The Experts below are selected from a list of 186 Experts worldwide ranked by ideXlab platform
Mirko Myllykoski - One of the best experts on this subject based on the ideXlab platform.
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A Task-based Multi-shift QR/QZ Algorithm with Aggressive Early Deflation.
arXiv: Mathematical Software, 2020Co-Authors: Mirko MyllykoskiAbstract:The QR algorithm is one of the three phases in the process of computing the eigenvalues and the eigenvectors of a dense nonsymmetric matrix. This paper describes a task-based QR algorithm for reducing an upper Hessenberg matrix to Real Schur Form. The task-based algorithm also supports generalized eigenvalue problems (QZ algorithm) but this paper focuses more on the standard case. The task-based algorithm inherits previous algorithmic improvements, such as tightly-coupled multi-shifts and Aggressive Early Deflation (AED), and also incorporates several new ideas that significantly improve the perFormance. This includes the elimination of several synchronization points, the dynamic merging of previously separate computational steps, the shorting and the prioritization of the critical path, and the introduction of an experimental GPU support. The task-based implementation is demonstrated to be significantly faster than multi-threaded LAPACK and ScaLAPACK in both single-node and multi-node configurations on two different machines based on Intel and AMD CPUs. The implementation is built on top of the StarPU runtime system and is part of an open-source StarNEig library.
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Parallel Robust Computation of Generalized Eigenvectors of Matrix Pencils
Parallel Processing and Applied Mathematics, 2020Co-Authors: Carl Christian Kjelgaard Mikkelsen, Mirko MyllykoskiAbstract:In this paper we consider the problem of computing generalized eigenvectors of a matrix pencil in Real Schur Form. In exact arithmetic, this problem can be solved using substitution. In practice, substitution is vulnerable to floating-point overflow. The robust solvers xTGEVC in LAPACK prevent overflow by dynamically scaling the eigenvectors. These subroutines are sequential scalar codes which compute the eigenvectors one by one. In this paper we discuss how to derive robust blocked algorithms. The new StarNEig library contains a robust task-parallel solver Zazamoukh which runs on top of StarPU. Our numerical experiments show that Zazamoukh achieves a super-linear speedup compared with DTGEVC for sufficiently large matrices.
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PPAM (1) - A Task-Based Algorithm for Reordering the Eigenvalues of a Matrix in Real Schur Form
Parallel Processing and Applied Mathematics, 2018Co-Authors: Mirko MyllykoskiAbstract:A task-based parallel algorithm for reordering the eigenvalues of a matrix in Real Schur Form is presented. The algorithm is Realized on top of the StarPU runtime system. Only the aspects which are relevant for shared memory machines are discussed here, but the implementation can be configured to run on distributed memory machines as well. Various techniques to reduce the overhead and the core idle time are discussed. Computational experiments indicate that the new algorithm is between 1.5 and 6.6 times faster than a state of the art MPI-based implementation found in ScaLAPACK. With medium to large matrices, strong scaling efficiencies above 60% up to 28 CPU cores are reported. The overhead and the core idle time are shown to be negligible with the exception of the smallest matrices and highest core counts.
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a task based algorithm for reordering the eigenvalues of a matrix in Real Schur Form
International Conference on Parallel Processing, 2017Co-Authors: Mirko MyllykoskiAbstract:A task-based parallel algorithm for reordering the eigenvalues of a matrix in Real Schur Form is presented. The algorithm is Realized on top of the StarPU runtime system. Only the aspects which are relevant for shared memory machines are discussed here, but the implementation can be configured to run on distributed memory machines as well. Various techniques to reduce the overhead and the core idle time are discussed. Computational experiments indicate that the new algorithm is between 1.5 and 6.6 times faster than a state of the art MPI-based implementation found in ScaLAPACK. With medium to large matrices, strong scaling efficiencies above 60% up to 28 CPU cores are reported. The overhead and the core idle time are shown to be negligible with the exception of the smallest matrices and highest core counts.
