Is this a bug in eig?

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Kenneth Johnson 2021-3-30

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https://ww2.mathworks.cn/matlabcentral/answers/788329-is-this-a-bug-in-eig

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 2021-3-30

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/788329-is-this-a-bug-in-eig#answer_662884

编辑：Matt J 2021-3-30

在 MATLAB Online 中打开

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 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 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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