A generalization of the numerical range of a matrix

Lenferink, H. W. J.; Spijker, M. N.

doi:10.1016/0024-3795(90)90232-2

Cited by 27 publications

(15 citation statements)

References 9 publications

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“…In the case of the maximum norm | • | = | • 1^ , it is known (cf. [16,21,26] In the following we use these characterizations to review in a coherent fashion some of the stability results to be found in the literature. The above conditions (2.6.b), (2.7.b) cannot be fulfilled by explicit methods (1.1) (i.e., methods where tp(Q is a polynomial).…”

Section: 3mentioning

confidence: 99%

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On the use of stability regions in the numerical analysis of initial value problems

Lenferink¹,

Spijker²

1991

Math. Comp.

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Abstract. This paper deals with the stability analysis of one-step methods in the numerical solution of initial (-boundary) value problems for linear, ordinary, and partial differential equations. Restrictions on the stepsize are derived which guarantee the rate of error growth in these methods to be of moderate size. These restrictions are related to the stability region of the method and to numerical ranges of matrices stemming from the differential equation under consideration.The errors in the one-step methods are measured in arbitrary norms (not necessarily generated by an inner product).The theory is illustrated in the numerical solution of the heat equation and some other differential equations, where the error growth is measured in the maximum norm.

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Section: 3mentioning

confidence: 99%

“…We shall derive stability estimates of type (1.2) which apply to some general situations not covered in the references cited above. We focus on modified versions of conditions (1.3), where a[A] is replaced by the so-called M-numerical range r[A], a subset of the complex plane recently introduced in [16].…”

mentioning

confidence: 99%

On the use of stability regions in the numerical analysis of initial value problems

Lenferink¹,

Spijker²

1991

Math. Comp.

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“…In the literature, many generalizations of the classical numerical range have been studied (see e.g. Horn & Johnson (1994), Lenferink & Spijker (1990) and the references cited therein). One of the possible generalizations of the classical numerical range is the so-called Mnumerical range, introduced by Lenferink & Spijker (1990).…”

Section: The Purpose Of the Papermentioning

confidence: 99%

An algorithm to compute sets enclosingM-numerical ranges with applications in numerical analysis

Dorsselaer

1996

Linear and Multilinear Algebra

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One of the possible generalizations of the classical numerical range studied in the literature is the so-called M-numerical range. In this paper we present an algorithm which can be used to construct a polygon V enclosing the M-numerical range of a given square matrix A. The algorithm can be executed such that V encloses the M-numerical range of A tightly.The concept of M-numerical ranges can be used to investigate stability properties of numerical methods for approximating the solution to linear initial value problems. To illustrate our algorithm, we consider a numerical method for a given initial-boundary value problem. A modified version of our algorithm will be used to obtain sufficient conditions for the stability of the numerical method under consideration.

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“…As an example, we considered the matrix of Lenferink and Spijker [12]. This is a non-normal tridiagonal matrix of the form…”

Section: Gm Res(a "0) == Gm Res(w L* Lw* "0)mentioning

confidence: 99%

Matrices that Generate the same Krylov Residual Spaces

Greenbaum

Strakoš

1994

Recent Advances in Iterative Methods

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Abstract. Given an n by n nonsingular matrix A and an n-vector v, we consider the spaces of the form AKk (A, v), k = 1, ... , n, where Kk(A, v) is the kth Krylov space, equal to span{ v, Av, ... , A k -1 v}. We characterize the set of matrices B that, with the given vector v, generate the same spaces; i.e., those matrices B for which BKk (B,v) = AKk(A,v), for all k = 1, ... ,n. It is shown that any such sequence of spaces can be generated by a unitary matrix. If zero is outside the field of values of A, then there is a Hermitian positive definite matrix that generates the same spaces, and, moreover, if A is close to Hermitian then there is a nearby Hermitian matrix that generates the same spaces. It is also shown that any such sequence of spaces can be generated by a matrix having any desired eigenvalues.Implications about the convergence rate of the GMRES method are discussed. A new proof is given that if zero is outside the field of values of A, then convergence of the GMRES algorithm is strictly monotonic. It is shown that if A is close to Hermitian, then the eigenvalues of A essentially determine the behavior of the GMRES iteration but that, in general, eigenvalue information alone is never sufficient to ensure rapid convergence of the GMRES algorithm.

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A generalization of the numerical range of a matrix

Cited by 27 publications

References 9 publications

On the use of stability regions in the numerical analysis of initial value problems

On the use of stability regions in the numerical analysis of initial value problems

An algorithm to compute sets enclosingM-numerical ranges with applications in numerical analysis

Matrices that Generate the same Krylov Residual Spaces

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