On the Complexity of Computing Error Bounds
 J. Demmel,
 B. Diament,
 G. Malajovich
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We consider the cost of estimating an error bound for the computed solution of a system of linear equations, i.e., estimating the norm of a matrix inverse. Under some technical assumptions we show that computing even a coarse error bound for the solution of a triangular system of equations costs at least as much as testing whether the product of two matrices is zero. The complexity of the latter problem is in turn conjectured to be the same as matrix multiplication, matrix inversion, etc. Since most error bounds in practical use have much lower complexity, this means they should sometimes exhibit large errors. In particular, it is shown that condition estimators that: (1) perform at least one operation on each matrix entry; and (2) are asymptotically faster than any zero tester, must sometimes over or underestimate the inverse norm by a factor of at least \(2^{\delta k + \tau(n)}\) , where n is the dimension of the input matrix, k is the bitsize, and where either \(\delta \geq 0.5\) or \(\tau(n)\) grows faster than any polynomial in n . Our results hold for the RAM model with bit complexity, as well as computations over rational and algebraic numbers, but not real or complex numbers. Our results also extend to estimating error bounds or condition numbers for other linear algebra problems such as computing eigenvectors.
 Title
 On the Complexity of Computing Error Bounds
 Journal

Foundations of Computational Mathematics
Volume 1, Issue 1 , pp 101125
 Cover Date
 20010101
 DOI
 10.1007/s10208001004
 Print ISSN
 16153375
 Publisher
 SpringerVerlag
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 Keywords

 AMS Classification. 65F35, 68Q25.
 Industry Sectors
 Authors

 J. Demmel ^{(A1)}
 B. Diament ^{(A2)}
 G. Malajovich ^{(A3)}
 Author Affiliations

 A1. Computer Science Division and Mathematics Department University of California Berkeley, CA 94720, USA demmel@cs.berkeley.edu, US
 A2. Computer Science Division University of California Berkeley, CA 94720, USA bjd@cs.berkeley.edu, US
 A3. Departamento de Matem&aacute;tica Aplicada Universidade Federal do Rio de Janeiro Caixa Postal 68530 CEP 21945, Rio de Janeiro, RJ, Brasil gregorio@labma.ufrj.br, BR