On inexact alternating direction implicit iteration for continuous Sylvester equations

Liu, Zhongyun; Zhou, Yang; Zhang, Y.

doi:10.1002/nla.2320

“…where In represents the initial step, G represents the generator, P represents the predictor at the fractional step k + 1/2, and C represents the corrector for approximating the Moore-Penrose inverse V k+1 of 􏽢 Λ at the k + 1 th step in (11).…”

Section: The Iterative Predictor-corrector Methods For Approximating ...mentioning

confidence: 99%

“…Sylvester Equation. Let col(X) k+1 � V k+1 col(C) be the approximate solution of (14), where V k+1 is the inverse of Λ obtained by the predictor-corrector method given in (11) at the k + 1 − th iteration by using initial approximation V 0 � αΛ T , and α satisfies 0 < α < 2/λ 1 (Λ T Λ) where…”

Section: Algorithm For Approximating Inverse Of Matrix λ To Solvementioning

confidence: 99%

“…erefore, the residual error at the corresponding step is τ k+1 � col(C) − Λcol(X) k+1 . Given a predescribed accuracy ε > 0, we propose the following algorithm based on Algorithm A in [24] that uses the predictor-corrector method (11) and approximates the pseudoinverse for Λ (See Algorithm 1).…”

Section: Algorithm For Approximating Inverse Of Matrix λ To Solvementioning

confidence: 99%

“…Later, a preconditioned positive-de nite skew-Hermitian iteration method of splitting for continuous Sylvester equations with positive-de nite/semi-de nite matrices was presented in [10]. e implicit iteration alternating direction for solving the continuous Sylvester equation, where the coe cient matrices are taking as positive semi-de nite matrices, and at least one of them to be positive de nite is given in [11]. e implicit iteration alternating direction method for solving the continuous Sylvester equation, when the coe cient matrices are taking as positive semide nite matrices (at least one of them to be positive de nite) is given in [12].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical Solution of Sylvester Equation Based on Iterative Predictor‐Corrector Method

Iyikal

¹

2022

Journal of Mathematics

0

View full text Add to dashboard Cite

The inspiration of the study concerns an iterative predictor-corrector method with order of convergence p = 45 for computing the inverse of the coefficient matrix Λ = I n ⊗ A + B T ⊗ I m , which is obtained by the Sylvester equation A X + X B = C . The numerical solutions of three examples by predictor-corrector algorithm are given. The final numerical results also support the applicability, fast convergency, and high accuracy of the method for finding matrix inverses.

show abstract

“…inn is computed according to [37,Equation (3.18b)]. See also [43] for similar results in case of Sylvester equations.…”

Section: (J)mentioning

confidence: 99%

On an integrated Krylov-ADI solver for large-scale Lyapunov equations

Benner¹,

Palitta²,

Saak³

2022

Preprint

0

View full text Add to dashboard Cite

One of the most computationally expensive steps of the low-rank ADI method for large-scale Lyapunov equations is the solution of a shifted linear system at each iteration. We propose the use of the extended Krylov subspace method for this task. In particular, we illustrate how a single approximation space can be constructed to solve all the shifted linear systems needed to achieve a prescribed accuracy in terms of Lyapunov residual norm. Moreover, we show how to fully merge the two iterative procedures in order to obtain a novel, efficient implementation of the low-rank ADI method, for an important class of equations. Many state-of-the-art algorithms for the shift computation can be easily incorporated into our new scheme, as well. Several numerical results illustrate the potential of our novel procedure when compared to an implementation of the low-rank ADI method based on sparse direct solvers for the shifted linear systems.

show abstract

On an integrated Krylov-ADI solver for large-scale Lyapunov equations

Benner

¹

,

Palitta

²

,

Saak

³

2022

View full text Add to dashboard Cite

One of the most computationally expensive steps of the low-rank ADI method for large-scale Lyapunov equations is the solution of a shifted linear system at each iteration. We propose the use of the extended Krylov subspace method for this task. In particular, we illustrate how a single approximation space can be constructed to solve all the shifted linear systems needed to achieve a prescribed accuracy in terms of Lyapunov residual norm. Moreover, we show how to fully merge the two iterative procedures in order to obtain a novel, efficient implementation of the low-rank ADI method, for an important class of equations. Many state-of-the-art algorithms for the shift computation can be easily incorporated into our new scheme, as well. Several numerical results illustrate the potential of our novel procedure when compared to an implementation of the low-rank ADI method based on sparse direct solvers for the shifted linear systems.

show abstract

On inexact alternating direction implicit iteration for continuous Sylvester equations

Cited by 9 publications

References 26 publications

Numerical Solution of Sylvester Equation Based on Iterative Predictor‐Corrector Method

Numerical Solution of Sylvester Equation Based on Iterative Predictor‐Corrector Method

On an integrated Krylov-ADI solver for large-scale Lyapunov equations

On an integrated Krylov-ADI solver for large-scale Lyapunov equations

Contact Info

Product

Resources

About