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Pseudospectra of matrix pencils for transient analysis of differential-algebraic equations. (English) Zbl 1379.65058

65L80 Numerical methods for differential-algebraic equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A09 Implicit ordinary differential equations, differential-algebraic equations
15A22 Matrix pencils
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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[1] S. S. Ahmad, R. Alam, and R. Byers, On pseudospectra, critical points, and multiple eigenvalues of matrix pencils, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1915–1933, . · Zbl 1206.65137
[2] R. Astudillo and Z. Castillo, Approximating the weighted pseudospectra of large matrices, Math. Comput. Modelling, 57 (2013), pp. 2169–2176. · Zbl 1286.65046
[3] J. S. Baggett, T. A. Driscoll, and L. N. Trefethen, A mostly linear model of transition to turbulence, Phys. Fluids, 7 (1995), pp. 833–838. · Zbl 1039.76509
[4] K. E. Brennan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Classics Appl. Math. 14, SIAM, Philadelphia, 1996, . · Zbl 0844.65058
[5] J. V. Burke, A. S. Lewis, and M. L. Overton, Robust stability and a criss-cross algorithm for pseudospectra, IMA J. Numer. Anal., 23 (2003), pp. 359–375. · Zbl 1042.65060
[6] K. M. Butler and B. F. Farrell, Three-dimensional optimal perturbations in viscous shear flow, Phys. Fluids A, 4 (1992), pp. 1637–1650.
[7] S. L. Campbell and C. D. Meyer, Jr., Generalized Inverses of Linear Transformations, Pitman, London, 1979.
[8] J.-M. Chomaz, Global instabilities in spatially developing flows: Non-normality and nonlinearity, Annu. Rev. Fluid Mech., 37 (2005), pp. 357–392. · Zbl 1117.76027
[9] K. A. Cliffe, T. J. Garratt, and A. Spence, Eigenvalues of block matrices arising from problems in fluid mechanics, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 1310–1318, . · Zbl 0807.65030
[10] P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge, UK, 1981.
[11] H. C. Elman, K. Meerbergen, A. Spence, and M. Wu, Lyapunov inverse iteration for identifying Hopf bifurcations in models of incompressible flow, SIAM J. Sci. Comput., 34 (2012), pp. A1584–A1606, . · Zbl 1247.65047
[12] H. C. Elman, A. Ramage, and D. J. Silvester, IFISS: A computational laboratory for investigating incompressible flow problems, SIAM Rev., 56 (2014), pp. 261–273, . · Zbl 1426.76645
[13] H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics, 2nd ed., Oxford University Press, Oxford, UK, 2014. · Zbl 1304.76002
[14] M. Embree and L. N. Trefethen, Generalizing eigenvalue theorems to pseudospectra theorems, SIAM J. Sci. Comput., 23 (2001), pp. 583–590, . · Zbl 0995.15003
[15] E. Emmrich and V. Mehrmann, Operator differential-algebraic equations arising in fluid dynamics, Comput. Methods Appl. Math., 13 (2013), pp. 443–470. · Zbl 1392.34070
[16] V. Frayssé, M. Gueury, F. Nicoud, and V. Toumazou, Spectral Portraits for Matrix Pencils, Technical Report TR/PA/96/19, CERFACS, Toulouse, France, 1996.
[17] Z. Gajić and M. T. J. Qureshi, Lyapunov Matrix Equation in System Stability and Control, Academic Press, San Diego, 1995.
[18] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Springer-Verlag, Berlin, 1986. · Zbl 0413.65081
[19] S. K. Godunov, Modern Aspects of Linear Algebra, American Mathematical Society, Providence, RI, 1998; translation of Russian original, Scientific Books, Novosibirsk, 1997. · Zbl 0960.15001
[20] K. Green and T. Wagenknecht, Pseudospectra of delay differential equations, J. Comput. Appl. Math., 196 (2006), pp. 567–578. · Zbl 1106.65068
[21] P. M. Gresho, D. K. Gartling, J. R. Torczysnski, K. A. Cliffe, K. H. Winters, T. J. Garratt, A. Spence, and J. W. Goodrich, Is the steady viscous incompressible two-dimensional flow over a backward-facing step at \({\rm Re} = 800\) stable?, Internat. J. Numer. Methods Fluids, 17 (1993), pp. 501–541. · Zbl 0784.76050
[22] N. Guglielmi and C. Lubich, Differential equations for roaming pseudospectra: Paths to extremal points and boundary tracking, SIAM J. Numer. Anal., 49 (2011), pp. 1194–1209, . · Zbl 1232.15009
[23] N. Guglielmi and C. Lubich, Erratum/addendum: Differential equations for roaming pseudospectra: Paths to extremal points and boundary tracking, SIAM J. Numer. Anal., 50 (2012), pp. 977–981, . · Zbl 1257.15007
[24] N. Guglielmi and M. L. Overton, Fast algorithms for the approximation of the pseudospectral abscissa and pseudospectral radius of a matrix, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166–1192, . · Zbl 1248.65034
[25] N. Guglielmi, M. L. Overton, and G. W. Stewart, An efficient algorithm for computing the generalized null space decomposition, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 38–54, . · Zbl 1327.65072
[26] M. D. Gunzburger, Finite Element Methods for Viscous Flows: A Guide to Theory, Practice, and Algorithms, Academic Press, Boston, 1989.
