Numerische Mathematik

, Volume 51, Issue 3, pp 251–289

On condition numbers and the distance to the nearest ill-posed problem

• James Weldon Demmel
Article

Summary

The condition number of a problem measures the sensitivity of the answer to small changes in the input. We call the problem ill-posed 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 ill-posed 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·κ2≤∥Dκ∥≤M·κ2, 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 ill-posed 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 ill-posed 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.

Subject Classifications

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

References

1. 1.
Demmel, J.: The condition number of equivalence transformations which block diagonalize matrix pencils. SIAM J. Numer. Anal.20, 599–601 (1983)Google Scholar
2. 2.
Demmel, J.: Computing stable eigendecompositions of matrices. Linear Algebra Appl.79:163–193 (1986)Google Scholar
3. 3.
Demmel, J.: On the conditioning of pole assignment. Computer Science Dept. Report # 150. Courant Institute of Mathematical Sciences. New York, Jan. 1985Google Scholar
4. 4.
Eckart, C., Young, G.: A principal axis transformation for non-hermitian matrices. Bull. Am. Math. Soc., New Ser45, 118–121 (1939)Google Scholar
5. 5.
Golub, G., Van Loan C: Matric computations. Baltimore: Johns Hopkins Press 1983Google Scholar
6. 6.
Hartman, P.: Ordinary differential equations. New York: John Wiley 1973Google Scholar
7. 7.
Hough, D.: Explaining and ameliorating the ill condition of zeros of polynomials. Thesis. Mathematics Department, University of California, Berkeley, CA 1977Google Scholar
8. 8.
Kahan, W.: Numerical linear algebra. Can. Math. Bull.9, 757–801 (1966)Google Scholar
9. 9.
Kahan, W.: Conserving confluence curbs ill-condition. Computer Science Dept. Report. University of California. Berkeley 1972Google Scholar
10. 10.
Kato, T.: Perturbation theory for linear operators. Berlin. Heidelberg, New York: Springer 1966.Google Scholar
11. 11.
Kautsky, J., Nichols, N., Van Dooren, P.: Robust pole assignment in linear state feedback. Int. J Control41, 1129–1155 (1985)Google Scholar
12. 12.
Parlett, B. N., Ng, K. C.: Development of an accurate algorithm for exp(Bτ). Report PAM-294. Center for Pure and Applied Mathematics. University of California. Berkeley. August 1985Google Scholar
13. 13.
Ruhe, A.: Properties of a matrix with a very ill-conditioned eigenproblem. Numer. Math.15, 57–60 (1970)Google Scholar
14. 14.
Stewart, G.W.: Error bounds for approximate invariant subspaces of closed linear operators SIAM J. Numer. Anal.8 796–808 (1971)Google Scholar
15. 15.
Wilkinson, J.H.: The algebraic eigenvalue problem. Oxford Clarendon Press 1965Google Scholar
16. 16.
Wilkinson, J.H.: Note on matrices with a very ill-conditioned eigenproblem. Numer. Math.19, 176–178 (1972)Google Scholar
17. 17.
Wilkinson, J.H.: On neighboring matrices with quadratic elementary divisors. Numer. Math.44, 1–21 (1984)Google Scholar
18. 18.
Wilkinson, J.H.: Sensitivity of eigenvalues. Util. Math.25, 5–76.Google Scholar
19. 19.
Wonham, W.M.: Linear multivariable control.: A geometric approach. 2nd edition. Berlin- Heidelberg, New York: Springer 1979Google Scholar 