Note on the Practical Computation of Proper Values

ABSTRACT

It has been suggested by Householder [1] and by Householder and Bauer [2] that orthogonal similarity transformations of matrices are particularly stable with respect to the practical computation of proper values. It is the purpose of this note to examine this question and to demonstrate in terms of a “condition number” to be defined below a sense in which this conjecture is true. Broadly speaking, any problem may be termed “ill-conditioned” for which the solution is acutely sensitive to slight variations in the parameters of the problem. Examples of ill-conditioning occur in many contexts. Computers are familiar with the phenomenon as it manifests itself, for example, in the study of matrix inversion. Since our purpose is to study the conditioning of matrices specifically for the proper value problem, it is convenient to have a nomenclature and notation which will avoid confusion with conditioning as it is used in other senses.

REFERENCES

• 1: A. S. HOUSEHOLDER, Hedrick Lectures delivered at the Summer Meeting of the Mathematical Association of America, August, 1958.
• 2: a~k. S. HOUSEHOLDER AND F. L. BAUER, On certain methods for expanding the characteristic polynomial. Numerische Ma~h. 1 (1959), pp. 29-37.
• 3: J. 1~. FR~XLIN, Computation of Eigenvalues by the Method of Iteration. California Institute of Technology Computing Center Technical Report No. 111, October, 1957.
• 4: LOTHAR COLLaTZ, E~genwertau, fgabe~ mit ~echnischen A~wendunger,, p. 289. Leipzig, 1949.
• 5: E. KREYSZ}O, Die Einschliessung yon Eigenwerten hermiteseher Matrizen beim Iterationsverfahren. Z. angew. Math. Mech. S~ (1954), 459-469.

INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computation of transforms (e.g., fast Fourier transform)

General Terms:
Theory, Verification