Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting

Betcke, Marta M.; Voß, Heinrich

doi:10.1007/s10492-007-0014-5

Cited by 7 publications

(8 citation statements)

References 16 publications

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“…The relation with the nonlinear Arnoldi method [18] is particularly noteworthy, since the residual inverse iteration is the motivation for the subspace expansion in [18] and it has been successfully used to solve many different types of nonlinear eigenvalue problems; see, e.g., [2,3,19,20] and related works. Residual inverse iteration has been used in [14] to study the stability of a time-delay system with periodic coefficients.…”

mentioning

confidence: 99%

Analyzing the convergence factor of residual inverse iteration

Jarlebring

Michiels

2011

Bit Numer Math

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We will establish here a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well-known inverse iteration. The formula for the convergence factor is explicit and involves quantities associated with the eigenvalue to which the iteration converges, in particular the eigenvalue and eigenvector. Residual inverse iteration allows for some freedom in the choice of a vector w k and we can use the formula for the convergence factor to analyze how it depends on the choice of w k . We also use the formula to illustrate the convergence when the shift is close to the eigenvalue. Finally, we explain the slow convergence for double eigenvalues by showing that under generic conditions, the convergence factor is one, unless the eigenvalue is semisimple. If the eigenvalue is semisimple, it turns out that we can expect convergence similar to the simple case.

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confidence: 99%

Analyzing the convergence factor of residual inverse iteration

Jarlebring

Michiels

2011

Bit Numer Math

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“…Example 3 For the example (2), the pair (X, S) with X = 1 1 1 1 and S = diag (3,4) is invariant and minimal with minimality index 2.…”

Section: Invariant Pairsmentioning

confidence: 99%

“…Voss and his co-authors [4,5,7,[27][28][29][30] have developed Arnoldi-type and Jacobi-Davidson-type methods that employ this numbering as a safety scheme for avoiding repeated convergence towards the same eigenvalue. Unfortunately, for many applications such minimum-maximum characterizations do not exist or are difficult to verify.…”

Section: Introductionmentioning

confidence: 99%

A block Newton method for nonlinear eigenvalue problems

Kreßner¹

2009

Numer. Math.

102

121

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We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences from the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The purpose of this work is to show that the concept of invariant pairs offers a way of representing eigenvalues and eigenvectors that is insensitive to this phenomenon. To demonstrate the use of this concept in the development of numerical methods, we have developed a novel block Newton method for computing such invariant pairs. Algorithmic aspects of this method are considered and a few academic examples demonstrate its viability.

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“…Depending on the application some additional condition for (6) may need to be imposed. For example, we typically select l = 0 and require h to be positive for our QD simulation.…”

Section: Inner Loopmentioning

confidence: 99%