A rational Arnoldi process with applications

Pranić, Miroslav S.; Reichel, Lothar; Rodriguez, Giuseppe; Wang, Zhengsheng; Yu, Xuebo

doi:10.1002/nla.2065

Cited by 8 publications

(10 citation statements)

References 27 publications

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“…However, the derivation and analysis is also more involved. Pranić et al [18] proposed a Lanczos-type method for non-symmetric linear equation that is essentially identical to the Lanczos method in this article, which can thus be considered a specialization of that in [18] to the symmetric case.…”

Section: Introductionmentioning

confidence: 99%

Projection Methods for Ill-Posed Problems Revisited

Kindermann

2015

Computational Methods in Applied Mathematics

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The discretization of least-squares problems for linear ill-posed operator equations in Hilbert spaces is considered. The main subject of this article concerns conditions for convergence of the associated discretized minimum-norm least-squares solution to the exact solution using exact attainable data. The two cases of global convergence (convergence for all exact solution) or local convergence (convergence for a specific exact solution) are investigated. We review the existing results and prove new equivalent condition when the discretized solution always converges to the exact solution. An important tool is to recognize the discrete solution operator as oblique projection. Hence, global convergence can be characterized by certain subspaces having uniformly bounded angles. We furthermore derive practically useful conditions when this holds and put them into the context of known results. For local convergence we generalize results on the characterization of weak or strong convergence and state some new sufficient conditions. We furthermore provide an example of a bounded sequence of discretized solutions which does not converge at all, not even weakly.

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Section: Introductionmentioning

confidence: 99%

Projection Methods for Ill-Posed Problems Revisited

Kindermann

2015

Computational Methods in Applied Mathematics

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“…Orthogonal rational bases have been used in system identification before, mostly for discrete‐time systems, 32,34,35 including approximations with real coefficients 36 . The rational Arnoldi decomposition by Ruhe 37 has also been successfully used to solve more general problems such as approximation of matrix functions or eigenvalues 38 . In particular, the RKFIT algorithm 39 is also based on an orthogonal rational basis to solve the rational approximation problem.…”

Section: Introductionmentioning

confidence: 99%

“…36 The rational Arnoldi decomposition by Ruhe 37 has also been successfully used to solve more general problems such as approximation of matrix functions or eigenvalues. 38 In particular, the RKFIT algorithm 39 is also based on an orthogonal rational basis to solve the rational approximation problem. Our method generates the orthogonal rational basis using a regular Arnoldi (or Lanczos) algorithm (unlike RKFIT that uses the rational Arnoldi decomposition algorithm), and it includes an efficient multiport implementation.…”

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confidence: 99%

Orthogonal rational approximation of transfer functions for high‐frequency circuits

Engin

2022

Circuit Theory & Apps

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This paper introduces the orthogonal rational approximation (ORA) algorithm for rational function approximation of transfer functions, based on data available from simulations or measurements. A rational function allows integration of such data in transient solvers for analysis of high-frequency circuits. In rational function approximation, the unknown denominator polynomial of the model results in a nonlinear problem, which can be replaced with successive solutions of linearized problems following the Sanathanan-Koerner (SK) iteration. An orthogonal basis can be obtained based on Arnoldi resulting in a stabilized SK iteration. We present an extension of the stabilized SK, called ORA, which ensures real polynomial coefficients and stable poles for realizability of electrical networks. We also introduce an efficient implementation of ORA for multiport networks based on a block QR decomposition.

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“…This process was first proposed by Ruhe [31] in the context of computing the eigenvalues and have been used during the last years for the approximation of matrix functions, see. [20,30,12,13,14,28]. In this paper, we present the global extended-rational Arnoldi method to approximate the matrix function (1).…”

Section: Introductionmentioning

confidence: 99%

The global extended-rational Arnoldi method for matrix function approximation

Bentbib,

Ghomari,

Jbilou

2020

Preprint

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The numerical computation of matrix functions such as f (A)V , where A is an n × n large and sparse square matrix, V is an n × p block with p ≪ n and f is a nonlinear matrix function, arises in various applications such as network analysis ( f (t) = exp(t) or f (t) = t 3 ), machine learning ( f (t) = log(t)), theory of quantum chromodynamics ( f (t) = t 1/2 ), electronic structure computation, and others. In this work, we propose the use of global extended-rational Arnoldi method for computing approximations of such expressions. The derived method projects the initial problem onto an global extended-rational Krylov sub-An adaptive procedure for the selection of shift parameters {s 1 , . . . , s m } is given. The proposed method is also applied to solve parameter dependent systems. Numerical examples are presented to show the performance of the global extended-rational Arnoldi for these problems.

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A rational Arnoldi process with applications

Cited by 8 publications

References 27 publications

Projection Methods for Ill-Posed Problems Revisited

Projection Methods for Ill-Posed Problems Revisited

Orthogonal rational approximation of transfer functions for high‐frequency circuits

The global extended-rational Arnoldi method for matrix function approximation

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