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I have a program and I need to take the Dot product of many matrices, which is quite effectively working for small NStep. But for large(=30000), it is failing and saying. Either I use ParallelTable or not, problem remains.

$IterationLimit::itlim: Iteration limit of 4096 exceeded.

This is my program,

a = 1;
tI = 0.0001;
dt = 0.0001;
NStep=30000;
T1[t_] = j1 (Cos[t]); 

T2[t_] = j2 ;

cond = {j1 -> 0.9, j2 -> 1.};
HSm[t_] = ({{0, -(T1[t] + T2[t] Cos[k])}, {-(T1[t] + T2[t] Cos[k]),0}})//. cond;
HStDig[t_] = MatrixExp[t *DiagonalMatrix[Eigenvalues[HSm[t]]]]


us = ParallelTable[HStDig[tI + j dt], {j, 0, NStep}];
us1 = Apply[Dot, us];

How to go about it? Is there a way to speed up the program?

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    $\begingroup$ Isn't the MatrixExp of a diagonal matrix just the diagonal matrix made up of the elements of that matrix exponentiated? You might save some time putting that in explicitly rather than using MatrixExp. $\endgroup$
    – march
    Commented Apr 6, 2017 at 17:57
  • $\begingroup$ @march great idea. Thanks $\endgroup$
    – L.K.
    Commented Apr 6, 2017 at 17:59
  • $\begingroup$ I think the big issue is that you're trying to multiply 30000 expressions rather than 3000 numbers (since k is left as a symbol). This is going to be tough for Mathematica's symbolic processing. I would recommend choosing a value for k, and then doing the multiplication. Finally, since matrix multiplication of diagonal matrices is the same as element-wise multiplication of the matrices, do Times@@us rather than Dot@@us. If you insist on using Dot, you can do Block[{$IterationLimit = 30002}, Dot @@ us], but it will take longer than the Times method. $\endgroup$
    – march
    Commented Apr 6, 2017 at 18:06
  • $\begingroup$ Once again thanks @march. Let me go through it meticulously. k is supposed to be given value from a list of numbers $\endgroup$
    – L.K.
    Commented Apr 6, 2017 at 18:17

1 Answer 1

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To answer the specific question, I think that instead of Apply[Dot, l], Fold[Dot, l] will not run into any problems. Of course, all the other comments are valid.

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