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.

The QR algorithm: 50 years later its genesis by John Francis and Vera Kublanovskaya and subsequent developments

“…On pages 335-339 Francis discusses the implementation of Givens rotations and Householder eliminations in detail for their stability properties and the overall backward stability of the algorithm. The inspiration and reference for the latter were Fike's 1959 paper (Fike, 1959) on the preservation of matrix and eigenvalue condition numbers under unitary similarities. On page 339 Francis details a close precursor of what later became known as the Wilkinson shift.…”

“…His knowledge of Fike's (1959) and Wilkinson's work on the stability of unitary matrix transformations helped John shift the emphasis from the Gaussian elimination-based LR algorithm of Rutishauser to the use of orthogonal Householder and Givens eliminations in QR. John was aware of the (slow) convergence of Jacobi's method that annihilates the largest subdiagonal entry in a symmetric matrix A via similarity until the matrix becomes diagonal and displays A's eigenvalues.…”

The Development and Use of Methods of LR Type

“…It is generally believed that, in numericM work, unitary transformations are very stable. Francis cites Fike's result [5] that the condition of an eigenvMue is unchanged (i.e. not made worse) when the matrix undergoes n orthogonal congruence.…”