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Task-Based Parallel Algorithms for Eigenvalue Reordering of Matrices in Real Schur Forms
2017Co-Authors: Mirko Myllykoski, Lars Karlsson, Carl Christian Kjelgaard Mikkelsen, Bo KagstromAbstract:We develop a task-based parallel algorithm for reordering eigenvalues of matrices in Real Schur Form. We describe how we implemented the algorithm using StarPU runtime system and report on experime ...
Bo Kagstrom - One of the best experts on this subject based on the ideXlab platform.
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Task-Based Parallel Algorithms for Eigenvalue Reordering of Matrices in Real Schur Forms
2017Co-Authors: Mirko Myllykoski, Lars Karlsson, Carl Christian Kjelgaard Mikkelsen, Bo KagstromAbstract:We develop a task-based parallel algorithm for reordering eigenvalues of matrices in Real Schur Form. We describe how we implemented the algorithm using StarPU runtime system and report on experime ...
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Task-Based Parallel Algorithms for Eigenvalue Reordering of Matrices in Real Schur Forms
'Umea University Library', 2017Co-Authors: Myllykoski Mirko, Kjelgaard Mikkelsen, Carl Christian, Karlsson Lars, Bo KagstromAbstract:We develop a task-based parallel algorithm for reordering eigenvalues of matrices in Real Schur Form. We describe how we implemented the algorithm using StarPU runtime system and report on experiments perFormed on a shared memory machine. Compared with ScaLAPACK we achieve average speedup of 3. We have strong and weak scaling efficiencies which are well above 50%. We are able to achieve more than 50% of the peak flop rate for all but the smallest matrices. The idle time and the overhead is negligible except for the smallest matrices. The next step is to reconfigure and further develop the code so that it can be applied to matrix pairs in generalized Schur Forms and run efficiently on distributed memory machines.NLAFE
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A numerical evaluation of solvers for the periodic Riccati differential equation
BIT Numerical Mathematics, 2010Co-Authors: Sergei Gusev, Bo Kagstrom, Stefan Johansson, Anton Shiriaev, Andras VargaAbstract:Efficient and accurate structure exploiting numerical methods for solving the periodic Riccati differential equation (PRDE) are addressed. Such methods are essential, for example, to design periodic feedback controllers for periodic control systems. Three recently proposed methods for solving the PRDE are presented and evaluated on challenging periodic linear artificial systems with known solutions and applied to the stabilization of periodic motions of mechanical systems. The first two methods are of the type multiple shooting and rely on computing the stable invariant subspace of an associated Hamiltonian system. The stable subspace is determined using either algorithms for computing an ordered periodic Real Schur Form of a cyclic matrix sequence, or a recently proposed method which implicitly constructs a stable deflating subspace from an associated lifted pencil. The third method reFormulates the PRDE as a convex optimization problem where the stabilizing solution is approximated by its truncated Fourier series. As known, this reFormulation leads to a semidefinite programming problem with linear matrix inequality constraints admitting an effective numerical Realization. The numerical evaluation of the PRDE methods, with focus on the number of states ( n ) and the length of the period ( T ) of the periodic systems considered, includes both quantitative and qualitative results.
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a parallel Schur method for solving continuous time algebraic riccati equations
IEEE International Symposium on Computer Aided Control System Design, 2008Co-Authors: Robert Granat, Bo Kagstrom, Daniel KressnerAbstract:Numerical algorithms for solving the continuous-time algebraic Riccati matrix equation on a distributed memory parallel computer are considered. In particular, it is shown that the Schur method, based on computing the stable invariant subspace of a Hamiltonian matrix, can be parallelized in an efficient and scalable way. Our implementation employs the state-of-the-art library ScaLAPACK as well as recently developed parallel methods for reordering the eigenvalues in a Real Schur Form. Some experimental results are presented, confirming the scalability of our implementation and comparing it with an existing implementation of the matrix sign iteration from the PLiCOC library.
