Abstract not found

. Expressions and bounds are derived for the residual norm in GMRES. It is shown that the minimal residual norm is large as long as the Krylov basis is well-conditioned.For scaled Jordan blocks the minimal residual norm is expressed in terms of eigenvalues and departure from normality.For normal matrices the minimal residual norm is expressed in terms of products of relative eigenvalue di#erences. Key words. linear system, Krylov methods, GMRES, MINRES, Vandermonde matrix, eigenvalues, departure from normality AMS subject classi#cation. 15A03, 15A06, 15A09, 15A12, 15A18, 15A60, 65F10, 65F15, 65F20, 65F35. 1. Introduction.. The generalised minimal residual method #GMRES# #31, 36# #and MINRES for Hermitian matrices #30## is an iterative method for solving systems of linear equations Ax = b. The approximate solution in iteration i minimises the two-norm of the residual b , Az over the Krylov space spanfb;Ab;:::;A i,1 bg. The goal of this paper is to express this minimal residual norm...

Abstract. We extend the concept of Lifshits–Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of (admissible) operators that are similar to self-adjoint operators. An operator H is called admissible if: (i) there is a bounded operator V with a bounded inverse such that H = V −1 ̂HV for some self-adjoint operator ̂H; (ii) the operators H and ̂ H are resolvent comparable, i.e., the difference of the resolvents of H and ̂ H is a trace class operator (for non-real values of the spectral parameter); (iii) tr(V R − RV) = 0 whenever R is bounded and the commutator V R − RV is a trace class operator. The spectral shift function ξ(λ, H, A) associated with the pair of resolvent comparable admissible operators (H, A) is introduced then by the equality ξ(λ, H, A) = ξ(λ, ̂ H, Â) where ξ(λ, ̂ H, Â) denotes the Lifshits– Krein spectral shift function associated with the pair ( ̂H, Â) of self-adjoint operators. Our main result is the following. Let H0 and H1 be separable Hilbert spaces, A0 a self-adjoint operator in H0, A1 a self-adjoint operator in H1, and Bij a bounded operator from Hj to Hi, i = 0, 1, j = 1 − i, and

. Let and x be a perturbed eigenpair of a diagonalisable matrix A. The problem is to bound the error in and x. We present one absolute perturbation bound and two relative perturbation bounds. The absolute perturbation bound implies that the condition number for x is the norm of an orthogonal projection of the reduced resolvent at . This condition number can be a lot less pessimistic than the traditional one, which is derived from a first-order analysis. A further upper bound leads to an extension of Davis and Kahan's sin ` Theorem from Hermitian to diagonalisable matrices. The two relative perturbation bounds assume that and x are an exact eigenpair of a perturbed matrix D1AD2 , where D1 and D2 are non-singular, but D1AD2 is not necessarily diagonalisable. We derive a bound on the relative error in and a sin ` theorem based on a relative eigenvalue separation. The perturbation bounds contain both the deviation of D 1 and D2 from similarity and the deviation of D2 from iden...

. Absolute and relative perturbation bounds are derived for angles between invariant subspaces of complex square matrices, in the two-norm and in the Frobenius norm. The absolute bounds are extensions of Davis and Kahan's sin ` theorem to general matrices and invariant subspaces of any dimension. When the perturbed subspace has dimension one, the relative bound is implied by the absolute bound. The relative bounds presented here are the most general relative bounds for invariant subspaces because they place no restrictions on the matrix or the perturbation. Key words. invariant subspace, condition number, separation between matrices, absolute error, relative error, eigenvalue separation, angle between subspaces AMS subject classification. 15A12, 15A18, 15A42, 15A69, 65F15, 65F35 1. Introduction. Absolute and relative perturbation bounds are derived for angles between invariant subspaces of a complex square matrix A and a perturbed matrix A+E, in the two-norm and in the Frobenius no...

In the context of Krylov methods for solving systems of linear equations, expressions and bounds are derived for the norm of the minimal residual, like the one produced by GMRES or MINRES. It is shown that the minimal residual norm is large as long as the Krylov basis is well-conditioned. In the context of non-normal matrices, examples are given where the minimal residual norm is a function of the departure of the matrix from normality, and where the decrease of the residual norm depends on how large the departure from normality is compared to the eigenvalues. With regard to normal matrices, the Krylov matrix is unitarily equivalent to a row-scaled Vandermonde matrix and the minimal residual norm in iteration i is proportional to a product of i relative eigenvalue separations. Arguments are given for why normal matrices with complex eigenvalues can produce larger residual norms than Hermitian matrices, and why indefinite matrices can produce larger residual norms than definite matric...

Abstract. We extend the concept of Lifshits–Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of (admissible) operators that are similar to self-adjoint operators. An operator H is called admissible if: (i) there is a bounded operator V with a bounded inverse such that H = V −1 ̂HV for some self-adjoint operator ̂H; (ii) the operators H and ̂ H are resolvent comparable, i.e., the difference of the resolvents of H and ̂ H is a trace class operator (for non-real values of the spectral parameter); (iii) tr(V R − RV) = 0 whenever R is bounded and the commutator V R − RV is a trace class operator. The spectral shift function ξ(λ, H, A) associated with the pair of resolvent comparable admissible operators (H, A) is introduced then by the equality ξ(λ, H, A) = ξ(λ, ̂ H, Â) where ξ(λ, ̂ H, Â) denotes the Lifshits– Krein spectral shift function associated with the pair ( ̂H, Â) of self-adjoint operators. Our main result is the following. Let H0 and H1 be separable Hilbert spaces, A0 a self-adjoint operator in H0, A1 a self-adjoint operator in H1, and Bij a bounded operator from Hj to Hi, i = 0, 1, j = 1 − i, and