How to compute the middle part of a sorted eigenvalues of an extremely large sparse matrix

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子冲
子冲 2025-11-21，2:23
评论： 子冲 2025-11-24，3:48
I currently have a large sparse matrix H of size k×k (which is symmetric and real). If its sorted eigenvalues are denoted as Ep, I need to compute the k/2th and k/2+1th values of Ep (which I mean middle part of the sorted eigenvalues). Is there a fast method for this?
Given that H is too large to use 'eig' directly, I have also attempted 'eigs', but this proved impractical. Because 'eigs' only computes the smallest few eigenvalues of H, whereas the two values I require are not near zero.
  1 个评论
子冲
子冲 2025-11-24，3:48
I have developed an algorithm that resolves this issue. My current approach efficiently computes the Nth eigenvalue of a sorted, extremely sparse matrix with minimal resource consumption. Please feel free to contact me via email to discuss further. 12224043@zju.edu.cn

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回答（1 个）

Steven Lord
Steven Lord 2025-11-21，14:08
移动：Steven Lord 2025-11-21，14:19
With the default options, eigs computes the largest eigenvalues and eigenvectors. But depending on what you specify as the sigma output argument, it can compute other quantities. If you know or can estimate a value of sigma between those two eigenvalues, you could get the eigenvalues closest to that value.
A = magic(501);
d = sort(eig(A));
S = sparse(A); % Yes, I know, Big Sparse.
format longg
d(20:30)
ans = 11×1
1.0e+00 * -1003326.70070185 -955806.306513294 -912618.017387523 -873196.789771618 -837071.754658116 -803847.382677599 -773188.99570839 -744811.498128658 -718470.522800272 -693955.408918091 -671083.584289529
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ds = eigs(S, 4, -800000) % 4 eigenvalues close to -800000
ds = 4×1
1.0e+00 * -803847.382677596 -773188.995708388 -837071.754658116 -744811.498128661
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