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Kokiopoulou, and E. Kokiopoulou, E. Gallopoulos, and V. Gallopoulos, and I. Koutis Towards the effective parallel computation of matrix pseudospectra Proc. Benilov, S. O'Brien, and I. Sazonov A new type of instability: Explosive disturbances in a liquid film inside a rotating horizontal cylinder. Bernhardsson, A. Rantzer, and L. Qiu Real perturbation values and real quadratic forms in a complex vector space Linear Algebra Appl. MR 99a Link to article. Bewley Flow control: New challenges for a new renaissance Prog. Aerospace Sci. Borba, K. Riedel, W. Kerner, G. Huysmans, M.

Ottaviani, and P. Plasmas 1 Borthwick and A. Analysis 10 Integral Equations Appl. MR 96a MR 96m Bulletin 45 Links to Article: [postscript] [PDF]. Embree, and L. MR g Grudsky, and B. MR 98d MR 1 Boulton Non-self-adjoint harmonic oscillator, compact semigroups, and pseudospectra arXiv.

Boulton, P. Lancaster, and P. Psarrakos On pseudospectra of matrix polynomials and their boundaries arXiv. Boutry, M. Elad, G. Golub, and P. Matrix Anal. Link to report gzipped postscript. Braconnier and N. Higham Computing the field of values and pseudospectra using the Lanczos method with continuation BIT 36 , MR 97d Braconnier, R. McCoy, and V. Brown and M. Marletta Spectral inclusion and spectral exactness for singular non-self-adjoint Sturm--Liouville problems Proc. A Bulgak Pseudoeigenvalues, spectral portrait of a matrix and their connections with different criteria of stability In Error Control and Adaptivity in Scientific Computing , pp.

Bulgak and C. Zenger, eds. MR b Bulgak and D. Math 2 Burke, A. Lewis, and M. Overton Optimization over pseudospectra Preprint, 15 February Link to preprint postscript. Carpraux, J. Erhel, and M. Sadkane Spectral portrait for non-Hermitian large sparse matrices Computing 53 , Chaitin-Chatelin Is nonnormality a serious computational difficulty in practice? Boisvert, ed. Chaitin-Chatelin and V. MR 97b Chaitin-Chatelin and S.

Numerical Analysis 2000 : Linear Algebra - Linear Systems and Eigenvalues (Numerical Analysis 2000)

Gratton Convergence in finite precision of successive iteration methods under high nonnormality. BIT 36 , MR 97h Chaitin-Chatelin and A. Chaitin-Chatelin, V. Toumazou, and E. Traviesas Accuracy assessment for eigencomputations: Variety of backward errors and pseudospectra Linear Algebra Appl. MR c Link to article. Chaitin-Chatelin and E. Champier and L.

Grammont Note on the norm involved in the definition of the epsilon-spectrum Applied Math. Christiansen and M. Zworski Pseudospectra in automorphic scattering Preprint, Link to Preprint postscript. Cossu and J. Chomaz Global measures of local convective instabilities Phys. Crawford Pseudospectra of the linearized Vlasov equation Bull.

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Cullum and A. Ruehli Pseudospectra analysis, nonlinear eigenvalue problems, and studying linear systems with delays BIT 41 Dar, M. Verma, and V. Eswaran Energy transfer in two-dimensional magnetohydrodynamic turbulence: formalism and numerical results Physica D Davis and S. Troian Influence of boundary slip on the optimal excitations in thermocapillary driven spreading Phys.

E 70 11 pages. Darmofal and P. Schmid The importance of eigenvectors for local preconditioners of the Euler equations J. Davies Pseudo-spectra, the harmonic oscillator and complex resonances Proc. A , MR d MR 99m Davies Pseudospectra of differential operators J. Operator Theory 43 , Davies Wild spectral behaviour of anharmonic oscillators Bull. London Math. Davies Non-self-adjoint differential operators Bull. Davies Semigroup growth bounds arXiv. Davies Approximating semigroups by using pseudospectra arXiv. Troian On a generalized approach to the linear stability of spatially nonuniform thin film flows Phys.

Fluids 15 Dedieu, M. Shub and F. FTP Link to report gzipped postscript. Costa, and C. Caramiello Energy growth of initial perturbations in two-dimensional gravitational jets Phys. Fluids 14 Control AC , Demmel The geometry of ill-conditioning J. Complexity 3 , MR 88h Dencker, J. Zworski Pseudospectra of semiclassical pseudo- differential operators Comm. Pure Appl.

MR 98m Kraaijevanger, and M. Spijker Linear stability analysis in the numerical solution of initial value problems In Acta Numerica , p. MR 94e Driscoll and L. Trefethen Pseudospectra for the wave equation with an absorbing boundary J. MR 97e Du and Y. Wei Structured pseudospectra and structured sensitivity of eigenvalues J. Edelman and N. Rao Random matrix theory In Acta Numerica , p.

Embree and L. Flaherty, C. Seyler, and L. Trefethen Large-amplitude transient growth in the linear evolution of equatorial spread F with a sheared zonal flow J. Giraud, and V. Liddell, A. Colbrook, B. Hertzberger, and P. Sloot, eds. Gueury, F. Nicoud, and V. Friswell and A. Champneys Defective systems and pseudospectra Materials Science Forum Link to abstract. Fuhrman, H.

Department of Mathematics, University of Manchester

Bingham, P. Madsen, and P.

