How to suppress errors in matrix algebra involving very large matrices ?

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Aditya Nanda
Aditya Nanda 2016-12-13
回答： Aditya Nanda 2017-5-24
I have two very large matrices, A and B which constitute an equation
A* x + B* y =0
I am interested in solving for x (denoted x_star) so
x_star = pinv(A) * (- B*y);
Then, I back- substitute x_star to check the accuracy
A* x_star + B* y
But this value is not zero. Its usually a large number. How do I ensure this is zero? The matrices A and B are given as
y= [1,0]';
A(1,1) = 4.15325070802278e+22 - .07633591784246e+18i ;
A(1,2) = 9.04710238303399e+20- 8.73556649303388e+16i ;
A(2,1) = -9.04710238303399e+20+ .73556649303387e+16i;
A(2,2) = -1.9707469469826e+19 + 1.87151868951167e+15i ;
B(1,1) = 2.57576471264519e+18
B(1,2) = -1.18245242413448e+20 ;
B(2,1) = -5.610835634053e+16 ;
B(2,2 ) = 2.57575785721108e+18;

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回答（2 个）

Matt J
Matt J 2016-12-13
编辑：Matt J 2016-12-13
You can't make it zero. To hope for that is to be in denial of the realities of floating point arithmetic. But as for your specific data, the substitution result looks pretty reasonable,
>> A*x_star+B*y
ans =
1.0e+07 *
0.0001 + 4.5513i
-0.0000 - 0.0991i
Looks pretty small to me - certainly when compared to the magnitude of the elements of A and B which are O(1e20).
Aditya Nanda
Aditya Nanda 2017-5-24
Just for the sake of documenting, I actually found a way to make this work. Matlab has vpa (variable precision arithmetic), using which it is possible to minimize the error to exactly zero (well, of the order of 1e-100 or even less).
Of course, theres no way to do this using double arithmetic.

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