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CACSD - A parallel Schur method for solving continuous-time algebraic Riccati equations
2008 IEEE International Conference on Computer-Aided Control Systems, 2008Co-Authors: Robert Granat, Bo Kagstrom, Daniel KressnerAbstract:Numerical algorithms for solving the continuous-time algebraic Riccati matrix equation on a distributed memory parallel computer are considered. In particular, it is shown that the Schur method, based on computing the stable invariant subspace of a Hamiltonian matrix, can be parallelized in an efficient and scalable way. Our implementation employs the state-of-the-art library ScaLAPACK as well as recently developed parallel methods for reordering the eigenvalues in a Real Schur Form. Some experimental results are presented, confirming the scalability of our implementation and comparing it with an existing implementation of the matrix sign iteration from the PLiCOC library.
Xu Shu-fang - One of the best experts on this subject based on the ideXlab platform.
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A modified Schur method for robust pole assignment in state feedback control
automatica, 2015Co-Authors: Guo Zhen-chen, Cai Yun-feng, Qian Jiang, Xu Shu-fangAbstract:Recently, a Schur method was proposed in Chu (2007) to solve the robust pole assignment problem in state feedback control. It takes the departure from normality of the closed-loop system matrix A(c) as the measure of robustness, and intends to minimize it via the Real Schur Form of A(c). The Schur method works well for Real poles, but when complex conjugate poles are involved, it does not produce the Real Schur Form of A(c) and can be problematic. In this paper, we propose a modified Schur method, which improves Schur when nonReal poles are to be assigned. Besides producing the Real Schur Form of A(c), our approach also leads to a relatively small departure from normality of A(c). Numerical examples show that our modified method produces better or at least comparable results than both place and robpole algorithms, with much less computational costs. (C) 2014 Elsevier Ltd. All rights reserved.Automation & Control SystemsEngineering, Electrical & ElectronicSCI(E)EI0ARTICLEguozhch06@gmail.com; yfcai@math.pku.edu.cn; jqian104@gmail.com; xsf@pku.edu.cn334-3395
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A Modified Schur Method for Robust Pole Assignment in State Feedback Control
arXiv: Optimization and Control, 2014Co-Authors: Guo Zhen-chen, Cai Yun-feng, Qian Jiang, Xu Shu-fangAbstract:Recently, a \textbf{Schur} method was proposed in \cite{Chu2} to solve the robust pole assignment problem in state feedback control. It takes the departure from normality of the closed-loop system matrix $A_c$ as the measure of robustness, and intends to minimize it via the Real Schur Form of $A_c$. The \textbf{Schur} method works well for Real poles, but when complex conjugate poles are involved, it does not produce the Real Schur Form of $A_c$ and can be problematic. In this paper, we put forward a modified Schur method, which improves the efficiency of \textbf{Schur} when complex conjugate poles are to be assigned. Besides producing the Real Schur Form of $A_c$, our approach also leads to a relatively small departure from normality of $A_c$. Numerical examples show that our modified method produces better or at least comparable results than both \textbf{place} and \textbf{robpole} algorithms, with much less computational costs.
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A Modified Schur Method for Robust Pole Assignment in State Feedback Control
2014Co-Authors: Zhen-chen Guo, Yunfeng Cai, Jiang Qian, Xu Shu-fangAbstract:Recently, a \textbf{Schur} method was proposed in \cite{Chu2} to solve the robust pole assignment problem in state feedback control. It takes the departure from normality of the closed-loop system matrix $A_c$ as the measure of robustness, and intends to minimize it via the Real Schur Form of $A_c$. The \textbf{Schur} method works well for Real poles, but when complex conjugate poles are involved, it does not produce the Real Schur Form of $A_c$ and can be problematic. In this paper, we put forward a modified Schur method, which improves the efficiency of \textbf{Schur} when complex conjugate poles are to be assigned. Besides producing the Real Schur Form of $A_c$, our approach also leads to a relatively small departure from normality of $A_c$. Numerical examples show that our modified method produces better or at least comparable results than both \textbf{place} and \textbf{robpole} algorithms, with much less computational costs.Comment: 24 pages, 4 figure
Jiang Qian - One of the best experts on this subject based on the ideXlab platform.
