Scaling, Sensitivity and Stability in Numerical Solution of the Quadratic Eigenvalue Problem
|
|
- Clinton Preston
- 5 years ago
- Views:
Transcription
1 Scaling, Sensitivity and Stability in Numerical Solution of the Quadratic Eigenvalue Problem Nick Higham School of Mathematics The University of Manchester Joint work with Seamus Garvey, Steve Mackey and Françoise Tisseur
2 NLEVP Toolbox with T. Betcke, V. Mehrmann, C. Schröder, F. Tisseur Collection of Nonlinear Eigenvalue Problems : F(λ)x = 0, where F : C C m n. Provided as a MATLAB Toolbox. Problems from real-life applications + specially constructed problems. Available from numerical-analysis/nlevp.html MIMS Nick Higham Quadratic Eigenproblem 2 / 31
3 NLEVP Toolbox with T. Betcke, V. Mehrmann, C. Schröder, F. Tisseur Collection of Nonlinear Eigenvalue Problems : F(λ)x = 0, where F : C C m n. Provided as a MATLAB Toolbox. Problems from real-life applications + specially constructed problems. Available from numerical-analysis/nlevp.html Further contributions are welcome. MIMS Nick Higham Quadratic Eigenproblem 2 / 31
4 Outline 1 QEP and Linearization Background 2 Conditioning of Linearizations 3 Scaling 4 Algorithm based on Linearization MIMS Nick Higham Quadratic Eigenproblem 3 / 31
5 Quadratic Eigenproblems Consider Q(λ) = λ 2 M + λd + K, M, D, K C n n. QEP: find scalars λ and nonzero x, y C n satisfying Q(λ)x = 0 and y Q(λ) = 0. λ is an e val, x, y are corresponding right and left e vecs. Q(λ) has 2n eigenvalues, solutions of det(q(λ)) = 0. MIMS Nick Higham Quadratic Eigenproblem 4 / 31
6 Quadratic Eigenproblems Consider Q(λ) = λ 2 M + λd + K, M, D, K C n n. QEP: find scalars λ and nonzero x, y C n satisfying Q(λ)x = 0 and y Q(λ) = 0. λ is an e val, x, y are corresponding right and left e vecs. Q(λ) has 2n eigenvalues, solutions of det(q(λ)) = 0. When λ =, consider homogeneous form of Q: Q(α,β) = α 2 M + αβd + β 2 K. E vals are pairs (α,β) (0, 0) s.t. det Q(α,β) = 0. MIMS Nick Higham Quadratic Eigenproblem 4 / 31
7 Linearizations L(λ) = λx + Y, X, Y C 2n 2n is a linearization of Q(λ) = λ 2 M + λd + K if [ ] Q(λ) 0 E(λ)L(λ)F(λ) = 0 I n for some unimodular E(λ) and F(λ). Example [ ] M 0 For companion pencil C 1 (λ) = λ + 0 I n ( ) holds with [ In λm + D E(λ) = 0 I n ( ) [ ] D K, I n 0 ] [ ] λin I, F(λ) = n. I n 0 MIMS Nick Higham Quadratic Eigenproblem 5 / 31
8 Solution Process for QEP Linearize Q(λ) into L(λ) = λx + Y. Solve generalized eigenproblem L(λ)z = 0. Recover eigenvectors of Q from those of L. Usual choice of linearization: companion linearization, [ ] [ ] M 0 D K C 1 (λ) = λ + 0 I I 0 for which right and left e vecs have the form [ ] λx z =, w = x x, y being right and left e vecs of Q(λ). [ y λk y MIMS Nick Higham Quadratic Eigenproblem 6 / 31 ],
9 Beam Problem /////// L /////// //////////// Transverse displacement u(x, t) governed by ρa 2 u t 2 + c(x) u t + EI 4 u x 4 = 0. Boundary conditions: u(x, t) = u (x, t) = 0 at x = 0, L. u(x, t) = e λt v(x,λ) yields e val problem for the free vibrations : λ 2 ρav(x,λ) + λc(x)v(x,λ) + EI 4 v(x,λ) = 0. x4 MIMS Nick Higham Quadratic Eigenproblem 7 / 31
10 Discretized Beam Problem Finite element method leads to Q(λ) = λ 2 M + λd + K with symmetric M, D, K R n n. Roots of x Q(λ)x = 0, λ = (x Dx) ± (x Dx) 2 4(x Mx)(x Kx). 2(x Mx) M > 0, K > 0, D 0 all ei vals have Re(λ) 0. D is rank 1. Can show n pure imaginary ei vals. MIMS Nick Higham Quadratic Eigenproblem 8 / 31
11 Eigenvalues of Q via First Companion C 1 Q(λ) = λ 2 M + λd + K, C 1 (λ) = λ [ ] [ ] M 0 D K +. 0 I I 0 nele = 100; coeffs = nlevp( damped_beam,nele); K = coeffs{1}; D = coeffs{2}; M = coeffs{3}; I = eye(2*nele); O = zeros(2*nele); eval = eig([d K; -I O],-[M O; O I]); plot(eval,.r ) MIMS Nick Higham Quadratic Eigenproblem 9 / 31
12 [ eig on Companion C 1 (λ)=λ M0 ] [ 0 I + D I ] K 0 4 x MIMS Nick Higham Quadratic Eigenproblem 10 / 31
13 eig on Linearization L 1 (λ)=λ[ M0 ] [ 0 K + DK ] K 0 4 x MIMS Nick Higham Quadratic Eigenproblem 11 / 31
14 [ eig on Linearization L 2 (λ)=λ 0M ] [ M M0 ] D + 0 K 4 x MIMS Nick Higham Quadratic Eigenproblem 12 / 31
15 Spectrum of Beam Problem 4 x MIMS Nick Higham Quadratic Eigenproblem 13 / 31
