On condition numbers and the distance to the nearest illposed problem
 James Weldon Demmel
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessSummary
The condition number of a problem measures the sensitivity of the answer to small changes in the input. We call the problem illposed if its condition number is infinite. It turns out that for many problems of numerical analysis, there is a simple relationship between the condition number of a problem and the shortest distance from that problem to an illposed one: the shortest distance is proportional to the reciprocal of the condition number (or bounded by the reciprocal of the condition number). This is true for matrix inversion, computing eigenvalues and eigenvectors, finding zeros of polynomials, and pole assignment in linear control systems. In this paper we explain this phenomenon by showing that in all these cases, the condition number κ satisfies one or both of the diffrential inequalitiesm·κ^{2}≤∥Dκ∥≤M·κ^{2}, where ‖Dκ‖ is the norm of the gradient of κ. The lower bound on ‖Dκ‖ leads to an upper bound 1/mκ(x) on the distance. fromx to the nearest illposed problem, and the upper bound on ‖Dκ‖ leads to a lower bound 1/(Mκ(X)) on the distance. The attraction of this approach is that it uses local information (the gradient of a condition number) to answer a global question: how far away is the nearest illposed problem? The above differential inequalities also have a simple interpretation: they imply that computing the condition number of a problem is approximately as hard as computing the solution of the problem itself. In addition to deriving many of the best known bounds for matrix inversion, eigendecompositions and polynomial zero finding, we derive new bounds on the distance to the nearest polynomial with multiple zeros and a new perturbation result on pole assignment.
 Demmel, J. (1983) The condition number of equivalence transformations which block diagonalize matrix pencils. SIAM J. Numer. Anal. 20: pp. 599601
 Demmel, J. (1986) Computing stable eigendecompositions of matrices. Linear Algebra Appl. 79: pp. 163193
 Demmel, J. (1985) On the conditioning of pole assignment. Courant Institute of Mathematical Sciences, New York
 Eckart, C., Young, G. (1939) A principal axis transformation for nonhermitian matrices. Bull. Am. Math. Soc., New Ser 45: pp. 118121
 Golub, G., Loan, C (1983) Matric computations. Johns Hopkins Press, Baltimore
 Hartman, P. (1973) Ordinary differential equations. John Wiley, New York
 Hough, D. (1977) Explaining and ameliorating the ill condition of zeros of polynomials. Mathematics Department, University of California, Berkeley, CA
 Kahan, W. (1966) Numerical linear algebra. Can. Math. Bull. 9: pp. 757801
 Kahan, W. (1972) Conserving confluence curbs illcondition. University of California, Berkeley
 Kato, T. (1966) Perturbation theory for linear operators. Springer, Berlin. Heidelberg, New York
 Kautsky, J., Nichols, N., Dooren, P. (1985) Robust pole assignment in linear state feedback. Int. J Control 41: pp. 11291155
 Parlett, B. N., Ng, K. C. (1985) Development of an accurate algorithm for exp(Bτ). Center for Pure and Applied Mathematics. University of California, Berkeley
 Ruhe, A. (1970) Properties of a matrix with a very illconditioned eigenproblem. Numer. Math. 15: pp. 5760
 Stewart, G.W. (1971) Error bounds for approximate invariant subspaces of closed linear operators. SIAM J. Numer. Anal. 8: pp. 796808
 Wilkinson, J.H. (1965) The algebraic eigenvalue problem. Clarendon Press, Oxford
 Wilkinson, J.H. (1972) Note on matrices with a very illconditioned eigenproblem. Numer. Math. 19: pp. 176178
 Wilkinson, J.H. (1984) On neighboring matrices with quadratic elementary divisors. Numer. Math. 44: pp. 121
 Wilkinson, J.H.: Sensitivity of eigenvalues. Util. Math.25, 5–76.
 Wonham, W.M. (1979) Linear multivariable control.: A geometric approach. Springer, Berlin Heidelberg, New York
 Title
 On condition numbers and the distance to the nearest illposed problem
 Journal

Numerische Mathematik
Volume 51, Issue 3 , pp 251289
 Cover Date
 19870501
 DOI
 10.1007/BF01400115
 Print ISSN
 0029599X
 Online ISSN
 09453245
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 AMS(MOS): 15A12, 15A60, 65F35
 CR: F.2.1, G.1.0
 Industry Sectors
 Authors

 James Weldon Demmel ^{(1)}
 Author Affiliations

 1. Courant Institute of Mathematical Sciences, 251 Mercer Str, 10012, New York, NY, USA