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Monday, May 11, 2020 | History

3 edition of Numerical solution of large Lyapunov equations found in the catalog.

Numerical solution of large Lyapunov equations

Y. Saad

Numerical solution of large Lyapunov equations

by Y. Saad

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Published by Research Institute for Advanced Computer Science, NASA Ames Research Center in [Moffett Field, Calif.] .
Written in English

    Subjects:
  • Control theory.,
  • Controllability.,
  • Galerkin method.,
  • Liapunov functions.,
  • Matrices (Mathematics),
  • Numerical analysis.,
  • Quadratures.

  • Edition Notes

    StatementYoucef Saad.
    SeriesNASA contractor report -- NASA CR-180359., RIACS technical report -- 89.20., RIACS technical report -- TR 89.20.
    ContributionsResearch Institute for Advanced Computer Science (U.S.)
    The Physical Object
    FormatMicroform
    Pagination1 v.
    ID Numbers
    Open LibraryOL16125603M

    e.g. [12], first compute the solution Y of the reduced Lyapunov equation (UT kAU) TY(UT k AU)−Y = −(UT k B)(U T k B) T, (8) where U k is an orthonormal basis of K k(A,B).Then, for suffi-ciently large k, the matrix U kYUT k is a good approximation of X, the solution of the unreduced Lyapunov equation (2). A collection of functions has been implemented as a Julia package to solve several classes of Lyapunov, Sylvester and Riccati matrix equations. The goal was to demonstrate that programs written in the Julia language can achieve high computational performance, which is comparable with the performance of efficient structure exploiting Fortran implementations, as those available.

    An efficient ADI-based solver for large Lyapunov equations is the “workhorse ” of LYAPACK, which also contains implementations of two model reduction methods and modifications of the Newton method for the solution of large Riccati equations and linear-quadratic optimal control problems. First, choose an appropriate and such that the Lyapunov equation yields a nonsingular solution: Then construct the observer as,, where is the observer state vector, is the output, is the input, and is the estimated state vector.

    Book Description. As a satellite conference of the International Mathematical Congress and part of the celebration of the th anniversary of Charles University, the Partial Differential Equations Theory and Numerical Solution conference was held in Prague in August, In this dissertation the author considers the numerical solution of large ( {le} n {le} ) and very large (n {ge} ), sparse Lyapunov equations AX-+ XA' + Q = 0. The author first presents a parallel version of the Hammarling algorithm for the solution of Lyapunov equations where the coefficient matrix A is large and dense.


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Numerical solution of large Lyapunov equations by Y. Saad Download PDF EPUB FB2

This paper introduces a novel framework for the solution of (large-scale) Lyapunov and Sylvester equations derived from numerical integration methods. Suitable systems of ordinary differential Author: Youcef Saad.

A Parameter Free ADI-Like Method for the Numerical Solution of Large Scale Lyapunov Equations Danny C. Sorensen Abstract This work presents an algorithm for constructing an approximate numer-ical solution to a large scale Lyapunov equation in low rank factored form.

The algorithm is based upon a synthesis of an approximate power method and an al. Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems Article in Numerical Linear Algebra with Applications 15(9) - iterative methods for the numerical solution of large Lyapunov equations where the coefficient matrix A is sparse [13, 26, 27, 14, 15, 24].

In this paper we consider the problem of computing an estimate of the dominant low-rank invariant subspace of the exact solution X of the Lya. In the present paper, we consider large-scale differential Lyapunov matrix equations having a low rank constant term.

We present two new approaches for the numerical resolution of such differential Numerical solution of large Lyapunov equations book equations. The first approach is based on the integral expression of the exact solution and an approximation method for the computation of the exponential of a matrix times a block of by: 7.

Large-scale Lyapunov differential equations (LDEs) arise in many fields like: model reduction, damping optimization, optimal control, numerical solution of stochastic partial differential equations (SPDEs), etc.

In particular, LDEs are the key ingredient to perform a simulation of systems governed by certain SPDEs. Numerical solution of large scale Lyapunov equations using Krylov subspace methods. [] Proceedings of the 31st IEEE Conference on Decision and Control, Solution of under-determined Sylvester equations in sensor array signal by: Get this from a library.