16 Sensitivity and Stability of Linearizations Condition number measures sensitivity of the solution of a problem to perturbations in the data. Backward error measures how well the problem has been solved. error in solution < condition number backward error. MIMS Nick Higham Quadratic Eigenproblem 14 / 31
17 Sensitivity and Stability of Linearizations Condition number measures sensitivity of the solution of a problem to perturbations in the data. Backward error measures how well the problem has been solved. error in solution < condition number backward error. For a given Q(λ), infinitely many linearizations exist: can have widely varying eigenvalue condition numbers, computed eigenpairs can have widely varying backward errors. MIMS Nick Higham Quadratic Eigenproblem 14 / 31
18 Desiderata for a Linearization Good conditioning. Backward stability. Suitable eigenvector recovery formulae. Preservation of structure, e.g. symmetry. Numerical preservation of key qualitative properties, including location and symmetries of spectrum. Preserve partial multiplicities of e vals (strong linearization). MIMS Nick Higham Quadratic Eigenproblem 15 / 31
19 Vector Spaces L 1, L 2 Mackey, Mackey, Mehl & Mehrmann (2006) define L 1 (Q) = { L(λ) : L(λ)(Λ I n ) = v Q(λ), v C } 2, L 2 (Q) = { L(λ) : (Λ T I n )L(λ) = ṽ T Q(λ), ṽ C } 2, where Λ := [λ, 1] T. MIMS Nick Higham Quadratic Eigenproblem 16 / 31
20 Vector Spaces L 1, L 2 Mackey, Mackey, Mehl & Mehrmann (2006) define L 1 (Q) = { L(λ) : L(λ)(Λ I n ) = v Q(λ), v C 2 }, L 2 (Q) = { L(λ) : (Λ T I n )L(λ) = ṽ T Q(λ), ṽ C 2 }, where Λ := [λ, 1] T. L(λ) = λx + Y L 1 (Q) with v C 2 iff [ ] [ v1 M v 1 D v 1 K X11 X = 12 + Y 11 Y 12 v 2 M v 2 D v 2 K X 21 X 22 + Y 21 Y 22 ]. MIMS Nick Higham Quadratic Eigenproblem 16 / 31
21 Vector Spaces L 1, L 2 Mackey, Mackey, Mehl & Mehrmann (2006) define L 1 (Q) = { L(λ) : L(λ)(Λ I n ) = v Q(λ), v C } 2, L 2 (Q) = { L(λ) : (Λ T I n )L(λ) = ṽ T Q(λ), ṽ C } 2, where Λ := [λ, 1] T. Dimensions: L 1, L 2 : 2n Almost all pencils in L 1 and L 2 are linearizations. MIMS Nick Higham Quadratic Eigenproblem 16 / 31
22 Eigenvector Recovery for L 1 (Q) L 1 (Q) = { L(λ) : L(λ)(Λ I n ) = v Q(λ), v C 2 } Λ := [λ, 1] T. If L L 1 (Q) with vector v then every right e vec of L with finite e val λ is of the form Λ x for some right e vec x of P, [M 4, 2006] if w is a left e vec of L with e val λ then y = (v I n )w is a left e vec of P with e val λ. [H, Li, Tisseur, 2007]. E vecs of Q easily recovered from e vecs of L L 1. MIMS Nick Higham Quadratic Eigenproblem 17 / 31
23 Outline 1 QEP and Linearization Background 2 Conditioning of Linearizations 3 Scaling 4 Algorithm based on Linearization MIMS Nick Higham Quadratic Eigenproblem 18 / 31
24 Eigenvalue Condition Numbers κ Q (λ) Q(λ)x = 0, y Q(λ) = 0. Q(λ) = λ 2 M + λ D + K For λ simple, nonzero and finite, { λ κ Q (λ) = lim sup : [ (Q + Q)(λ + λ) ] (x + x) = 0, ǫ 0 ǫ λ } M 2 ǫm, D 2 ǫd, K 2 ǫk, κ Q (λ) = ( λ 2 m + λ d + k ) y 2 x 2. (Tisseur, 2000) λ y (2λM + D)x MIMS Nick Higham Quadratic Eigenproblem 19 / 31
25 Eigenvalue Conditioning of Linearizations For L(λ) = λx + Y, L(λ)z = 0, w L(λ) = 0, κ L (λ) = ( λ X 2 + Y 2 ) w 2 z 2. λ w Xz Define growth factor φ L : κ L (λ) = φ L (λ) κ Q (λ). Theorem (H, Mackey, Tisseur, 2006) Let L(λ) = λx + Y B(Q) with vector v. For λ simple, nonzero and finite, where Λ = [λ, 1] T. φ L (λ; v) = λ X 2 + Y 2 λ 2 m + λd + k Λ 2 2 Λ T v, MIMS Nick Higham Quadratic Eigenproblem 20 / 31
26 Sufficient conditions for κ Q κ L ρ = max(m, d, k)/ min(m, k), Linearization Eigenvalue Condition [ ] [ M 0 D K L 1 (λ) = λ + 0 K K 0 C 1 No restriction m d k 1 λ > L 1 1 ρ 1 λ 1 not available" λ > L 2 1 not available" λ 1 ρ 1 [ ] [ M 0 D K C 1 (λ) = λ + 0 I I 0 ], L 2 (λ) = λ [ 0 M M D ], ] [ ] M K MIMS Nick Higham Quadratic Eigenproblem 21 / 31
27 Beam Problem M 2 = , D 2 = 5, K 2 = Thus ρ = beam problem is badly scaled. Approximations to growth factors φ L (λ) = κ L (λ)/κ Q (λ): φ C1 (λ) φ L1 (λ) φ L2 (λ) λ = λ = λ = For λ = 10 6, ǫ 10 16, λ < ǫ λ κ Li (λ) = ǫ λ φ Li (λ)κ Q (λ) = O(1), i = 1, 2. E vals on imaginary axis can be perturbed by distance O(1) into the right half-plane. MIMS Nick Higham Quadratic Eigenproblem 22 / 31
28 Computed Spectrum of L 1, L 2 and C 1 4 x x x MIMS Nick Higham Quadratic Eigenproblem 23 / 31
