Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems

The final publication is available at Springer via http://dx.doi.org/ 10.1007/s10543-016-0601-5.We investigate how to adapt the Q-Arnoldi method for the case of symmetric quadratic eigenvalue problems, that is, we are interested in computing a few eigenpairs of with M, C, K symmetric matrices. This problem has no particular structure, in the sense that eigenvalues can be complex or even defective. Still, symmetry of the matrices can be exploited to some extent. For this, we perform a symmetric linearization , where A, B are symmetric matrices but the pair (A, B) is indefinite and hence standard Lanczos methods are not applicable. We implement a symmetric-indefinite Lanczos method and enrich it with a thick-restart technique. This method uses pseudo inner products induced by matrix B for the orthogonalization of vectors (indefinite Gram-Schmidt). The projected problem is also an indefinite matrix pair. The next step is to write a specialized, memory-efficient version that exploits the block structure of A and B, referring only to the original problem matrices M, C, K as in the Q-Arnoldi method. This results in what we have called the Q-Lanczos method. Furthermore, we define a stabilized variant analog of the TOAR method. We show results obtained with parallel implementations in SLEPc.This work was supported by the Spanish Ministry of Economy and Competitiveness under Grant TIN2013-41049-P. Carmen Campos was supported by the Spanish Ministry of Education, Culture and Sport through an FPU Grant with reference AP2012-0608.Campos, C.; Román Moltó, JE. (2016). Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems. BIT Numerical Mathematics. 56(4):1213-1236. https://doi.org/10.1007/s10543-016-0601-5S12131236564Bai, Z., Su, Y.: SOAR: a second-order Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. 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Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method

Given the $n\times n$ matrix polynomial $P(x)=\sum_{i=0}^kP_i x^i$, we consider the associated polynomial eigenvalue problem. This problem, viewed in terms of computing the roots of the scalar polynomial $\det P(x)$, is treated in polynomial form rather than in matrix form by means of the Ehrlich-Aberth iteration. The main computational issues are discussed, namely, the choice of the starting approximations needed to start the Ehrlich-Aberth iteration, the computation of the Newton correction, the halting criterion, and the treatment of eigenvalues at infinity. We arrive at an effective implementation which provides more accurate approximations to the eigenvalues with respect to the methods based on the QZ algorithm. The case of polynomials having special structures, like palindromic, Hamiltonian, symplectic, etc., where the eigenvalues have special symmetries in the complex plane, is considered. A general way to adapt the Ehrlich-Aberth iteration to structured matrix polynomial is introduced. Numerical experiments which confirm the effectiveness of this approach are reported.Comment: Submitted to Linear Algebra App

The asymptotic spectra of banded Toeplitz and quasi-Toeplitz matrices

Toeplitz matrices occur in many mathematical, as well as, scientific and engineering investigations. This paper considers the spectra of banded Toeplitz and quasi-Toeplitz matrices with emphasis on non-normal matrices of arbitrarily large order and relatively small bandwidth. These are the type of matrices that appear in the investigation of stability and convergence of difference approximations to partial differential equations. Quasi-Toeplitz matrices are the result of non-Dirichlet boundary conditions for the difference approximations. The eigenvalue problem for a banded Toeplitz or quasi-Toeplitz matrix of large order is, in general, analytically intractable and (for non-normal matrices) numerically unreliable. An asymptotic (matrix order approaches infinity) approach partitions the eigenvalue analysis of a quasi-Toeplitz matrix into two parts, namely the analysis for the boundary condition independent spectrum and the analysis for the boundary condition dependent spectrum. The boundary condition independent spectrum is the same as the pure Toeplitz matrix spectrum. Algorithms for computing both parts of the spectrum are presented. Examples are used to demonstrate the utility of the algorithms, to present some interesting spectra, and to point out some of the numerical difficulties encountered when conventional matrix eigenvalue routines are employed for non-normal matrices of large order. The analysis for the Toeplitz spectrum also leads to a diagonal similarity transformation that improves conventional numerical eigenvalue computations. Finally, the algorithm for the asymptotic spectrum is extended to the Toeplitz generalized eigenvalue problem which occurs, for example, in the stability of Pade type difference approximations to differential equations