Numerical solution of large Lyapunov equations. [Y Saad; Research Institute for Advanced Computer Science (U.S.)]. The numerical solution of large-scale continuous-time Lyapunov matrix equations is of great importance in many application areas.

Assuming that the coefficient matrix is positive definite, but not necessarily symmetric, in this paper we analyze the convergence of projection-type methods for approximating the solution matrix. Under suitable hypotheses on the coefficient matrix, we provide new Cited by: However, all our methods and techniques naturally restrict to the differential Lyapunov equation (DLE) (see comments in Section ).

A more detailed explanation and extensive numerical experiments for the DLE will be presented elsewhere to keep the presentation within usual page limits.

Direct Numerical Solution of Algebraic Lyapunov Equations For Large-Scale Systems Using Quantized Tensor Trains Michael Nip Center for Control, Dynamical Systems, and Computation University of California, Santa Barbara California, USA [email protected] Joao P. Hespanha˜ Center for Control, Dynamical Systems, and Computation.

Experience with numerical experiments reveal that solving Lyapunov equations of order higher than 20 using a companion form of A yields solutions with errors as large as the solutions themselves. Example in Chapter 8 illustrates the danger of solving a Lyapunov equation using the JCFs of A. Lecture notes in numerical linear algebra Numerical methods for Lyapunov equations Methods for Lyapunov equations This chapter is about numerical methods for a particular type of equa-tion expressed as a matrix equality.

The Lyapunov equation is the most com-mon problem in the class of problems called matrix equations. Other examplesFile Size: KB. Sylvester equations. Hodel (U. Auburn, Alabama, US) and B. Tenison have been very helpful answering some questions on their algorithms for solving Lyapunov equations.

Marlis Hochbruck verified a couple of my questions on preconditioned Krylov subspace methods for Lyapunov equations in the Kronecker product form. Numerical Solution of Large Lyapunov Equations Youcef Saad* RIACS, Mail Stop NASA Ames Research Center Moffet Field, California 9._ May, Abstract In this paper we propose a few methods for solving large Lyapunov equations that arise in control problems.

We consider the common case where the right hand side is. The book can be used in a semester course on algebraic Riccati equations or as a reference in a course on advanced numerical linear algebra and applications. Cited By Lin M and Chiang C () An accelerated technique for solving one type of discrete-time algebraic Riccati equations, Journal of Computational and Applied Mathematics, C.

We then present a novel parallel algorithm for the solution of Lyapunov equations where A is large and banded. We provide a detailed analysis of the computational requirements in tandem with the results of numerical experiments with these algorithms on an Alliant FX-8 multiprocessor. To implement the balancing based model reduction of large-scale dynamical systems we need to compute the low-rank (controllability and observability) Gramian factors by solving Lyapunov equations.

In recent time, Rational Krylov Subspace Method (RKSM) is considered as one of the efficient methods for solving the Lyapunov equations of large-scale sparse dynamical by: 1.

P. Benner and J. Saak, Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey, GAMM Mitteilungen 36 (), No. 1, 32– Crossref Google ScholarCited by: We present the approximate power iteration (API) algorithm for the computation of the dominant invariant subspace of the solution X of large-order Lyapunov equations AX + XA T + Q = 0 without first computing the matrix X itself.

The API algorithm is an iterative procedure that uses Krylov subspace bases in computing estimates of matrix-vector products X v in a power iteration by:.

Sorensen D.C. () A Parameter Free ADI-Like Method for the Numerical Solution of Large Scale Lyapunov Equations. In: Glowinski R., Osher S., Yin W. (eds) Splitting Methods in Communication, Imaging, Science, and by: 9.SOLUTION OF THE LYAPUNOV EQUATION where cu and xn are scalars and s, c and x are (n-1) element vectors.

Then Equation () gives the three equations (AI+XJXH = -cn and hence () () Once xu has been found from Equation (), Equation () can be solved, by forward substitution, for x and then Equation () is of the same form as (), but of.P.

Benner and J. Saak, Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey, GAMM Mitteilungen 36 (), No.

1, 32– [18]Author: Peter Benner.