29 Outline 1 QEP and Linearization Background 2 Conditioning of Linearizations 3 Scaling 4 Algorithm based on Linearization MIMS Nick Higham Quadratic Eigenproblem 24 / 31
30 Scaling Q(λ) = λ 2 M + λd + K Let λ = µγ and convert Q(λ) = λ 2 M + λd + K δq(µγ) = µ 2 (δγ 2 M) + µ(δγd) + δk = µ 2 M + µ D + K =: Q(µ), where γ = K 2 / M 2, δ = 2/( K 2 + D 2 γ). Fan, Lin and Van Dooren (2004). 2/3 max( M 2, D 2, K 2 ) 2. Does not affect sparsity of M, D, K. Has no effect on κ Q and η Q. γ minimizes scaling factor ρ. MIMS Nick Higham Quadratic Eigenproblem 25 / 31
31 Effect of Scaling on Beam Problem Before scaling After scaling M D K ρ = ρ = 1 Our theory guarantees optimal conditioning and stability for the companion linearization, E val bound µ 1 2 τκ 2(M) ( /(τ 2 κ 2 (M)) ) = Can show this implies symm linearization L 2 optimal in terms of both conditioning and stability. MIMS Nick Higham Quadratic Eigenproblem 26 / 31
32 Spectrum of C 1, L 2 before/after Scaling 4 x x x x MIMS Nick Higham Quadratic Eigenproblem 27 / 31
33 Outline 1 QEP and Linearization Background 2 Conditioning of Linearizations 3 Scaling 4 Algorithm based on Linearization MIMS Nick Higham Quadratic Eigenproblem 28 / 31
34 Meta-Algorithm for PEP 1 Balance, scale P (Fan, Lin & Van Dooren, 2004) 2 for one or more (scaled) linearizations L 3 Deflate L 4 Balance, scale L 5 Apply QZ to L (maybe HZ if structured) 6 Obtain relevant e vals 7 Recover left and right e vecs 8 Iteratively refine e vecs 9 Compute/estimate b errs and condition numbers 10 Detect nonregular problem 11 end MIMS Nick Higham Quadratic Eigenproblem 29 / 31
35 Balancing Ward (1981) for pencils. Lemonnier & Van Dooren (2006) for pencils. Betcke (2009) for polynomials. MIMS Nick Higham Quadratic Eigenproblem 30 / 31
36 Concluding Remarks Analysis of conditioning & backward error for wide variety of linearizations. E vector recovery formulae crucial. Scaling crucial. Favour L = companion form for general QEPs. Results useful to develop a general QEP algorithm & code. New version ofpolyeig in preparation. For papers and Eprints, MIMS Nick Higham Quadratic Eigenproblem 31 / 31
37 References I T. Betcke. Optimal scaling of generalized and polynomial eigenvalue problems. SIAM J. Matrix Anal. Appl., 30(4): , T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder, and F. Tisseur. NLEVP: A collection of nonlinear eigenvalue problems. research/numerical-analysis/nlevp.html. MIMS Nick Higham Quadratic Eigenproblem 27 / 31
38 References II T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder, and F. Tisseur. NLEVP: A collection of nonlinear eigenvalue problems. MIMS EPrint , Manchester Institute for Mathematical Sciences, The University of Manchester, UK, Apr pp. N. J. Higham, R.-C. Li, and F. Tisseur. Backward error of polynomial eigenproblems solved by linearization. SIAM J. Matrix Anal. Appl., 29(4): , MIMS Nick Higham Quadratic Eigenproblem 28 / 31
39 References III N. J. Higham, D. S. Mackey, N. Mackey, and F. Tisseur. Symmetric linearizations for matrix polynomials. SIAM J. Matrix Anal. Appl., 29(1): , N. J. Higham, D. S. Mackey, and F. Tisseur. The conditioning of linearizations of matrix polynomials. SIAM J. Matrix Anal. Appl., 28(4): , N. J. Higham, D. S. Mackey, F. Tisseur, and S. D. Garvey. Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems. Internat. J. Numer. Methods Eng., 73(3): , MIMS Nick Higham Quadratic Eigenproblem 29 / 31
40 References IV D. Lemonnier and P. M. Van Dooren. Balancing regular matrix pencils. SIAM J. Matrix Anal. Appl., 28(1): , D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann. Structured polynomial eigenvalue problems: Good vibrations from good linearizations. SIAM J. Matrix Anal. Appl., 28(4): , D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann. Vector spaces of linearizations for matrix polynomials. SIAM J. Matrix Anal. Appl., 28(4): , MIMS Nick Higham Quadratic Eigenproblem 30 / 31
41 References V R. C. Ward. Balancing the generalized eigenvalue problem. SIAM J. Sci. Statist. Comput., 2(2): , MIMS Nick Higham Quadratic Eigenproblem 31 / 31
An Algorithm for. Nick Higham. Françoise Tisseur. Director of Research School of Mathematics.
