How to avoid rounding error
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When I use lu[A] for A = [10^(-20) 1 ; 1 2] I get 2 matrices (L and U). When I multiply them, the result is not the same as the original matrix A. What method can I use in order to get the correct matrix A?
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Rik
2021-4-9
You could try vpa.
The more fundamental problem is that computers have finite precision. If you want infinite precision, you will need to use algebraic tools. Not every problem can be solved perfectly. The general solution for this is to avoid problems that span more than 20 orders of magnitude, so you can rely on eps to estimate if your results are close enough.
采纳的回答
A = [sym(10)^(-20) 1 ; 1 2]
[L,U] = lu(A)
L*U - A
You can see from this that in order to get back A exactly, then you need a system that can distinguish 99999999999999999998 from 100000000000000000000, but
eps(100000000000000000000)
it certainly is not double precision arithmetic.
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syms N real
A = [sym(10)^(-N) 1 ; 1 2]
[L,U] = lu(A)
eqn = U(2,2) == -1/eps
solve(eqn)
vpa(ans)
So beyond about 10^15.65 you go beyond what can be represented exactly in double precision.
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