On the convergence of second order spectra and multiplicity

Let A be a self-adjoint operator acting on a Hilbert space. The notion of second order spectrum of A relative to a given finite-dimensional subspace L has been studied recently in connection with the phenomenon of spectral pollution in the Galerkin method. We establish in this paper a general framework allowing us to determine how the second order spectrum encodes precise information about the multiplicity of the isolated eigenvalues of A. Our theoretical findings are supported by various numerical experiments on the computation of inclusions for eigenvalues of benchmark differential operators via finite element bases.Comment: 22 pages, 2 figures, 4 tables, research paper

A modification of the Dewilde–van der Veen method for inversion of finite structured matrices

AbstractWe study a class of block structured matrices R={Rij}i,j=1N with a property that the solution of the corresponding system Rx=y of linear algebraic equations may be performed for O(N) arithmetic operations. In this paper for finite invertible matrices we analyze in detail factorization and inversion algorithms. These algorithms are related to those suggested by P.M. Dewilde and A.J. van der Veen (Time-varying Systems and Computations, Kluwer Academic Publishers, New York, 1998) for a class of finite and infinite matrices with a small Hankel rank. The algorithms presented here are more transparent and are a modification of the algorithms from the above reference. The approach and the proofs are essentially different from those in the above-mentioned reference. The paper contains also analysis of complexity and results of numerical experiments

A trace formula for canonical differential expressions

AbstractWe prove a trace formula for pairs of self-adjoint operators associated to canonical differential expressions. An important role is played by the associated Weyl function

Analyticity and uniform stability in the inverse spectral problem for Dirac operators

We prove that the inverse spectral mapping reconstructing the square integrable potentials on [0,1] of Dirac operators in the AKNS form from their spectral data (two spectra or one spectrum and the corresponding norming constants) is analytic and uniformly stable in a certain sense.Comment: 19 page

Inverse Spectral Problem for Differential Operators With Rational Scattering Matrix Functions, Journal of Differential Equations

In this paper we obtain explicit formulas for the potential of an ordinary differential operator if its spectral function or its scattering functions are rational matrix functions which are analytic and invertible on the real line including infinity. The solution is given in terms of a realization of the spectral function or of the scattering function

Krein systems

In the present paper we extend results of M.G. Krein associated to the spectral problem for Krein systems to systems with matrix valued accelerants with a possible jump discontinuity at the origin. Explicit formulas for the accelerant are given in terms of the matrizant of the system in question. Recent developments in the theory of continuous analogs of the resultant operator play an essential role

Monotone power method in indefinite metric and inertia theorem for matrices

AbstractThe usual power method for matrices is generalized for contractions in indefinite metric spaces. This generalization unifies the power method and the inertia theorem in a natural way

Strings in five-dimensional anti-de Sitter space with a symmetry

The equation of motion of an extended object in spacetime reduces to an ordinary differential equation in the presence of symmetry. By properly defining of the symmetry with notion of cohomogeneity, we discuss the method for classifying all these extended objects. We carry out the classification for the strings in the five-dimensional anti-de Sitter space by the effective use of the local isomorphism between \SO(4,2) and \SU(2,2). We present a general method for solving the trajectory of the Nambu-Goto string and apply to a case obtained by the classification, thereby find a new solution which has properties unique to odd-dimensional anti-de Sitter spaces. The geometry of the solution is analized and found to be a timelike helicoid-like surface

Fermion Systems in Discrete Space-Time Exemplifying the Spontaneous Generation of a Causal Structure

As toy models for space-time on the Planck scale, we consider examples of fermion systems in discrete space-time which are composed of one or two particles defined on two up to nine space-time points. We study the self-organization of the particles as described by a variational principle both analytically and numerically. We find an effect of spontaneous symmetry breaking which leads to the emergence of a discrete causal structure.Comment: 37 pages, LaTeX, 12 figures, minor changes (published version

