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Sample Problem of a large block matrix:

I want to find the lowest eigenvalue of a blockdiagonal matrix h[x] for a range of values of x

  • To save the time, I do so by finding the lowest eigenvalue of each block individually and then lowest out of them but

Individual block diagonalizations take more time than single diagonalization of block matrix h[x] for a range of values of x.

b0 = RandomReal[{-1, 1}, {16, 16}]; c0 = RandomReal[{-1, 1}, {64, 64}]; d0 = RandomReal[{-1, 1}, {128, 128}]; e0 = RandomReal[{-1, 1}, {32, 32}];
b = b0 + Transpose[b0];
c = c0 + Transpose[c0];
d = d0 + Transpose[d0];
e = e0 + Transpose[e0];
h[x_] := x*SparseArray[Band[{1, 1}] -> {b, c, d, e}];
AbsoluteTiming[ParallelTable[-Eigenvalues[-h[x], 1, Method -> {"Arnoldi", "Criteria" -> "RealPart"}][[1]], {x, 0.1,5., 0.1}];][[1]]
blocks[x_] :=ConnectedComponents@AdjacencyGraph@Unitize@h[x]
eigbloc0[x_, bl_] := -Eigenvalues[-h[x][[#, #]], 1, Method -> {"Arnoldi", "Criteria" -> "RealPart"}][[1]] & /@ {blocks[x][[bl]]}
eigbloc1[x_] := Table[eigbloc0[x, bl], {bl, 1, 4, 1}]
AbsoluteTiming[ParallelTable[Min[Flatten[{eigbloc1[x]}]], {x, 0.1, 5., 0.1}];][[1]]

Output:

7.98266

11.7006

I was expecting lower time for individual block diagonalization, Could anyone suggest me the resolution ?

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  • $\begingroup$ One issue is that you rebuild the full matrix each time you want to compute only a block, because eigbloc0 calls h. Ideally, you know the blocks already independently of the construction of the matrix -- because you know how to construct the matrix. $\endgroup$
    – Henrik Schumacher
    Sep 11, 2021 at 7:59
  • $\begingroup$ @HenrikSchumacher Yes, For a single computation, say h[x=1.0], Individual digonalization procedure is faster. Could you please tell me how to avoid the repetitive computations in above program ? $\endgroup$
    – Quantum_Magnet
    Sep 11, 2021 at 8:57
  • 1
    $\begingroup$ Try this: ClearAll[blocks]; blocks = Range[#1 + 1, #2] & @@@ Partition[Accumulate[{0, 16, 64, 128, 32}], 2, 1]; eig[x_] := Module[{A}, A = -h[x]; Min@Table[ -Eigenvalues[A[[block, block]], 1, Method -> {"Arnoldi", "Criteria" -> "RealPart"}], {block, blocks} ] ]; $\endgroup$
    – Henrik Schumacher
    Sep 11, 2021 at 9:09
  • $\begingroup$ @HenrikSchumacher, You are wonderful, Worked! $\endgroup$
    – Quantum_Magnet
    Sep 11, 2021 at 9:20

1 Answer 1

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First issue is that you compute the matrix each time you try to compute within a block. Second issue is that you recompute all the blocks themselves.

So this was just not a fair comparison. Have a look at this example. Even without parallelization, computing the eigenvalues of the dense diagonal blocks is two orders of magnitude faster:

n = 100;
diagonal = # + #\[Transpose] &[RandomReal[{-1, 1}, {#, #}]] & /@ 
   RandomInteger[{120, 240}, {n}];
A0 = SparseArray[Band[{1, 1}] -> diagonal];


val1 = -Max[Eigenvalues[-A0, 1, 
       Method -> {"Arnoldi", "Criteria" -> "RealPart"}]]; // 
  AbsoluteTiming // First
val2 = -Max[Eigenvalues[-#, 1, 
         Method -> {"Arnoldi", "Criteria" -> "RealPart"}] & /@ 
       diagonal]; // AbsoluteTiming // First
val1 == val2

5.35384

0.063925

True

I guess the main speedup here comes from the fact that replace a the sparse matrix vector multiplication kernel by many small, highly efficient dense matrix-vector multiplication kernels. (Sparse matrix multiplication typically has quite a "random" memory access pattern, so the compute kernels do not optimize for contiguous memory access (because they cannot).)

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