large matrix sizes. This is not true for modern architectures due to Strassen algorithm's superior cache behavior. For example, on a 2GHz Core Duo processor, Strassen's algorithm is three
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An example of this actually being implemented would be a nice addition to the article. I searched the web and didn't find a single example of this algorithm actually being implemented. Sure you can talk about the algorithm all day, but it means nothing if no one can put it into code.
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The first
Strassen-like algorithm — in the sense of a computational scheme for multiplication of matrices of a fixed size that uses clever tricks to reduce the number of element multiplications needed — to achieve better asymptotic complexity than Strassen's original algorithm
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or "double"s then multiplications and additions take the same amount of time on most processors (as far as i know) today. I'm going to change the article to say, "if we only consider multiplications" without the statement bout multiplications being inherently slower.... :)
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Yes, it depends on the sizes of the numbers being multiplied. Perhaps a clarification of this point should be added to the article? I'm not familiar with the size of the matrix components used in most matrix algorithms so I will defer this point to the experts. -
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section claims that the form has 15 additions or subtractions, but the algorithm shown has 22. I could combine identical operations by introducing new variables but this might be seen as original research. How should we proceed? Cheers,
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It might be true, but it does need a reference, especially since these kind of measurements are typically very dependent on details and it is contradicted by other references like Golub and Van Loan's Matrix
Computations. --
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This article may require cleanup to meet
Knowledge's quality standards. The specific problem is: the article contains vague statements and lacks references to work it mentions and to the considerable relevant work that it
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I am unfamiliar with the use of blackboard bold to denote an arbitrary field, as I understand it to mean one of the canonical sets of numbers (e.g., the complex numbers, the integers, etc.). Shouldn't the article use
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I was looking at the code and see that this version is a memory hog. Could it be more optimized if it were using location references instead of creating a new submatrix to perform each of the operations on?
1505:(numerical stability) but there is no actual analysis, just a mention and a general link. I also think that there is some problem with the performance analysis, but I will go into that separately. --
779:). Hence the number of additions in a Strassen-type matrix multiplication algorithm only affects the constant factor, but the number of multiplications affect the exponent. See section 2.4 of
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The larger importance of multiplications becomes apparent when one starts to use the algorithm recursively, since the additions and multiplications at all but the bottom levels are then
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The "Ultra-fast matrix multiplication" reference in the article describes another matrix multiplication attributed to
Winograd, which has asymptotic complexity O(
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https://www.technologyreview.com/2022/10/05/1060717/deepmind-uses-its-game-playing-ai-to-best-a-50-year-old-record-in-computer-science/
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Yes, the reduction in complexity is a purely mathematical result independent of any computer implementation. I've fixed the article.
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I"m sorry, but I don't see where the formula at MathWorld differs from the formula here. Could you please be more specific? --
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related articles on
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DeepMind has used its board-game playing AI AlphaZero to discover a faster way to multiply matrices, see
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Agreed. A crossover point certainly exists, but is hardware dependent. I'll put something fluffier in.
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1531:(MCA) by Joachim von zur Gathen and Jurgen Gerhard for EXTENSIVE discussion of Strassen algorithm.
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at hand so I can check; possibly
Strassen's contribution was to push the exponent below 2.5.)
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This is mathematical thing, not a technical article. It states theoretical bounds.--
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That time period also saw many other developments which are worth mentioning:
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It is asymptotically faster than the standard matrix multiplication algorithm
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for integer multiplication), which reduced the exponent to below 2.5.
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1277:), not 1980. It has the same asymptotic complexity as Strassen's:
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Schönhage, A. (1981). "Partial and total matrix multiplication".
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Pan, V. Ya. (1980). "New fast algorithms for matrix operations".
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Little bit necro but I think the formula is wrong indeed. Either
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additions and multiplications. A matrix multiplication is near O(
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The Strassen-Schönhage algorithm (which is not the same as the
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2.807 is a blunder. Right 2.8074 or 2.81. Very simple Russion
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Find pictures for the biographies of computer scientists (see
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Doing a simple case by hand, the wikipedia ones seem wrong.
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Winograd, S. (1971). "On Multiplication of 2×2 Matrices".
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instead. I changed the article accordingly. And I changed
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Looking at Mathworld, I see a different formula for M1:
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Put in citations to MCA and to relevant work it cites.
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Thanks for your comments. I fixed the references. --
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7 multiplications and fewer additions than Strassen
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353:Tag all relevant articles in
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1442:
1441:81.231.33.217
1433:
1431:3-540-17205-X
1428:
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1396:(3): 434–455.
1395:
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503:(assessed as
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1393:
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1328:
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1260:
1233:Jitse Niesen
1229:
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1193:formula. --
942:Jitse Niesen
925:
908:
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657:
654:Broken links
630:
606:Low-priority
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531:Low‑priority
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40:WikiProjects
1507:RBarryYoung
1261:Winograd's
1215:I reverted
833:instead of
634:—Preceding
581:Mathematics
572:mathematics
528:Mathematics
1804:Categories
1465:What does
1275:: 381–388.
987:should be
913:MathMartin
1413:ignored (
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1379:0097-5397
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452:Computing
439:computing
435:computers
401:Computing
216:Computing
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636:unsigned
406:Software
264:Maintain
207:Copyedit
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1559:ignores
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1520:cleanup
1158:in the
608:on the
479:on the
245:Infobox
183:Cleanup
123:on the
30:C-class
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