An Algorithm for the Research Complete Matters Solution of Quadratic February Eigenvalue 25, 2009 Problems Nick Higham Françoise Tisseur Director of Research School of Mathematics The School University
More informationSolving the Polynomial Eigenvalue Problem by Linearization
Solving the Polynomial Eigenvalue Problem by Linearization Nick Higham School of Mathematics The University of Manchester higham@ma.man.ac.uk http://www.ma.man.ac.uk/~higham/ Joint work with Ren-Cang Li,
More informationAlgorithms for Solving the Polynomial Eigenvalue Problem
Algorithms for Solving the Polynomial Eigenvalue Problem Nick Higham School of Mathematics The University of Manchester higham@ma.man.ac.uk http://www.ma.man.ac.uk/~higham/ Joint work with D. Steven Mackey
More informationRecent Advances in the Numerical Solution of Quadratic Eigenvalue Problems
Recent Advances in the Numerical Solution of Quadratic Eigenvalue Problems Françoise Tisseur School of Mathematics The University of Manchester ftisseur@ma.man.ac.uk http://www.ma.man.ac.uk/~ftisseur/
More informationSolving Polynomial Eigenproblems by Linearization
Solving Polynomial Eigenproblems by Linearization Nick Higham School of Mathematics University of Manchester higham@ma.man.ac.uk http://www.ma.man.ac.uk/~higham/ Joint work with D. Steven Mackey and Françoise
More informationThe quadratic eigenvalue problem (QEP) is to find scalars λ and nonzero vectors u satisfying
I.2 Quadratic Eigenvalue Problems 1 Introduction The quadratic eigenvalue problem QEP is to find scalars λ and nonzero vectors u satisfying where Qλx = 0, 1.1 Qλ = λ 2 M + λd + K, M, D and K are given
More informationResearch Matters. February 25, The Nonlinear Eigenvalue Problem. Nick Higham. Part III. Director of Research School of Mathematics
Research Matters February 25, 2009 The Nonlinear Eigenvalue Problem Nick Higham Part III Director of Research School of Mathematics Françoise Tisseur School of Mathematics The University of Manchester
More informationAn Algorithm for the Complete Solution of Quadratic Eigenvalue Problems. Hammarling, Sven and Munro, Christopher J. and Tisseur, Francoise
An Algorithm for the Complete Solution of Quadratic Eigenvalue Problems Hammarling, Sven and Munro, Christopher J. and Tisseur, Francoise 2011 MIMS EPrint: 2011.86 Manchester Institute for Mathematical
More informationQuadratic Matrix Polynomials
Research Triangularization Matters of Quadratic Matrix Polynomials February 25, 2009 Nick Françoise Higham Tisseur Director School of of Research Mathematics The University of Manchester School of Mathematics
More informationSOLVING RATIONAL EIGENVALUE PROBLEMS VIA LINEARIZATION
SOLVNG RATONAL EGENVALUE PROBLEMS VA LNEARZATON YANGFENG SU AND ZHAOJUN BA Abstract Rational eigenvalue problem is an emerging class of nonlinear eigenvalue problems arising from a variety of physical
More informationAn Algorithm for the Complete Solution of Quadratic Eigenvalue Problems
An Algorithm for the Complete Solution of Quadratic Eigenvalue Problems SVEN HAMMARLING, Numerical Algorithms Group Ltd. and The University of Manchester CHRISTOPHER J. MUNRO, Rutherford Appleton Laboratory
More informationBackward Error of Polynomial Eigenproblems Solved by Linearization. Higham, Nicholas J. and Li, Ren-Cang and Tisseur, Françoise. MIMS EPrint: 2006.
Backward Error of Polynomial Eigenproblems Solved by Linearization Higham, Nicholas J and Li, Ren-Cang and Tisseur, Françoise 2007 MIMS EPrint: 2006137 Manchester Institute for Mathematical Sciences School
More informationPolynomial eigenvalue solver based on tropically scaled Lagrange linearization. Van Barel, Marc and Tisseur, Francoise. MIMS EPrint: 2016.
Polynomial eigenvalue solver based on tropically scaled Lagrange linearization Van Barel, Marc and Tisseur, Francoise 2016 MIMS EPrint: 201661 Manchester Institute for Mathematical Sciences School of Mathematics
More informationNonlinear eigenvalue problems - A Review
Nonlinear eigenvalue problems - A Review Namita Behera Department of Electrical Engineering Indian Institute of Technology Bombay Mumbai 8 th March, 2016 Outline 1 Nonlinear Eigenvalue Problems 2 Polynomial
More informationKU Leuven Department of Computer Science
Backward error of polynomial eigenvalue problems solved by linearization of Lagrange interpolants Piers W. Lawrence Robert M. Corless Report TW 655, September 214 KU Leuven Department of Computer Science
More informationResearch Matters. February 25, The Nonlinear Eigenvalue. Director of Research School of Mathematics
Research Matters February 25, 2009 The Nonlinear Eigenvalue Nick Problem: HighamPart I Director of Research School of Mathematics Françoise Tisseur School of Mathematics The University of Manchester Woudschoten
More informationEigenvector error bound and perturbation for nonlinear eigenvalue problems
Eigenvector error bound and perturbation for nonlinear eigenvalue problems Yuji Nakatsukasa School of Mathematics University of Tokyo Joint work with Françoise Tisseur Workshop on Nonlinear Eigenvalue
More informationDefinite Matrix Polynomials and their Linearization by Definite Pencils. Higham, Nicholas J. and Mackey, D. Steven and Tisseur, Françoise
Definite Matrix Polynomials and their Linearization by Definite Pencils Higham Nicholas J and Mackey D Steven and Tisseur Françoise 2007 MIMS EPrint: 200797 Manchester Institute for Mathematical Sciences
More informationHermitian Matrix Polynomials with Real Eigenvalues of Definite Type. Part I: Classification. Al-Ammari, Maha and Tisseur, Francoise
Hermitian Matrix Polynomials with Real Eigenvalues of Definite Type. Part I: Classification Al-Ammari, Maha and Tisseur, Francoise 2010 MIMS EPrint: 2010.9 Manchester Institute for Mathematical Sciences
More informationNonlinear palindromic eigenvalue problems and their numerical solution
Nonlinear palindromic eigenvalue problems and their numerical solution TU Berlin DFG Research Center Institut für Mathematik MATHEON IN MEMORIAM RALPH BYERS Polynomial eigenvalue problems k P(λ) x = (
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis Volume 38, pp 75-30, 011 Copyright 011, ISSN 1068-9613 ETNA PERTURBATION ANALYSIS FOR COMPLEX SYMMETRIC, SKEW SYMMETRIC, EVEN AND ODD MATRIX POLYNOMIALS SK
More informationA numerical method for polynomial eigenvalue problems using contour integral
A numerical method for polynomial eigenvalue problems using contour integral Junko Asakura a Tetsuya Sakurai b Hiroto Tadano b Tsutomu Ikegami c Kinji Kimura d a Graduate School of Systems and Information
More informationPolynomial Jacobi Davidson Method for Large/Sparse Eigenvalue Problems
Polynomial Jacobi Davidson Method for Large/Sparse Eigenvalue Problems Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan April 28, 2011 T.M. Huang (Taiwan Normal Univ.)
More informationPolynomial Eigenvalue Problems: Theory, Computation, and Structure. Mackey, D. S. and Mackey, N. and Tisseur, F. MIMS EPrint: 2015.
