Wrong eigenvalues from a sparse matrix

Question

Bug introduced after 5.0, in or before 8.0 and persisting through 13.3.

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I notice in the following example that wrong smallest 2 eigenvalues are resulted if calculating from a sparse matrix. But it gives correct result if we

• calculate the smallest 3,4,... eigenvalues from the sparse matrix
• calculate any smallest eigenvalues from the corresponding normal matrix

I found many cases with this behavior. Why and any remedy?

In the code below, the 104×104 matrix in the minimal example I found is imported from Pastebin because it exceeds the length limit of a post.

Import["https://pastebin.com/raw/PpDfY3EQ", "Package"];
mysparsemat = mymat;
mymat = Normal[mysparsemat];
m = 2;
Reverse@First[Eigensystem[mymat, -m]]
Reverse@First[Eigensystem[mysparsemat, -m]]
m = 4;
Reverse@First[Eigensystem[mymat, -m]]
Reverse@First[Eigensystem[mysparsemat, -m]]

The result of the above code is

{-0.712477 + 1.02863*10^-16 I, 0.712477 + 1.46577*10^-16 I}

{0.712477 - 1.44294*10^-11 I, 0.712656 - 2.12258*10^-11 I}

{-0.712477 + 1.02863*10^-16 I, 0.712477 + 1.46577*10^-16 I, 
-0.712656 - 1.05578*10^-16 I, 0.712656 + 6.49144*10^-16 I}

{-0.712477 - 5.10777*10^-10 I, 0.712477 - 5.44863*10^-12 I, 
 0.712656 + 3.10198*10^-11 I, -0.712656 + 3.64677*10^-10 I}

So the resultant wrong 2nd smallest eigenvalue is actually the correct 3rd or 4th smallest eigenvalue. (The eigenvalues should be real and doubly degenerate in absolute value as expected from the original problem's nature).

Update
As you might have noticed in the comments below, a first guess accusing the Arnoldi algorithm is irrelevant. (And Matlab gives the correct result.)

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A bug report has been filed for this in the Wolfram community.

Answer 1

It appears that the proximity of the eigenvalues causes Eigensystem with the default parameters to be inaccurate. This can be fixed by increasing the basis size to 30

Import["https://pastebin.com/raw/PpDfY3EQ", "Package"]; 
mysparsemat = mymat; 
mymat = Normal[mysparsemat]; 
m = 2; 
Reverse[First[Eigensystem[mymat, -m]]]
Reverse[First[Eigensystem[mysparsemat, -m, Method -> {"Arnoldi", "BasisSize" -> 30}]]]

which then gives the desired result

{-0.712477 + 1.02679*10^-16 I, 0.712477 + 2.53294*10^-16 I}

{-0.712477 + 3.34996*10^-16 I, 0.712477 + 1.11772*10^-15 I}

Answer 2

I found a similar issue in a non-numeric caluclation:

sol = C[1] Cos[\[Lambda] x] + C[2] Sin[\[Lambda] x] + C[3] Cosh[\[Lambda] x] + C[4] Sinh[\[Lambda] x];
row1 = CoefficientArrays[sol /. {x -> 0}, {C[1], C[2], C[3], C[4]}][[2]];
row2 = CoefficientArrays[D[sol, x] /. {x -> 0}, {C[1], C[2], C[3], C[4]}][[2]];
row3 = CoefficientArrays[D[sol, x, x] /. {x -> l}, {C[1], C[2], C[3], C[4]}][[2]];
row4 = CoefficientArrays[D[sol, x, x, x] /. {x -> l}, {C[1], C[2], C[3], C[4]}][[2]];

sparsematrix = {row1, row2, row3, row4};
Det[sparsematrix]
normalmatrix = Normal@sparsematrix;
Det[normalmatrix]// FullSimplify

It gives 0 and 2 \[Lambda]^6 (1 + Cos[l \[Lambda]] Cosh[l \[Lambda]]). The first result is unexpected. It took me a while to figure that out that Det[] does different things and fails when sparse vectors are collected as normalform-matrix. It can be fixed by Det[SparseArray[sparsematrix]]. I consider this a bug. Det[] apparently understands that the matrix is 4 by 4 (leaving out one row in the above example causes Det[] to complain about the matrix not being square). So one assumes that it is handled properly, but the result is just wrong.