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Thomsen Linear and non-linear stability analysis for finite difference discretizations of high-order Boussinesq equations Int. Methods Fluids 45 Gallestey Computing the spectral value sets using the subharmonicity of the norm of rational matrices BIT 38 , MR 99b Gallestey, D. Hinrichsen, and A. Pritchard Spectral value sets of infinite dimensional systems In V. Blondel, D. Sontag, M. Vidyasagar, and J. Willems, eds. Springer-Verlag, London, Pritchard Spectral value sets of closed linear operators Proc. Godunov, O. Kirilyuk, and V.

Godunov and M. Sadkane Computation of pseudospectra via spectral projectors Linear Algebra Appl. MR 99i Link to article. Gracia Nearest derogatory matrix when varying into a submatrix Manuscript, July Link to manuscript. Gracia Nearest submatrix that increases the geometric multiplicity Manuscript, September Gracia and I.

Graillat A note on structured pseudospectra J. Grammont and A. Largillier On epsilon-spectra and stability radii J. Math Green, A. Champneys, and M. Friswell Analysis of the transient response of an automatic dynamic balancer for eccentric rotors Int. Sciences 48 Link to article.

Green and T. Wagenknecht Pseudospectra of delay differential equations Bristol Centre for Applied Nonlinear Mathematics preprint Greenbaum Generalizations of the the field of values useful in the study of polynomial functions of a matrix Linear Algebra Appl. Greenbaum and L. Trefethen Do the pseudospectra of a matrix determine its behavior? Hagen, S. Roch, and B. Hager Bound on the number of eigenvalues near the boundary of the pseudospectrum Link to article. MR 99i Methods Eng. Heuveline, B. Philippe, and M. Sadkane Parallel computation of spectral portrait of large matrices by Davidson type methods Numer.

Algorithms 16 , Higham and B. Owren Nonnormality effects in a discretised nonlinear reaction-convection-diffusion equation J. MR 97a Higham and L. Higham Accuracy and Stability of Numerical Algorithms , 2nd ed. SIAM, Philadelphia, Higham and P. Higham and F.

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MR d Link to article. Tisseur More on pseudospectra for polynomial eigenvalue problems and applications in control theory Linear Algebra Appl. Hinrichsen and B. Kelb Spectral value sets: a graphical tool for robustness analysis Systems Control Lett. Hinrichsen and A. Pritchard On spectral variations under bounded real matrix perturbations Numer.

MR 93a MR 95e Link to website at Springer. Hitrik Boundary spectral behavior for semiclassical operators in dimension one Intern. Notices Link to article. Hochbruck and C. MR 97m Hochstenbach and B.

Lecture 47 Matrix Eigenvalue Problems - 2 Power Method - 2

Plestenjak Backward error, condition numbers, and pseudospectra for the multiparameter eigenvalue problem Linear Algebra Appl. Hristova, S. Roch, P. Schmid, L. Tuckerman Transient growth in exactly counter-rotating Couette-Taylor flow Theoret. Fluid Dynamics 16 Tuckerman Transient growth in Taylor-Couette flow Phys.

Jackiewicz, B. Owren, and B. Welfert Pseudospectra of waveform relaxation operators Comput. MR 99g Jackiewicz and R. However, Microsoft Word alike are not recommended because of the equations. Hand-written homework is due in class on the due date. All programming part of the homework are also due at pm on the due date, through email submission of the source code and the report to the TA.

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References back to top. The following is an excellent text for reviewing fundamental concepts and some applications of linear algebra. The following books are graduate-level textbooks on numerical linear algebra, similar to the main textbook for this course. The following books are for additional readings on iterative methods and multigrid methods, which are increasingly important but not covered in this course due to time constraint. Kernighan, D. Ritchie, C Programming Language 2nd edition.

Prentice Hall, There are some free online books linked at this "C Programming Language " page.

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Among these, the following book might be most appropriate. Banahan, D. Brady and M. C Programming, Wikibooks. Policies and Academic Integrity back to top. Required Textbook Lloyd N. Trefethen and D. Golub and C. ISBN AMS co-requisite for students without programming experience. Learning Objectives The objective of this course is to introduce the key concepts and algorithms in numerical linear algebra, including direct and iterative methods for solving simultaneous linear equations, least squares problems, computation of eigenvalues and eigenvectors, and singular value decomposition.

Outline Fundamentals matrix notation and basic operations; vector spaces; algorithmic considerations; norms and condition numbers; decomposition of matrices Linear systems triangular systems; Gaussian elimination; accuracy and stability; Cholesky factorization; sparse linear systems QR factorization and least squares Gram-Schmidt orthogonalization; QR factorization with Householder reflection; updating QR factorization with Givens rotation; stability of QR factorization; least squares problems; rank-revealing QR factorization; SVD and low-rank approximations Eigenvalue problems eigenvalues and invariant spaces; classical eigenvalue methods; QR algorithms; two-stage methods; Arnoldi and Lanczos iterations Iterative Methods for linear systems basic iterative methods; conjugate gradient methods; minimal residual style methods; bi-Lanczos iterations; preconditioners Special topics multigrid methods; under-determined linear systems etc.

Assignments Homework assignments are due in class typically two weeks after they are assigned. Exams The exams including two tests and the final exam are closed-book , but you are allowed to bring a single-sided, one-page, letter-size cheat sheet , which you must prepare by yourself. Attendance All students are expected to attend all the lectures and exams. Assignments You are allowed to discuss course materials and homework problems in small groups, but limited to discussion of general ideas only. Homework 1 due Sept. Homework 2 due Sept. Sample Tests. References on Undergraduate-Level Linear Algebra The following is an excellent text for reviewing fundamental concepts and some applications of linear algebra.