Why does matrix power give wrong answer?

2014 2 26
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更新时间：2014 3 1
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I am trying to do a MCMC simulation and need to calculate a small frequency event over a long time. To do this I need to raise
A=[1-1.01E-11,1.01E-11,0;1-3.024E-11,0,3.024E-11;0,0,1];
to a large power (10^10). I am particularly interested in the (1,3) element of the result (the probability of transferring between states 1 and 3 after 10 billion iterations). After 10^9 the (1,3) element is 3.05423999691491e-13. However after 10^10 it is -3.05423999991124e-22. Which is impossible because I am multiplying matrices with strictly non-negative values. As a check I tried (A^(10^9))^10, and this gave a (1,3) entry of 3.05423999966373e-12. Doing the calculation (A^(10^10) using repeated squaring and multiplication) in Excel gave 3.05424E-12.
I am using MATLAB 2013b on windows 7 64-bit.
Why does A^(10^10) fail but (A^(10^9))^10 work?

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John D'Errico
John D'Errico 2014-2-26
Time to learn about floating point arithmetic. When your numbers are only held accurately to roughly 16 digits, what do you expect?

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per isakson
per isakson 2014-2-27
编辑：per isakson 2014-2-27

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Joseph
Joseph 2014-2-27
Try
A=[1-1e-12,1e-12,0;1-1e-12,0,1e-12;0,0,1];
B=A^(2^31-1)*A
You get
0.999999999999000 9.99999999999000e-13 2.14748364699785e-15
0.999999999998000 9.99999999998000e-13 1.00214748364600e-12
0 0 1
Now try
B=A^(2^31)
You get
0.999999999999000 9.99999999999000e-13 -9.99999999997652e-25
0.999999999998000 9.99999999998000e-13 9.99999999997652e-13
0 0 1
The answers should be identical. As far as the floating point arithmetic argument, the second calculation can be done with 31 squarings, yet some other algorithm is used and the value in the (1,3) element is horribly wrong. Floating point arithmetic errors alone do not explain the difference in values between A^(2^31-1)*A and A^(2^31).
per isakson
per isakson 2014-2-27
编辑：per isakson 2014-2-27
"The answers should be identical." If you think Matlab does not behave as promised, you might want to report it as a bug.

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Joseph
Joseph
2014-2-26

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per isakson
per isakson
2014-3-1

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