expm function problem for stiff matrix

Question

Michal 2021-6-10

1
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/852920-expm-function-problem-for-stiff-matrix

回答： Noorolhuda wyal 2022-11-22

For very specific matrix A:

a = -1e20;
b = eps;
c = 1;
A = [a,0,b;0,c,0;-b,0,a];
disp('A:'), disp(num2str(A))
A:
                -1e+20                      0  2.220446049250313e-16
                     0                      1                      0
-2.220446049250313e-16                      0                 -1e+20

is known exact matrix exponential as:

expA = exp(a)*( ...
    [1,0,0;0,0,0;0,0,1]*cos(b)+ ...
    [0,0,1;0,0,0;-1,0,0]*sin(b))+ ...
    [0,0,0;0,exp(c),0;0,0,0];
expA =
0           0           0
0      2.7183           0
0           0           0

the Matlab function expm give wrong result:

expm(A)
ans =
     0     0     0
     0     1     0
     0     0     0

but direct computing of expm(A) via definition gives again right result:

[V,D] = eig(A);
expmA = V*diag(exp(diag(D)))/V
expmA =
         0         0         0
         0    2.7183         0
         0         0         0

So, what is wrong with expm function? Bad implementation of Pade's approximation?

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Matt J 2021-6-10

If they are linear ODEs, maybe you could solve them symbolically?

Michal 2021-6-10

Symbolic solutions always ends on matrix exponentials and integration, which must be finally evaluated always numerically, so in this case by multi-precision arithmetic, which is sometimes very slow (especially with VPA in MATLAB). So, this problem is really hard ... :)

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Answer 1

Shadaab Siddiqie 2021-6-18

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https://ww2.mathworks.cn/matlabcentral/answers/852920-expm-function-problem-for-stiff-matrix#answer_727530

From my understanding you are getting wrong result for certain cases wile using expm function. This issue has been forwarded to the development team for further investigation.

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Michal 2021-6-18

OK ... great! I am looking forward for any news regarding this topic.

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Answer 2

Bobby Cheng 2021-8-12

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https://ww2.mathworks.cn/matlabcentral/answers/852920-expm-function-problem-for-stiff-matrix#answer_765717

在 MATLAB Online 中打开

This is a weakness of the scaling and squaring algorithm. Inside EXPM, which you can read the implementation, there are special treatments for diagonal to deal with extreme cases, but it is only triggered if the input is of the Schur form due to performance. You can call SCHUR to create the Schur factorization, and pass the Schur form to EXPM to trigger the special diagonal treatment.

>> a = -1e20;
>> b = eps;
>> c = 1;
>> A = [a,0,b;0,c,0;-b,0,a];
>> [Q T] = schur(A);
>> Q*expm(T)*Q'
ans =
         0         0         0
         0    2.7183         0
         0         0         0