An extension of A-stability to alternating direction implicit methods

An alternating direction implicit (ADI) scheme was constructed by the method of approximate factorization. An A-stable linear multistep method (LMM) was used to integrate a model two-dimensional hyperbolic-parabolic partial differential equation. Sufficient conditions for the A-stability of the LMM were determined by applying the theory of positive real functions to reduce the stability analysis of the partial differential equations to a simple algebraic test. A linear test equation for partial differential equations is defined and then used to analyze the stability of approximate factorization schemes. An ADI method for the three-dimensional heat equation is also presented

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. SIAM Journal on Scientific Computing, 38(4), A2512-A2534. doi:10.1137/15m1054493Guo, C.-H., Higham, N. J., & Tisseur, F. (2010). An Improved Arc Algorithm for Detecting Definite Hermitian Pairs. SIAM Journal on Matrix Analysis and Applications, 31(3), 1131-1151. doi:10.1137/08074218xNiendorf, V., & Voss, H. (2010). Detecting hyperbolic and definite matrix polynomials. Linear Algebra and its Applications, 432(4), 1017-1035. doi:10.1016/j.laa.2009.10.014NakatsukasaY NoferiniV. Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems;2017. Preprint arXiv:1711.00495.Parlett, B. N., & Chen, H. C. (1990). Use of indefinite pencils for computing damped natural modes. Linear Algebra and its Applications, 140, 53-88. doi:10.1016/0024-3795(90)90222-xCampos, C., & Roman, J. E. (2016). Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems. 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. Journal of Computational Chemistry, 39(22), 1806-1814. doi:10.1002/jcc.25350Voss, H., Werner, B., & Hadeler, K. P. (1982). A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems. Mathematical Methods in the Applied Sciences, 4(1), 415-424. doi:10.1002/mma.1670040126Higham, N. J., Mackey, D. S., & Tisseur, F. (2009). Definite Matrix Polynomials and their Linearization by Definite Pencils. SIAM Journal on Matrix Analysis and Applications, 31(2), 478-502. doi:10.1137/080721406Al-Ammari, M., & Tisseur, F. (2012). Hermitian matrix polynomials with real eigenvalues of definite type. Part I: Classification. Linear Algebra and its Applications, 436(10), 3954-3973. doi:10.1016/j.laa.2010.08.035Gohberg, I., Lancaster, P., & Rodman, L. (1980). Spectral Analysis of Selfadjoint Matrix Polynomials. The Annals of Mathematics, 112(1), 33. doi:10.2307/1971320RozložnÍk, M., Okulicka-DŁużewska, F., & Smoktunowicz, A. (2015). 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

Tridiagonal substitution Hamiltonians

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the fractal structure and fractal dimensions of the spectrum, exact dimensionality of the integrated density of states, and the gap structure. We present a review of previous results, some applications, and open problems. Our investigation is based largely on the dynamics of trace maps. This work is an extension of similar results on Schroedinger operators, although some of the results that we obtain differ qualitatively and quantitatively from those for the Schoedinger operators. The nontrivialities of this extension lie in the dynamics of the associated trace map as one attempts to extend the trace map formalism from the Schroedinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are, in a sense, a test item, as many other models can be attacked via the same techniques, and we present an extensive discussion on this.Comment: 41 pages, 5 figures, 81 reference

Eigenvalue Problems

In natural sciences and engineering, are often used differential equations and systems of differential equations. Their solution leads to the problem of eigenvalues. Because of that, problem of eigenvalues occupies an important place in linear algebra. In this caption we will consider the problem of eigenvalues, and to linear and quadratic problems of eigenvalues. During the studying of linear problem of eigenvalues, we put emphasis on QR algorithm for unsymmetrical case and on minmax characterization of symmetric case. During the studying of quadratic problems of eingenvalue, we consider the linearization and variational characterization. We illustrate all with practical examples

Nonlinear potential analysis techniques for supersonic-hypersonic configuration design

Approximate nonlinear inviscid theoretical techniques for predicting aerodynamic characteristics and surface pressures for relatively slender vehicles at moderate hypersonic speeds were developed. Emphasis was placed on approaches that would be responsive to preliminary configuration design level of effort. Second order small disturbance and full potential theory was utilized to meet this objective. Numerical pilot codes were developed for relatively general three dimensional geometries to evaluate the capability of the approximate equations of motion considered. Results from the computations indicate good agreement with higher order solutions and experimental results for a variety of wing, body and wing-body shapes for values of the hypersonic similarity parameter M delta approaching one. Case computational times of a minute were achieved for practical aircraft arrangements