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    $\begingroup$ @ShutaoTang does the update answer your question about "how to store and organize the intermediate result?" (Namely, don't copy it around.) $\endgroup$ Commented Apr 4, 2015 at 4:15
  • $\begingroup$ You can use Compile`GetElement to speed up doolittleDecomposition by a factor of 2. (See here for an example. ) $\endgroup$ Commented Apr 4, 2015 at 4:25
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    $\begingroup$ +1Nice answer,thanks a lot. In fact, I'd like to implement this algorithm in functional paradigm(like Nest, because I think functional paradigm is more efficient) at the beginning. However, I didn't know how to organize the intermediate result(like $u_{11},u_{12} \cdots u_{1n}$). So I use the Do(I am always thinking it is not efficient in Mathematica) to implement it. In addition, @Daniel Lichtblau gave the suggestion(changing the Sum to dot products), but I forgot to use the built-in functionSpan(;;) . So your edit fo my original code is very elegant:) $\endgroup$ Commented Apr 4, 2015 at 4:35
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    $\begingroup$ @ShutaoTang The Doolittle algorithm is essentially procedural: shoehorning it into a functional construct will not make it faster. The reason we advise a functional style most of the time is that builtins are usually faster; but there are no builtins that apply to this case. I believe my testing shows that here, Do is faster than Nest. $\endgroup$ Commented Apr 4, 2015 at 4:40
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    $\begingroup$ @Alittle LU decomposition is in general not unique, so if the built-in function uses a different algorithm it can yield a different result. $\endgroup$ Commented Sep 20, 2020 at 18:13