Polynomial Eigenvalue Problems: Theory, Computation, and Structure Mackey, D. S. and Mackey, N. and Tisseur, F. 2015 MIMS EPrint: 2015.29 Manchester Institute for Mathematical Sciences School of Mathematics
More informationTropical roots as approximations to eigenvalues of matrix polynomials. Noferini, Vanni and Sharify, Meisam and Tisseur, Francoise
Tropical roots as approximations to eigenvalues of matrix polynomials Noferini, Vanni and Sharify, Meisam and Tisseur, Francoise 2014 MIMS EPrint: 2014.16 Manchester Institute for Mathematical Sciences
More informationHow to Detect Definite Hermitian Pairs
How to Detect Definite Hermitian Pairs Françoise Tisseur School of Mathematics The University of Manchester ftisseur@ma.man.ac.uk http://www.ma.man.ac.uk/~ftisseur/ Joint work with Chun-Hua Guo and Nick
More informationPerturbation theory for eigenvalues of Hermitian pencils. Christian Mehl Institut für Mathematik TU Berlin, Germany. 9th Elgersburg Workshop
Perturbation theory for eigenvalues of Hermitian pencils Christian Mehl Institut für Mathematik TU Berlin, Germany 9th Elgersburg Workshop Elgersburg, March 3, 2014 joint work with Shreemayee Bora, Michael
More informationMATHEMATICAL ENGINEERING TECHNICAL REPORTS. Eigenvector Error Bound and Perturbation for Polynomial and Rational Eigenvalue Problems
MATHEMATICAL ENGINEERING TECHNICAL REPORTS Eigenvector Error Bound and Perturbation for Polynomial and Rational Eigenvalue Problems Yuji NAKATSUKASA and Françoise TISSEUR METR 2016 04 April 2016 DEPARTMENT
More informationELA
SHARP LOWER BOUNDS FOR THE DIMENSION OF LINEARIZATIONS OF MATRIX POLYNOMIALS FERNANDO DE TERÁN AND FROILÁN M. DOPICO Abstract. A standard way of dealing with matrixpolynomial eigenvalue problems is to
More informationACCURATE SOLUTIONS OF POLYNOMIAL EIGENVALUE PROBLEMS
ACCURATE SOLUTIONS OF POLYNOMIAL EIGENVALUE PROBLEMS YILING YOU, JOSE ISRAEL RODRIGUEZ, AND LEK-HENG LIM Abstract. Quadratic eigenvalue problems (QEP) and more generally polynomial eigenvalue problems
More informationA Structure-Preserving Doubling Algorithm for Quadratic Eigenvalue Problems Arising from Time-Delay Systems
A Structure-Preserving Doubling Algorithm for Quadratic Eigenvalue Problems Arising from Time-Delay Systems Tie-xiang Li Eric King-wah Chu Wen-Wei Lin Abstract We propose a structure-preserving doubling
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 436 (2012) 3954 3973 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Hermitian matrix polynomials
More informationA Framework for Structured Linearizations of Matrix Polynomials in Various Bases
A Framework for Structured Linearizations of Matrix Polynomials in Various Bases Leonardo Robol Joint work with Raf Vandebril and Paul Van Dooren, KU Leuven and Université
More informationA Structure-Preserving Method for Large Scale Eigenproblems. of Skew-Hamiltonian/Hamiltonian (SHH) Pencils
A Structure-Preserving Method for Large Scale Eigenproblems of Skew-Hamiltonian/Hamiltonian (SHH) Pencils Yangfeng Su Department of Mathematics, Fudan University Zhaojun Bai Department of Computer Science,
More informationStructured Backward Error for Palindromic Polynomial Eigenvalue Problems
Structured Backward Error for Palindromic Polynomial Eigenvalue Problems Ren-Cang Li Wen-Wei Lin Chern-Shuh Wang Technical Report 2008-06 http://www.uta.edu/math/preprint/ Structured Backward Error for
More informationInverse Eigenvalue Problem with Non-simple Eigenvalues for Damped Vibration Systems
Journal of Informatics Mathematical Sciences Volume 1 (2009), Numbers 2 & 3, pp. 91 97 RGN Publications (Invited paper) Inverse Eigenvalue Problem with Non-simple Eigenvalues for Damped Vibration Systems
More informationarxiv: v2 [math.na] 8 Aug 2018
THE CONDITIONING OF BLOCK KRONECKER l-ifications OF MATRIX POLYNOMIALS JAVIER PÉREZ arxiv:1808.01078v2 math.na] 8 Aug 2018 Abstract. A strong l-ification of a matrix polynomial P(λ) = A i λ i of degree
More informationNick Higham. Director of Research School of Mathematics
Exploiting Research Tropical Matters Algebra in Numerical February 25, Linear 2009 Algebra Nick Higham Françoise Tisseur Director of Research School of Mathematics The School University of Mathematics
More informationKU Leuven Department of Computer Science
Compact rational Krylov methods for nonlinear eigenvalue problems Roel Van Beeumen Karl Meerbergen Wim Michiels Report TW 651, July 214 KU Leuven Department of Computer Science Celestinenlaan 2A B-31 Heverlee
More informationANALYSIS OF STRUCTURED POLYNOMIAL EIGENVALUE PROBLEMS
ANALYSIS OF STRUCTURED POLYNOMIAL EIGENVALUE PROBLEMS A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2011
More informationResearch Matters. February 25, The Nonlinear Eigenvalue. Director of Research School of Mathematics
Research Matters February 25, 2009 The Nonlinear Eigenvalue Nick Problem: HighamPart II Director of Research School of Mathematics Françoise Tisseur School of Mathematics The University of Manchester Woudschoten
More informationComputing Unstructured and Structured Polynomial Pseudospectrum Approximations
Computing Unstructured and Structured Polynomial Pseudospectrum Approximations Silvia Noschese 1 and Lothar Reichel 2 1 Dipartimento di Matematica, SAPIENZA Università di Roma, P.le Aldo Moro 5, 185 Roma,
More informationin Numerical Linear Algebra
Exploiting ResearchTropical MattersAlgebra in Numerical Linear Algebra February 25, 2009 Nick Françoise Higham Tisseur Director Schoolof ofresearch Mathematics The School University of Mathematics of Manchester
More informationTrimmed linearizations for structured matrix polynomials
Trimmed linearizations for structured matrix polynomials Ralph Byers Volker Mehrmann Hongguo Xu January 5 28 Dedicated to Richard S Varga on the occasion of his 8th birthday Abstract We discuss the eigenvalue
More informationAvailable online at ScienceDirect. Procedia Engineering 100 (2015 ) 56 63
Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 100 (2015 ) 56 63 25th DAAAM International Symposium on Intelligent Manufacturing and Automation, DAAAM 2014 Definite Quadratic
More informationSOLVING A STRUCTURED QUADRATIC EIGENVALUE PROBLEM BY A STRUCTURE-PRESERVING DOUBLING ALGORITHM
SOLVING A STRUCTURED QUADRATIC EIGENVALUE PROBLEM BY A STRUCTURE-PRESERVING DOUBLING ALGORITHM CHUN-HUA GUO AND WEN-WEI LIN Abstract In studying the vibration of fast trains, we encounter a palindromic
More informationthat determines x up to a complex scalar of modulus 1, in the real case ±1. Another condition to normalize x is by requesting that
Chapter 3 Newton methods 3. Linear and nonlinear eigenvalue problems When solving linear eigenvalue problems we want to find values λ C such that λi A is singular. Here A F n n is a given real or complex
More informationNonlinear Eigenvalue Problems: An Introduction
Nonlinear Eigenvalue Problems: An Introduction Cedric Effenberger Seminar for Applied Mathematics ETH Zurich Pro*Doc Workshop Disentis, August 18 21, 2010 Cedric Effenberger (SAM, ETHZ) NLEVPs: An Introduction
More information1. Introduction. Throughout this work we consider n n matrix polynomials with degree k of the form
LINEARIZATIONS OF SINGULAR MATRIX POLYNOMIALS AND THE RECOVERY OF MINIMAL INDICES FERNANDO DE TERÁN, FROILÁN M. DOPICO, AND D. STEVEN MACKEY Abstract. A standard way of dealing with a regular matrix polynomial
More informationComputing the Action of the Matrix Exponential
Computing the Action of the Matrix Exponential Nick Higham School of Mathematics The University of Manchester higham@ma.man.ac.uk http://www.ma.man.ac.uk/~higham/ Joint work with Awad H. Al-Mohy 16th ILAS
More informationMultiparameter eigenvalue problem as a structured eigenproblem
Multiparameter eigenvalue problem as a structured eigenproblem Bor Plestenjak Department of Mathematics University of Ljubljana This is joint work with M Hochstenbach Będlewo, 2932007 1/28 Overview Introduction
More informationDefinite versus Indefinite Linear Algebra. Christian Mehl Institut für Mathematik TU Berlin Germany. 10th SIAM Conference on Applied Linear Algebra
Definite versus Indefinite Linear Algebra Christian Mehl Institut für Mathematik TU Berlin Germany 10th SIAM Conference on Applied Linear Algebra Monterey Bay Seaside, October 26-29, 2009 Indefinite Linear
More informationPALINDROMIC LINEARIZATIONS OF A MATRIX POLYNOMIAL OF ODD DEGREE OBTAINED FROM FIEDLER PENCILS WITH REPETITION.
PALINDROMIC LINEARIZATIONS OF A MATRIX POLYNOMIAL OF ODD DEGREE OBTAINED FROM FIEDLER PENCILS WITH REPETITION. M.I. BUENO AND S. FURTADO Abstract. Many applications give rise to structured, in particular
More informationStructured Condition Numbers and Backward Errors in Scalar Product Spaces. February MIMS EPrint:
Structured Condition Numbers and Backward Errors in Scalar Product Spaces Françoise Tisseur and Stef Graillat February 2006 MIMS EPrint: 2006.16 Manchester Institute for Mathematical Sciences School of
More informationNLEVP: A Collection of Nonlinear Eigenvalue Problems. Users Guide
NLEVP: A Collection of Nonlinear Eigenvalue Problems. Users Guide Betcke, Timo and Higham, Nicholas J. and Mehrmann, Volker and Schröder, Christian and Tisseur, Françoise 2010 MIMS EPrint: 2010.99 Manchester
More informationStructure preserving stratification of skew-symmetric matrix polynomials. Andrii Dmytryshyn
Structure preserving stratification of skew-symmetric matrix polynomials by Andrii Dmytryshyn UMINF 15.16 UMEÅ UNIVERSITY DEPARTMENT OF COMPUTING SCIENCE SE- 901 87 UMEÅ SWEDEN Structure preserving stratification
More informationSTRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES IN STRUCTURED PENCILS
Electronic Transactions on Numerical Analysis Volume 44 pp 1 24 215 Copyright c 215 ISSN 168 9613 ETNA STRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES IN STRUCTURED PENCILS VOLKER MEHRMANN AND HONGGUO
More informationPerturbation of Palindromic Eigenvalue Problems
Numerische Mathematik manuscript No. (will be inserted by the editor) Eric King-wah Chu Wen-Wei Lin Chern-Shuh Wang Perturbation of Palindromic Eigenvalue Problems Received: date / Revised version: date
More informationComputing Matrix Functions by Iteration: Convergence, Stability and the Role of Padé Approximants
Computing Matrix Functions by Iteration: Convergence, Stability and the Role of Padé Approximants Nick Higham School of Mathematics The University of Manchester higham@ma.man.ac.uk http://www.ma.man.ac.uk/~higham/
More informationc 2006 Society for Industrial and Applied Mathematics
SIAM J MATRIX ANAL APPL Vol 28, No 4, pp 971 1004 c 2006 Society for Industrial and Applied Mathematics VECTOR SPACES OF LINEARIZATIONS FOR MATRIX POLYNOMIALS D STEVEN MACKEY, NILOUFER MACKEY, CHRISTIAN
More informationON SYLVESTER S LAW OF INERTIA FOR NONLINEAR EIGENVALUE PROBLEMS
ON SYLVESTER S LAW OF INERTIA FOR NONLINEAR EIGENVALUE PROBLEMS ALEKSANDRA KOSTIĆ AND HEINRICH VOSS Key words. eigenvalue, variational characterization, principle, Sylvester s law of inertia AMS subject
More informationStructured eigenvalue/eigenvector backward errors of matrix pencils arising in optimal control
Electronic Journal of Linear Algebra Volume 34 Volume 34 08) Article 39 08 Structured eigenvalue/eigenvector backward errors of matrix pencils arising in optimal control Christian Mehl Technische Universitaet
More informationOn a root-finding approach to the polynomial eigenvalue problem
On a root-finding approach to the polynomial eigenvalue problem Dipartimento di Matematica, Università di Pisa www.dm.unipi.it/ bini Joint work with Vanni Noferini and Meisam Sharify Limoges, May, 10,
More informationPolynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations
Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations Elias Jarlebring a, Michiel E. Hochstenbach b,1, a Technische Universität Braunschweig,
More informationFiedler Companion Linearizations and the Recovery of Minimal Indices. De Teran, Fernando and Dopico, Froilan M. and Mackey, D.
