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I have been trying to compute eigenvalues of a rather sizable matrix A, about $500 \times 500$ (but sparse). I asked Mathematica to compute Eigenvalues[A], and left it to work. After a night of computation, Mathematica still failed to produce the answer. Just out of curiousity, I tried to see what happens if I replace A by it's numeric approximation (before, A had integer entries, hence exact). (I didn't quite expect it to make a difference, but I was trying some random things). Much to my surprise, the answer appeared immediately. Hence my question: How does it happen that Mathematica produces the answer so much faster for inexact input? To what extent can this output be trusted?

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A simple answer is that inexact values make use of built-in hardware support for arithmetic. –  image_doctor May 7 '13 at 7:28
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For a matrix that large, it is often infeasible to ask for exact solutions. "To what extent can this output be trusted?" - I don't trust output entirely, but there is the procedure of using Mathematica's arbitrary-precision capabilities, asking for the eigenvalues to 20 digits (Eigenvalues[N[A, 20]]), 30 digits, ... and so on. If there is some agreement between these results, then I trust the results a little bit more. –  J. M. May 7 '13 at 7:42
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In addition to the complexity of solving roots, manipulation of exact values, in general, is time-consuming. On efficiently vectorized code of machine-precision arithmetic, a CPU may perform up to something like 16 basic arithmetic operations per a clock cycle, thanks to direct machine representation of data. With exact non-integer values, cost of a single arithmetic operation probably starts at low dozens of clock cycles. So, unless you specifically require exact symbolic results, avoid them. Often finding them is even infeasible. Sometimes tools such as RootApproximant may save you though. –  kirma May 7 '13 at 9:03
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Try a simple experiment. Do these computations on a smaller matrix, one for which the exact computation finishes in reasonable time. Then look at the results. Look very hard at the results. They will give a clue as to why one takes hugely longer than the other. –  Daniel Lichtblau May 7 '13 at 14:00

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Calculating eigenvalues involves solving for the roots of the characteristic polynomial, which is of degree equal to the order of the size of the matrix. When you input real numbers, it can search for the roots of the polynomial using numerical techniques. When you input exact integers (or rationals, probably) it tries to find exact answers for the roots of the polynomials. This is a much more difficult task, since closed form solutions only exist up to order 5 or so.

In terms of trusting the answers, J.M.'s suggestions are good. It would also make sense to check the answers using alternative methods. For instance, the product of the eigenvalues is equal to the determinant. So you could take the answer you get for the eigenvalues and multiply them together; if they are close to the Det then this provides some confidence. If they are not the same, then the answer can be discarded.

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Thank you! This clarifies a lot. –  Feanor May 7 '13 at 9:31
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Actually, for quintics and higher degrees, Mathematica is certainly forced to express the results in terms of Root[], whereupon the effort is in determining the (coefficients of the) minimal polynomial corresponding to the eigenvalue, plus root isolation. All this takes a nontrivial amount of work. –  J. M. May 7 '13 at 11:45
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At least for inexact arithmetic, I'm pretty sure Mathematica never looks for the roots of a polynomial (because that's not numerically stable). It probably uses an iterative method like Jacobi's. –  nikie May 7 '13 at 12:34
    
I remember taking a class on numerical methods. We read a paper called "18 Ways not to Calculate the Matrix Exponential" -- the various complexities of eigenvector calculations also played a large role. So yes, I'm sure they are doing all sorts of clever things. –  bill s May 7 '13 at 12:44
    
@nikie, at least for eigenvalues, it uses the QR algorithm in the inexact case. –  J. M. May 7 '13 at 12:46

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