I'd like to calculate the following one-dimensional array:

OneDimArray = Table[(Norm[Sum[ MatrixA[[i,j]]VectorB[[j]],{j,N}]])^2,{i,N}]

However this takes a very long time when N goes up; for N = 1500 it already takes ~40 minutes (on my MacBook Pro).

I've heard there are certain functions in Mathematica that are optimized for speed; could I employ those (and if so: which ones?) to speed up this calculation?

Thanks a lot in advance for any help!


  • 1
    $\begingroup$ I'm pretty sure you can just do Abs[Dot[MatrixA, VectorB]]^2. By the way, don't use capitalized words for user-defined symbols so as not to conflict with Mathematica's built-in functions, which all start with capital letters. $\endgroup$ – march Jan 4 '16 at 18:19

You have plenty of redundant operations here:

The equivalent code is (where a is a matrix and b is a list of 3-vectors):

Map[Norm[#]^2 &, a . b];

I have:

n = 500;
a = RandomReal[1, {n, n}];
b = RandomReal[1, {n, 3}];

oneDimArray = 
   Table[(Norm[Sum[a[[i, j]] b[[j]], {j, n}]])^2, {i, 
     n}]; // AbsoluteTiming
(* {46.5618, Null} *)

oneDimArrayMap = Map[Norm[#]^2 &, a.b]; // AbsoluteTiming
(* {0.000621823, Null} *)

oneDimArray == oneDimArrayMap
(* True *)

Thus, using Dot almost 70000 times faster than manual loops.

| improve this answer | |
  • $\begingroup$ Apologies; my question wasn't complete/correct: the vectorB I'm working with in this case is actually an array of 3-dimensional vectors -- and, to be as complete as possible now, the MatrixA the eigenvectors of a Hamiltonian of an excitonic vibrational system. (The physics, for context: To calculate the IR absorption intensities of the normal modes of such a system one has to first multiply the local mode vectors in VectorB with their respective eigenvector contribution, then sum these products to get the normal mode vector; the intensity of the mode is then given by norm[NormalModeVector]^2.) $\endgroup$ – Steven R Jan 5 '16 at 7:59
  • $\begingroup$ @StevenR Than you still can utilize the Dot with a simple Map on it. On random matrix-vector I have 70000 speed up. Try this code on your data and let us know. $\endgroup$ – m0nhawk Jan 5 '16 at 8:31
  • $\begingroup$ Wow, indeed, that gives a 45000-fold speed up for my specific matrices, thank you very much for the great advice m0nhawk! $\endgroup$ – Steven R Jan 5 '16 at 21:29
  • $\begingroup$ If you replace Norm[#]^2 & with #.# &, it should even be quicker. $\endgroup$ – J. M.'s technical difficulties Feb 22 '16 at 12:28
  • $\begingroup$ @J.M. I have tried it, but no performance improvements, timing is the same for me (in Mathematica 10.3). $\endgroup$ – m0nhawk Feb 22 '16 at 12:44

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