Is this a bug in eig?

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Kenneth Johnson
Kenneth Johnson 2021-3-30
评论： Kenneth Johnson 2021-3-31
This issue came up with the roots function (see roots_ on File Exchange), where eig was failing on the companion matrix. Following are two test cases; the first one works and the second one doesn't. (I reported this to tech support, but it's not listed in the official Matlab bug reports.)
a = [-1,-1,0;1,0,0;0,1,0];
a(1,3) = 10e-32;
[v,d] = eig(a);
disp(abs(a*v-v*d))
% 1.0e-15 *
% 0.055511151231258 0.055511151231258 0.000000000000000
% 0.166533453693773 0.166533453693773 0.000000000000000
% 0.138777878078145 0.138777878078145 0
disp(num2str(diag(d)))
% -0.5+0.86603i
% -0.5-0.86603i
% 1e-31+0i
a(1,3) = 9e-32;
[v,d] = eig(a);
disp(abs(a*v-v*d))
% 0.000000000000000 0.000000000000000 0.000000000000000
% 0.000000000000000 0.000000000000000 0.000000000000000
% 0.707106781186548 0.707106781186548 0.000000000000000
disp(num2str(diag(d)))
% -0.5+0.86603i
% -0.5-0.86603i
% 0+0i

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采纳的回答

Matt J
Matt J 2021-3-30
编辑：Matt J 2021-3-30
Ran in:
It's fine. You just need to disable balancing:
a = [-1,-1,0;1,0,0;0,1,0];
for e=[10e-32, 9e-32]
a(1,3) = e;
[v,d] = eig(a,'nobalance');
Discrepancy = abs(a*v-v*d)
end
Discrepancy = 3×3
1.0e+-15 * 0.3608 0.3608 0.0342 0.3554 0.3554 0.0220 0.2355 0.2355 0.2282
Discrepancy = 3×3
1.0e+-15 * 0.3608 0.3608 0.0342 0.3554 0.3554 0.0220 0.2355 0.2355 0.2282
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Bruno Luong
Bruno Luong 2021-3-31
编辑：Bruno Luong 2021-3-31
Google I found this page where there is a paper discuss issue with banancing. Someone claims (last post) he has developped a correct balancing method in Lapack 3.5.0.
Kenneth Johnson
Kenneth Johnson 2021-3-31
For my test case it doesn't work without balancing, but Matlab's balancing doesn't work. There is a discussion of matrix balancing in Numerical Recipes in C++.

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