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I want to tridagonalize a sparse matrix using the Lanczos algorithm. I am working with a very large sparse matrix and I need some speed. In the following for example H=SparseArray[#->RandomComplex[{-1-I,1+I}]&/@RandomInteger[{1,1000000},{5000000,2}]]// (# + #\[ConjugateTranspose])/2 &; is my sparse matrix. I cannot use Compail as my code include a sparse array. Is there any way to speed up the code?

abGenerator[niter_] := 
 Module[{ket0, ket1, ketm, ketk, alist, blist},
  ket0 =Normalize@RandomComplex[{-1-I,1+I},Length@H];
  alist = Table[0, niter];
  blist = Table[0, niter];
  ketk = H.ket0;
  alist[[1]] = ket0\[Conjugate].ketk;
  ket1 = ketk - alist[[1]] ket0;
  blist[[1]] = Norm@ket1;
  ket1 = ket1/blist[[1]];
  Do[
   ketm = ket0;
   ket0 = ket1;
   ketk = H.ket0;
   alist[[i]] = ket0\[Conjugate].ketk;
   ket1 = ketk - alist[[i]] ket0 - blist[[i - 1]]\[Conjugate] ketm;
   blist[[i]] = Norm@ket1;
   ket1 = ket1/blist[[i]];, {i, 2, niter}];
  {alist,
   blist}]
abGenerator[100]//AbsoluteTiming

For me, this takes around 16.28 seconds! Note that I don't need this code for diagonalization and I need it for other purposes!

Update:

For confirmation consider the following matrix,

H = SparseArray[# -> RandomComplex[{-1 - I, 1 + I}] & /@ 
 RandomInteger[{1, 10000}, {50000, 
   2}]] // (# + #\[ConjugateTranspose])/2 &;

Now I can find its eigenvalue using Eigenvalue

Eigenvalues[H, 3, Method -> "Arnoldi"]

Which for me it gives [each run is different because the H is different!]

{-2.7889, 2.78153, 2.76776}

However, I can get the same result if I reconstruct the three-diagonal matrix and get its eigenvalue!

    Eigenvalues[
 abGenerator[100] // 
  DiagonalMatrix[SparseArray@#[[1]]] + 
    DiagonalMatrix[SparseArray@#[[2, 1 ;; -2]], 1] + 
    DiagonalMatrix[SparseArray@#[[2, 1 ;; -2]]\[Conjugate], -1] &, 3]

which gives,

{-2.7889, 2.78153, 2.76776}
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  • $\begingroup$ How we know that your code doing right job? $\endgroup$
    – Alex Trounev
    Commented Oct 22, 2021 at 4:07
  • $\begingroup$ @AlexTrounev, you are right. I update my question! $\endgroup$
    – Rasoul-Ghadimi
    Commented Oct 25, 2021 at 0:55
  • 1
    $\begingroup$ Now it is good looking post (+1). Nevertheless, you question is not so clear, since you mentioned that you don't need this code for diagonalization and you need it for other purposes. What are these purposes for? $\endgroup$
    – Alex Trounev
    Commented Oct 25, 2021 at 6:09
  • $\begingroup$ Thanks, @AlexTrounev Actually this code is used for obtaining Green Function. $\endgroup$
    – Rasoul-Ghadimi
    Commented Oct 26, 2021 at 6:42
  • $\begingroup$ Could your show an example of Green function computed with this code? $\endgroup$
    – Alex Trounev
    Commented Oct 26, 2021 at 16:07

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