[27] M. Heinkenschloss, D. C. Sorensen, and K. Sun, Balanced truncation model reduction for a class of descriptor systems with application to the Oseen equations, SIAM J. Sci. Comput., 30 (2008), pp. 1038–1063, . · Zbl 1216.76015
[28] N. J. Higham and F. Tisseur, More on pseudospectra for polynomial eigenvalue problems and applications in control theory, Linear Algebra Appl., 351/352 (2002), pp. 435–453. · Zbl 1004.65046
[29] M. E. Hochstenbach, Fields of values and inclusion regions for matrix pencils, Electron. Trans. Numer. Anal., 38 (2011), pp. 98–112. · Zbl 1287.65025
[30] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0729.15001
[31] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin, 1976. · Zbl 0342.47009
[32] D. Kressner and B. Vandereycken, Subspace methods for computing the pseudospectral abscissa and the stability radius, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 292–313, . · Zbl 1306.65187
[33] P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution, European Mathematical Society, Zürich, 2006. · Zbl 1095.34004
[34] P.-F. Lavallée and M. Sadkane, Pseudospectra of linear matrix pencils by block diagonalization, Computing, 60 (1998), pp. 133–156. · Zbl 0892.65026
[35] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM, Philadelphia, 1998, . · Zbl 0901.65021
[36] C.-K. Li and L. Rodman, Numerical range of matrix polynomials, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 1256–1265, . · Zbl 0814.15023
[37] K. Meerbergen, A. Spence, and D. Roose, Shift-invert and Cayley transforms for detection of rightmost eigenvalues of nonsymmetric matrices, BIT, 34 (1994), pp. 409–423. · Zbl 0814.65037
[38] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. · Zbl 0516.47023
[39] P. J. Psarrakos, Numerical range of linear pencils, Linear Algebra Appl., 317 (2000), pp. 127–141. · Zbl 0966.15014
[40] K. S. Riedel, Generalized epsilon-pseudospectra, SIAM J. Numer. Anal., 31 (1994), pp. 1219–1225, . · Zbl 0805.15005
[41] A. Ruhe, The Rational Krylov Algorithm for Large Nonsymmetric Eigenvalues—Mapping the Resolvent Norms (Pseudospectrum), unpublished manuscript, 1995.
[42] P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows, Springer-Verlag, New York, 2001. · Zbl 0966.76003
[43] P. Sirković, A Reduced Basis Approach to Large-Scale Pseudospectra Computation, MATHICSE Technical Report 35.2016, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 2016.
[44] G. Söderlind, The logarithmic norm: History and modern theory, BIT, 46 (2006), pp. 631–652. · Zbl 1102.65088
[45] M. H. Stone, Linear Transformations in Hilbert Space, American Mathematical Society, New York, 1932. · JFM 58.0420.02
[46] T. Stykel, Gramian-based model reduction for descriptor systems, Math. Control Signals Systems, 16 (2004), pp. 297–319. · Zbl 1067.93011
[47] F. Tisseur and N. J. Higham, Structured pseudospectra for polynomial eigenvalue problems, with applications, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 187–208, . · Zbl 0996.65042
[48] L. N. Trefethen, Computation of pseudospectra, Acta Numer., 8 (1999), pp. 247–295. · Zbl 0945.65039
[49] L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, Princeton, NJ, 2005. · Zbl 1085.15009
[50] L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science, 261 (1993), pp. 578–584. · Zbl 1226.76013
[51] J. L. M. van Dorsselaer, Pseudospectra for matrix pencils and stability of equilibria, BIT, 37 (1997), pp. 833–845. · Zbl 0894.65022
[52] K. Veselić, Bounds for exponentially stable semigroups, Linear Algebra Appl., 358 (2003), pp. 309–333. · Zbl 1046.47041
[53] K. Veselić, Damped Oscillations of Linear Systems: A Mathematical Introduction, Lecture Notes in Math. 2023, Springer-Verlag, Berlin, 2011.
[54] E. Wegert and L. N. Trefethen, From the Buffon needle problem to the Kreiss matrix theorem, Amer. Math. Monthly, 101 (1994), pp. 132–139. · Zbl 0799.30002
[55] T. G. Wright, Algorithms and Software for Pseudospectra, D.Phil. thesis, Oxford University, Oxford, UK, 2002.
[56] T. G. Wright, EigTool, 2002; software available online at .
[57] T. G. Wright and L. N. Trefethen, Large-scale computation of pseudospectra using ARPACK and eigs, SIAM J. Sci. Comput., 23 (2001), pp. 591–605, . · Zbl 0992.65030
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