On condition numbers and the distance to the nearest illposed problem
 James Weldon Demmel
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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.: The condition number of equivalence transformations which block diagonalize matrix pencils. SIAM J. Numer. Anal.20, 599–601 (1983)
 Demmel, J.: Computing stable eigendecompositions of matrices. Linear Algebra Appl.79:163–193 (1986)
 Demmel, J.: On the conditioning of pole assignment. Computer Science Dept. Report # 150. Courant Institute of Mathematical Sciences. New York, Jan. 1985
 Eckart, C., Young, G.: A principal axis transformation for nonhermitian matrices. Bull. Am. Math. Soc., New Ser45, 118–121 (1939)
 Golub, G., Van Loan C: Matric computations. Baltimore: Johns Hopkins Press 1983
 Hartman, P.: Ordinary differential equations. New York: John Wiley 1973
 Hough, D.: Explaining and ameliorating the ill condition of zeros of polynomials. Thesis. Mathematics Department, University of California, Berkeley, CA 1977
 Kahan, W.: Numerical linear algebra. Can. Math. Bull.9, 757–801 (1966)
 Kahan, W.: Conserving confluence curbs illcondition. Computer Science Dept. Report. University of California. Berkeley 1972
 Kato, T.: Perturbation theory for linear operators. Berlin. Heidelberg, New York: Springer 1966.
 Kautsky, J., Nichols, N., Van Dooren, P.: Robust pole assignment in linear state feedback. Int. J Control41, 1129–1155 (1985)
 Parlett, B. N., Ng, K. C.: Development of an accurate algorithm for exp(Bτ). Report PAM294. Center for Pure and Applied Mathematics. University of California. Berkeley. August 1985
 Ruhe, A.: Properties of a matrix with a very illconditioned eigenproblem. Numer. Math.15, 57–60 (1970)
 Stewart, G.W.: Error bounds for approximate invariant subspaces of closed linear operators SIAM J. Numer. Anal.8 796–808 (1971)
 Wilkinson, J.H.: The algebraic eigenvalue problem. Oxford Clarendon Press 1965
 Wilkinson, J.H.: Note on matrices with a very illconditioned eigenproblem. Numer. Math.19, 176–178 (1972)
 Wilkinson, J.H.: On neighboring matrices with quadratic elementary divisors. Numer. Math.44, 1–21 (1984)
 Wilkinson, J.H.: Sensitivity of eigenvalues. Util. Math.25, 5–76.
 Wonham, W.M.: Linear multivariable control.: A geometric approach. 2nd edition. Berlin Heidelberg, New York: Springer 1979
 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