Fiedler Companion Linearizations and the Recovery of Minimal Indices De Teran, Fernando and Dopico, Froilan M and Mackey, D Steven 2009 MIMS EPrint: 200977 Manchester Institute for Mathematical Sciences
More informationA Jacobi Davidson Method for Nonlinear Eigenproblems
A Jacobi Davidson Method for Nonlinear Eigenproblems Heinrich Voss Section of Mathematics, Hamburg University of Technology, D 21071 Hamburg voss @ tu-harburg.de http://www.tu-harburg.de/mat/hp/voss Abstract.
More informationJordan Structures of Alternating Matrix Polynomials
Jordan Structures of Alternating Matrix Polynomials D. Steven Mackey Niloufer Mackey Christian Mehl Volker Mehrmann August 17, 2009 Abstract Alternating matrix polynomials, that is, polynomials whose coefficients
More informationFinite and Infinite Elementary Divisors of Matrix Polynomials: A Global Approach. Zaballa, Ion and Tisseur, Francoise. MIMS EPrint: 2012.
Finite and Infinite Elementary Divisors of Matrix Polynomials: A Global Approach Zaballa, Ion and Tisseur, Francoise 2012 MIMS EPrint: 201278 Manchester Institute for Mathematical Sciences School of Mathematics
More informationA SIMPLIFIED APPROACH TO FIEDLER-LIKE PENCILS VIA STRONG BLOCK MINIMAL BASES PENCILS.
A SIMPLIFIED APPROACH TO FIEDLER-LIKE PENCILS VIA STRONG BLOCK MINIMAL BASES PENCILS. M. I. BUENO, F. M. DOPICO, J. PÉREZ, R. SAAVEDRA, AND B. ZYKOSKI Abstract. The standard way of solving the polynomial
More informationComputational Methods for Feedback Control in Damped Gyroscopic Second-order Systems 1
Computational Methods for Feedback Control in Damped Gyroscopic Second-order Systems 1 B. N. Datta, IEEE Fellow 2 D. R. Sarkissian 3 Abstract Two new computationally viable algorithms are proposed for
More informationChap 3. Linear Algebra
Chap 3. Linear Algebra Outlines 1. Introduction 2. Basis, Representation, and Orthonormalization 3. Linear Algebraic Equations 4. Similarity Transformation 5. Diagonal Form and Jordan Form 6. Functions
More informationA UNIFIED APPROACH TO FIEDLER-LIKE PENCILS VIA STRONG BLOCK MINIMAL BASES PENCILS.
A UNIFIED APPROACH TO FIEDLER-LIKE PENCILS VIA STRONG BLOCK MINIMAL BASES PENCILS M I BUENO, F M DOPICO, J PÉREZ, R SAAVEDRA, AND B ZYKOSKI Abstract The standard way of solving the polynomial eigenvalue
More informationNonlinear Eigenvalue Problems and Contour Integrals
Nonlinear Eigenvalue Problems and Contour Integrals Marc Van Barel a,,, Peter Kravanja a a KU Leuven, Department of Computer Science, Celestijnenlaan 200A, B-300 Leuven (Heverlee), Belgium Abstract In
More informationNLEVP: A Collection of Nonlinear Eigenvalue Problems
NLEVP: A Collection of Nonlinear Eigenvalue Problems Betcke, Timo and Higham, Nicholas J. and Mehrmann, Volker and Schröder, Christian and Tisseur, Françoise 2 MIMS EPrint: 2.6 Manchester Institute for
More informationEXPLICIT BLOCK-STRUCTURES FOR BLOCK-SYMMETRIC FIEDLER-LIKE PENCILS
EXPLICIT BLOCK-STRUCTURES FOR BLOCK-SYMMETRIC FIEDLER-LIKE PENCILS M I BUENO, M MARTIN, J PÉREZ, A SONG, AND I VIVIANO Abstract In the last decade, there has been a continued effort to produce families
More informationA block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error
Calcolo manuscript No. (will be inserted by the editor) A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error M. I Bueno F. M. Dopico
More informationStructured Condition Numbers and Backward Errors in Scalar Product Spaces
Structured Condition Numbers and Backward Errors in Scalar Product Spaces Françoise Tisseur Department of Mathematics University of Manchester ftisseur@ma.man.ac.uk http://www.ma.man.ac.uk/~ftisseur/ Joint
More informationPolynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations
Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations Elias Jarlebring a, Michiel E. Hochstenbach b,1, a Technische Universität Braunschweig,
More informationNonlinear eigenvalue problems: Analysis and numerical solution
Nonlinear eigenvalue problems: Analysis and numerical solution Volker Mehrmann TU Berlin, Institut für Mathematik DFG Research Center MATHEON Mathematics for key technologies MATTRIAD 2011 July 2011 Outline
More informationTriangularizing matrix polynomials. Taslaman, Leo and Tisseur, Francoise and Zaballa, Ion. MIMS EPrint:
Triangularizing matrix polynomials Taslaman, Leo and Tisseur, Francoise and Zaballa, Ion 2012 MIMS EPrint: 2012.77 Manchester Institute for Mathematical Sciences School of Mathematics The University of
More informationLinearizing Symmetric Matrix Polynomials via Fiedler pencils with Repetition
Linearizing Symmetric Matrix Polynomials via Fiedler pencils with Repetition Kyle Curlett Maribel Bueno Cachadina, Advisor March, 2012 Department of Mathematics Abstract Strong linearizations of a matrix
More informationEfficient computation of transfer function dominant poles of large second-order dynamical systems
Chapter 6 Efficient computation of transfer function dominant poles of large second-order dynamical systems Abstract This chapter presents a new algorithm for the computation of dominant poles of transfer
More informationSkew-Symmetric Matrix Polynomials and their Smith Forms
