Spectrum through pseudospectrum

This report was written in 2001 and it is a translation of work that was originally published in Greek, in the author’s diploma thesis in July 1998. It may contain minor mistakes and should not be considered a complete study. It however touches upon several of the considerations that will be included in the complete paper. The work was done jointly with E. Gallopoulos and was presented at the FOCM99 conference, at the 5th IMACS conference on iterative methods in scientific computing, and at the 50th annual meeting of SIAM. We present an iteration for the computation of simple eigenvalues using a pseudospectrum approach. The most appealing characteristic of the proposed iteration is that it reduces the computation of a single eigenvalue to a small number of eigenvalue computations on Hermitian matrices. We show that this number is directly associated with the matrix pseudospectrum. We present numerical results and we discuss advantages and drawbacks of the method. We also discuss its relationship with an iteration that was proposed independently in [Stewart, O’ Leary, ETNA, Vol 8, 1998]. 1 A pseudospectrum setting We shortly describe some well known characteristics of matrix pseudospectrum. Denote with Λ(A) the spectrum of a matrix A, with Λǫ(A) the ǫ-pseudospectrum of A, with σmin = σN ≤ σN−1 . . . ≤ σ1 the singular values of A, and with D(z, ̺) and D(z, ̺) a closed disk and an open disk respectively, with center z and radius ̺. Let N denote a normal matrix. Theorem 1 • Λ(A) ⊂ Λǫ(A) for ǫ > 0. • ǫ < ǫ1 ⇔ Λǫ(A) ⊂ Λǫ1(A). • Λ(N) = Λ(A)⇒ Λǫ(N) ⊆ Λǫ(A). These properties state that ǫ-pseudospectrum forms closed curves around eigenvalues. The crucial property of subharmonicity of the norm of the resolvent ||(zI −A)|| (see [2]), assures that σmin(zI −A) = local minimum = 0 ⇔ z ∈ Λ(A) (1) This alternative characterization of an eigenvalue together with the properties of matrix pseudospectrum appeal for the corresponding optimization problem: min f(z) = σmin(zI −A), z ∈ C Notice that in this formulation we are seeking for a minimizer over the complex plane. This is not usually the case in other methods for the computation of eigenvalues . The ”heart” of any competent minimization algorithm can naturally be the following theorem by Sun [11].