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A modified Schur method for robust pole assignment in state feedback control
Automatica, 2015Co-Authors: Zhen-chen Guo, Yunfeng Cai, Jiang QianAbstract:Recently, a Schur method was proposed in Chu (2007) to solve the robust pole assignment problem in state feedback control. It takes the departure from normality of the closed-loop system matrix A c as the measure of robustness, and intends to minimize it via the Real Schur Form of A c . The Schur method works well for Real poles, but when complex conjugate poles are involved, it does not produce the Real Schur Form of A c and can be problematic. In this paper, we propose a modified Schur method, which improves Schur when nonReal poles are to be assigned. Besides producing the Real Schur Form of A c , our approach also leads to a relatively small departure from normality of A c . Numerical examples show that our modified method produces better or at least comparable results than both place and robpole algorithms, with much less computational costs.
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A Modified Schur Method for Robust Pole Assignment in State Feedback Control
2014Co-Authors: Zhen-chen Guo, Yunfeng Cai, Jiang Qian, Xu Shu-fangAbstract:Recently, a \textbf{Schur} method was proposed in \cite{Chu2} to solve the robust pole assignment problem in state feedback control. It takes the departure from normality of the closed-loop system matrix $A_c$ as the measure of robustness, and intends to minimize it via the Real Schur Form of $A_c$. The \textbf{Schur} method works well for Real poles, but when complex conjugate poles are involved, it does not produce the Real Schur Form of $A_c$ and can be problematic. In this paper, we put forward a modified Schur method, which improves the efficiency of \textbf{Schur} when complex conjugate poles are to be assigned. Besides producing the Real Schur Form of $A_c$, our approach also leads to a relatively small departure from normality of $A_c$. Numerical examples show that our modified method produces better or at least comparable results than both \textbf{place} and \textbf{robpole} algorithms, with much less computational costs.Comment: 24 pages, 4 figure
Andreas Varga - One of the best experts on this subject based on the ideXlab platform.
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Computation of transfer function matrices of periodic systems
International Journal of Control, 2003Co-Authors: Andreas VargaAbstract:We present a numerical approach to evaluate the transfer function matrices of a periodic system corresponding to lifted state-space representations as constant systems. The proposed pole-zero method determines each entry of the transfer function matrix in a minimal zeros-poles-gain representation. A basic computation is the minimal Realization of special single-input single-output periodic systems, for which both balancing-related as well as orthogonal periodic Kalman Forms based algorithms can be employed. The main computational ingredient to compute poles is the extended periodic Real Schur Form of a periodic matrix. This Form also underlies the solution of periodic Lyapunov equations when computing minimal Realizations via balancing-related techniques. To compute zeros and gains, numerically stable fast algorithms are proposed, which are specially tailored to particular single-input single-output periodic systems. The new method relies exclusively on reliable numerical computations and is well suited f...
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Computation of transfer function matrices of periodic systems
Proceedings of the 41st IEEE Conference on Decision and Control 2002., 1Co-Authors: Andreas VargaAbstract:We present a numerical approach to evaluate the transfer function matrices of a periodic system corresponding to lifted state-space representations as constant systems. The proposed pole-zero method determines each entry of the transfer function matrix in a minimal zeros-poles-gain representation. A basic computational ingredient for this method is the extended periodic Real Schur Form of a periodic matrix, which underlies the computation of minimal Realizations and system poles. To compute zeros and gains, fast algorithms are proposed, which are specially tailored to particular single-input single-output periodic systems. The new method relies exclusively on reliable numerical computations and is well suited for robust software implementations.
