Block variants of Hammarling's method for solving Lyapunov equations

Kreßner, Daniel

doi:10.1145/1322436.1322437

Cited by 30 publications

(17 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Automatic generation of algorithms for the continuous-time Sylvester equations was developed in Quintana-Orti and van de Geijn [2003]. Kressner developed stable block algorithms based on Hammarling's [1982] method for Lyapunov-type equations in Kressner [2008].…”

Section: Basic Solution Methods For Matrix Equationsmentioning

confidence: 99%

Parallel Solvers for Sylvester-Type Matrix Equations with Applications in Condition Estimation, Part I

Granat

Kågström

2010

ACM Trans. Math. Softw.

View full text Add to dashboard Cite

Parallel ScaLAPACK-style algorithms for solving eight common standard and generalized Sylvester-type matrix equations and various sign and transposed variants are presented. All algorithms are blocked variants based on the Bartels-Stewart method and involve four major steps: reduction to triangular form, updating the right-hand side with respect to the reduction, computing the solution to the reduced triangular problem, and transforming the solution back to the original coordinate system. Novel parallel algorithms for solving reduced triangular matrix equations based on wavefront-like traversal of the right-hand side matrices are presented together with a generic scalability analysis. These algorithms are used in condition estimation and new robust parallel sep~'-estimators are developed. Experimental results from three parallel platforms, including results from a mixed OpenMP/MPI platform, are presented and analyzed using several performance and accuracy metrics. The analysis includes results regarding general and triangular parallel solvers as well as parallel condition estimators. ACM Reference Format:Granat, R. and Kagström, B. 2010. Parallel solvers for Sylvester-type matrix equations with applications in condition estimation. Part I: Theory and algorithms.

show abstract

Section: Basic Solution Methods For Matrix Equationsmentioning

confidence: 99%

Parallel Solvers for Sylvester-Type Matrix Equations with Applications in Condition Estimation, Part I

Granat

Kågström

2010

ACM Trans. Math. Softw.

View full text Add to dashboard Cite

show abstract

“…The computation of the Cholesky factor has some advantages when X is nonsingular but severely ill-conditioned, so that dealing with L significantly improves the accuracy and robustness of computations with X; we refer to [293] for a comparison between Hammarling's and Bartel-Stewart methods. A block variant of the Hammarling's method for the discrete-time Lyapunov equation is suggested in [161], which can dramatically improve the performance of the original scalar (unpartitioned) algorithm on specific machine architectures, while preserving the stability of the original method.…”

Section: Lyapunov Equation Small Scale Computationmentioning

confidence: 99%

Computational Methods for Linear Matrix Equations

Simoncini¹

2016

SIAM Rev.

412

434

View full text Add to dashboard Cite

Given the matrices A, B, C, D and E of conforming dimensions, we consider the linear matrix equation AXE + DXB = C in the unknown matrix X. Our aim is to provide an overview of the major algorithmic developments that have taken place in the past few decades in the numerical solution of this and of related problems, which are becoming a reliable tool in the numerical formulation of advanced application models.

show abstract

“…The , [20] is such a direct method. However, for a Lyapunov equation of the form (1) a variant called Hammarling's method [15], [24] is more suitable, as it takes the low rank structure of the right-hand side into account and directly produces a Cholesky factorization of X. As n increases, these methods exceed the capacities of a serial computing environment.…”

Section: Introductionmentioning

confidence: 99%

Memory-efficient Krylov subspace techniques for solving large-scale Lyapunov equations

Kreßner¹

2008

2008 IEEE International Conference on Computer-Aided Control Systems

Self Cite

View full text Add to dashboard Cite

This paper considers the solution of large-scale Lyapunov matrix equations of the form AX +XA T = −bb T. The Arnoldi method is a simple but sometimes ineffective approach to deal with such equations. One of its major drawbacks is excessive memory consumption caused by slow convergence. To overcome this disadvantage, we propose two-pass Krylov subspace methods, which only compute the solution of the compressed equation in the first pass. The second pass computes the product of the Krylov subspace basis with a low-rank approximation of this solution. For symmetric A, we employ the Lanczos method; for nonsymmetric A, we extend a recently developed restarted Arnoldi method for the approximation of matrix functions. Preliminary numerical experiments reveal that the resulting algorithms require significantly less memory at the expense of extra matrix-vector products.

show abstract

Block variants of Hammarling's method for solving Lyapunov equations

Cited by 30 publications

References 26 publications

Parallel Solvers for Sylvester-Type Matrix Equations with Applications in Condition Estimation, Part I

Parallel Solvers for Sylvester-Type Matrix Equations with Applications in Condition Estimation, Part I

Computational Methods for Linear Matrix Equations

Memory-efficient Krylov subspace techniques for solving large-scale Lyapunov equations

Contact Info

Product

Resources

About