Skew-Symmetric Matrix Polynomials and their Smith Forms D. Steven Mackey Niloufer Mackey Christian Mehl Volker Mehrmann March 23, 2013 Abstract Two canonical forms for skew-symmetric matrix polynomials
More informationNLEVP: A Collection of Nonlinear Eigenvalue Problems
School of Mathematical and Physical Sciences Department of Mathematics and Statistics Preprint MPS_2-33 5 November 2 NLEVP: A Collection of Nonlinear Eigenvalue Problems by Timo Betcke, Nicholas J. Higham,
More informationA Multi-Step Hybrid Method for Multi-Input Partial Quadratic Eigenvalue Assignment with Time Delay
A Multi-Step Hybrid Method for Multi-Input Partial Quadratic Eigenvalue Assignment with Time Delay Zheng-Jian Bai Mei-Xiang Chen Jin-Ku Yang April 14, 2012 Abstract A hybrid method was given by Ram, Mottershead,
More informationStructured eigenvalue condition numbers and linearizations for matrix polynomials
Eidgenössische Technische Hochschule Zürich Ecole polytechnique fédérale de Zurich Politecnico federale di Zurigo Swiss Federal Institute of Technology Zurich Structured eigenvalue condition numbers and
More informationReduction of nonlinear eigenproblems with JD
Reduction of nonlinear eigenproblems with JD Henk van der Vorst H.A.vanderVorst@math.uu.nl Mathematical Institute Utrecht University July 13, 2005, SIAM Annual New Orleans p.1/16 Outline Polynomial Eigenproblems
More informationStructured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations
Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations Mackey, D. Steven and Mackey, Niloufer and Mehl, Christian and Mehrmann, Volker 006 MIMS EPrint: 006.8 Manchester Institute
More informationNonlinear Eigenvalue Problems: A Challenge for Modern Eigenvalue Methods
Nonlinear Eigenvalue Problems: A Challenge for Modern Eigenvalue Methods Volker Mehrmann Heinrich Voss November 29, 2004 Abstract We discuss the state of the art in numerical solution methods for large
More informationEigenvalues and Eigenvectors
Eigenvalues and Eigenvectors Definition 0 Let A R n n be an n n real matrix A number λ R is a real eigenvalue of A if there exists a nonzero vector v R n such that A v = λ v The vector v is called an eigenvector
More informationMaths for Signals and Systems Linear Algebra in Engineering
Maths for Signals and Systems Linear Algebra in Engineering Lectures 13 15, Tuesday 8 th and Friday 11 th November 016 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE
More informationMore on pseudospectra for polynomial eigenvalue problems and applications in control theory
Linear Algebra and its Applications 351 352 (2002) 435 453 www.elsevier.com/locate/laa More on pseudospectra for polynomial eigenvalue problems and applications in control theory Nicholas J. Higham,1,
More information7 NLEVP: A Collection of Nonlinear Eigenvalue Problems
7 NLEVP: A Collection of Nonlinear Eigenvalue Problems TIMO BETCKE, University College London NICHOLAS J. HIGHAM, The University of Manchester VOLKER MEHRMANN and CHRISTIAN SCHRÖDER, Technische Universität
More informationOn an Inverse Problem for a Quadratic Eigenvalue Problem
International Journal of Difference Equations ISSN 0973-6069, Volume 12, Number 1, pp. 13 26 (2017) http://campus.mst.edu/ijde On an Inverse Problem for a Quadratic Eigenvalue Problem Ebru Ergun and Adil
More informationComputable error bounds for nonlinear eigenvalue problems allowing for a minmax characterization
Computable error bounds for nonlinear eigenvalue problems allowing for a minmax characterization Heinrich Voss voss@tuhh.de Joint work with Kemal Yildiztekin Hamburg University of Technology Institute
More informationInduced Dimension Reduction method to solve the Quadratic Eigenvalue Problem
Induced Dimension Reduction method to solve the Quadratic Eigenvalue Problem R. Astudillo and M. B. van Gijzen Email: R.A.Astudillo@tudelft.nl Delft Institute of Applied Mathematics Delft University of
More informationHOMOGENEOUS JACOBI DAVIDSON. 1. Introduction. We study a homogeneous Jacobi Davidson variant for the polynomial eigenproblem
HOMOGENEOUS JACOBI DAVIDSON MICHIEL E. HOCHSTENBACH AND YVAN NOTAY Abstract. We study a homogeneous variant of the Jacobi Davidson method for the generalized and polynomial eigenvalue problem. Special
More information1. Introduction. Throughout this work we consider n n matrix polynomials with degree k 2 of the form
FIEDLER COMPANION LINEARIZATIONS AND THE RECOVERY OF MINIMAL INDICES FERNANDO DE TERÁN, FROILÁN M DOPICO, AND D STEVEN MACKEY Abstract A standard way of dealing with a matrix polynomial P (λ) is to convert
More informationPH.D. PRELIMINARY EXAMINATION MATHEMATICS
UNIVERSITY OF CALIFORNIA, BERKELEY SPRING SEMESTER 207 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials NAME PH.D. PRELIMINARY EXAMINATION MATHEMATICS Problem
More informationTaylor s Theorem for Matrix Functions with Applications to Condition Number Estimation. Deadman, Edvin and Relton, Samuel. MIMS EPrint: 2015.
Taylor s Theorem for Matrix Functions with Applications to Condition Number Estimation Deadman, Edvin and Relton, Samuel 215 MIMS EPrint: 215.27 Manchester Institute for Mathematical Sciences School of
More information