Inertia-based spectrum slicing for symmetric quadratic eigenvalue problems

[EN] In the quadratic eigenvalue problem (QEP) with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos-based method for computing all real eigenvalues contained in a given interval of large-scale symmetric QEPs. The method uses matrix inertias of the quadratic polynomial evaluated at different shift values. In this way, for hyperbolic problems, it is possible to make sure that all eigenvalues in the interval have been computed. We also discuss the general nonhyperbolic case. Our implementation is memory-efficient by representing the computed pseudo-Lanczos basis in a compact tensor product representation. We show results of computational experiments with a parallel implementation in the SLEPc library.Agencia Estatal de Investigacion, Grant/Award Number: TIN2016-75985-PCampos, C.; Román Moltó, JE. (2020). Inertia-based spectrum slicing for symmetric quadratic eigenvalue problems. Numerical Linear Algebra with Applications. 27(4):1-17. https://doi.org/10.1002/nla.2293S117274Tisseur, F., & Meerbergen, K. (2001). The Quadratic Eigenvalue Problem. SIAM Review, 43(2), 235-286. doi:10.1137/s0036144500381988Veselić, K. (2011). Damped Oscillations of Linear Systems. Lecture Notes in Mathematics. doi:10.1007/978-3-642-21335-9Grimes, R. G., Lewis, J. G., & Simon, H. D. (1994). A Shifted Block Lanczos Algorithm for Solving Sparse Symmetric Generalized Eigenproblems. SIAM Journal on Matrix Analysis and Applications, 15(1), 228-272. doi:10.1137/s0895479888151111Campos, C., & Roman, J. E. (2012). Strategies for spectrum slicing based on restarted Lanczos methods. Numerical Algorithms, 60(2), 279-295. doi:10.1007/s11075-012-9564-zLi, R., Xi, Y., Vecharynski, E., Yang, C., & Saad, Y. (2016). A Thick-Restart Lanczos Algorithm with Polynomial Filtering for Hermitian Eigenvalue Problems. 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BIT Numerical Mathematics, 56(4), 1213-1236. doi:10.1007/s10543-016-0601-5Guo, C.-H., & Lancaster, P. (2005). Algorithms for hyperbolic quadratic eigenvalue problems. Mathematics of Computation, 74(252), 1777-1792. doi:10.1090/s0025-5718-05-01748-5Li, H., & Cai, Y. (2015). Solving the real eigenvalues of hermitian quadratic eigenvalue problems via bisection. The Electronic Journal of Linear Algebra, 30, 721-743. doi:10.13001/1081-3810.1979RomanJE CamposC RomeroE andTomasA. SLEPc users manual. DSIC‐II/24/02–Revision 3.9. D. Sistemes Informàtics i Computació Universitat Politècnica de València;2018.Hernandez, V., Roman, J. E., & Vidal, V. (2005). SLEPc. ACM Transactions on Mathematical Software, 31(3), 351-362. doi:10.1145/1089014.1089019Guo, J.-S., Lin, W.-W., & Wang, C.-S. (1995). Numerical solutions for large sparse quadratic eigenvalue problems. Linear Algebra and its Applications, 225, 57-89. doi:10.1016/0024-3795(93)00318-tSleijpen, G. L. G., Booten, A. G. L., Fokkema, D. R., & van der Vorst, H. A. (1996). Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT Numerical Mathematics, 36(3), 595-633. doi:10.1007/bf01731936Bai, Z., & Su, Y. (2005). SOAR: A Second-order Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem. SIAM Journal on Matrix Analysis and Applications, 26(3), 640-659. doi:10.1137/s0895479803438523Güttel, S., & Tisseur, F. (2017). The nonlinear eigenvalue problem. Acta Numerica, 26, 1-94. doi:10.1017/s0962492917000034Yang, L., Sun, Y., & Gong, F. (2018). The inexact residual iteration method for quadratic eigenvalue problem and the analysis of convergence. Journal of Computational and Applied Mathematics, 332, 45-55. doi:10.1016/j.cam.2017.10.003Keçeli, M., Corsetti, F., Campos, C., Roman, J. E., Zhang, H., Vázquez-Mayagoitia, Á., … Wagner, A. F. (2018). SIESTA-SIPs: Massively parallel spectrum-slicing eigensolver for an ab initio molecular dynamics package. 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Cholesky-Like Factorization of Symmetric Indefinite Matrices and Orthogonalization with Respect to Bilinear Forms. SIAM Journal on Matrix Analysis and Applications, 36(2), 727-751. doi:10.1137/130947003Lu, D., Su, Y., & Bai, Z. (2016). Stability Analysis of the Two-level Orthogonal Arnoldi Procedure. SIAM Journal on Matrix Analysis and Applications, 37(1), 195-214. doi:10.1137/151005142Campos, C., & Roman, J. E. (2016). Parallel Krylov Solvers for the Polynomial Eigenvalue Problem in SLEPc. SIAM Journal on Scientific Computing, 38(5), S385-S411. doi:10.1137/15m1022458Higham, N. J., Mackey, D. S., Mackey, N., & Tisseur, F. (2007). Symmetric Linearizations for Matrix Polynomials. SIAM Journal on Matrix Analysis and Applications, 29(1), 143-159. doi:10.1137/050646202BalayS AbhyankarS AdamsM et al. PETSc users manual. ANL‐95/11 ‐ Revision 3.10. Argonne National Laboratory;2018.Betcke, T., Higham, N. J., Mehrmann, V., Schröder, C., & Tisseur, F. (2013). NLEVP. ACM Transactions on Mathematical Software, 39(2), 1-28. doi:10.1145/2427023.2427024Assink, J., Waxler, R., & Velea, D. (2017). A wide-angle high Mach number modal expansion for infrasound propagation. The Journal of the Acoustical Society of America, 141(3), 1781-1792. doi:10.1121/1.497757