Matrix structures and applications
- [1] Dipartimento di Matematica, Università di Pisa, Italy
Les cours du CIRM (2014)
- Volume: 4, Issue: 1, page 1-45
- ISSN: 2108-7164
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topBini, Dario A.. "Matrix structures and applications." Les cours du CIRM 4.1 (2014): 1-45. <http://eudml.org/doc/275637>.
@article{Bini2014,
affiliation = {Dipartimento di Matematica, Università di Pisa, Italy},
author = {Bini, Dario A.},
journal = {Les cours du CIRM},
language = {eng},
number = {1},
pages = {1-45},
publisher = {CIRM},
title = {Matrix structures and applications},
url = {http://eudml.org/doc/275637},
volume = {4},
year = {2014},
}
TY - JOUR
AU - Bini, Dario A.
TI - Matrix structures and applications
JO - Les cours du CIRM
PY - 2014
PB - CIRM
VL - 4
IS - 1
SP - 1
EP - 45
LA - eng
UR - http://eudml.org/doc/275637
ER -
References
top- A. H. Al-Mohy and N. J. Higham. The complex step approximation to the Fréchet derivative of a matrix function. Numer. Algorithms, 53:113–148 (2010). Zbl1188.65054MR2566131
- A. Amiraslani, R. M. Corless, and P. Lancaster. Linearization of matrix polynomials expressed in polynomial bases. IMA J. Numer. Anal., 29(1):141–157, 2009. Zbl1158.15022MR2470944
- G.S. Ammar and W.B. Gragg, Superfast solution of real positive definite Toeplitz systems, SIAM J. Matrix Anal. Appl. 9, 61–76 (1988). Zbl0658.65022MR938136
- J.L. Aurentz, T. Mach, R. Vandebril, D.S. Watkins, Fast and backward stable computation of roots of polynomials, TW654, 36 pages, Department of Computer Science, KU Leuven, Leuven, Belgium, August 2014 Zbl1319.65034MR3366913
- J.L. Aurentz, R. Vandebril, D.S. Watkins, Fast computation of eigenvalues of companion, comrade, and related matrices, BIT Numer Math 54, 7–30 (2014). Zbl1293.65052MR3177953
- F. Avram, On bilinear forms on Gaussian random variables and Toeplitz matrices, Probab. Theory Related Fields 79, 37–45 (1988). Zbl0648.60043MR952991
- O. Axelsson and G. Lindskög, On the rate of convergence of the preconditioned conjugate gradient method, Numerische Mathematik, 48, 499–523, 1986. Zbl0564.65017MR839614
- S. Barnett, A companion matrix analogue for orthogonal polynomials, Linear Algebra Appl., 12, 197–202 (1975). Zbl0327.15026MR387310
- T. Betcke, N.J. Higham, V. Mehrmann, C. Schröder, and F. Tisseur. NLEVP: A collection of nonlinear eigenvalue problems. http://www.mims.manchester.ac.uk/research/numerical-analysis/nlevp.html. Zbl1295.65140
- D. Bini. Parallel solution of certain Toeplitz linear systems. SIAM J. Comput., 13, 268–276 (1984). Zbl0534.68026MR739989
- D. Bini, M. Capovani, Spectral and computational properties of band symmetric Toeplitz matrices, Linear Algebra Appl. 52/53, 99–126 (1983). Zbl0549.15005MR709346
- D.A. Bini, Y. Eidelman, L. Gemignani, I. Gohberg, Fast QR eigenvalue algorithms for Hessenberg matrices which are rank-one perturbations of unitary matrices SIAM J. Matrix Anal. Appl., 29, 566–585 (2007). Zbl1147.65031MR2318363
- D. Bini, P. Favati, On a matrix algebra related to the discrete Hartley transform, SIAM J. Matrix Anal. Appl., 14, 500–507 (1993). Zbl0773.65029MR1211803
- D.A. Bini, G. Fiorentino, Design, Analysis and Implementation of a Multiprecision Polynomial Rootfinder, Numer. Algorithms, 23, 127–219 (2000). Zbl1018.65061MR1772050
- D.A. Bini, B. Iannazzo, B. Meini, Numerical Solution of Algebraic Riccati Equations, Fundamentals of Algorithms, SIAM, Philadelphia 2012. Zbl1244.65058MR2896454
- D. A. Bini, S. Dendievel, G. Latouche, and B. Meini. Computing the Exponential of Large Block-Triangular Block-Toeplitz Matrices Encountered in Fluid Queues, Linear Algebra Appl., in Press, doi:10.1016/j.laa.2015.03.035. Zbl06579448
- D. A. Bini, G. Latouche, and B. Meini. Numerical methods for structured Markov chains. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2005. Oxford Science Publications. Zbl1076.60002MR2132031
- D.A. Bini, B. Meini, The cyclic reduction algorithm: from Poisson equation to stochastic processes and beyond, Numerical Algorithms, 51, 23–60 (2009). Zbl1170.65021MR2505832
- D. A. Bini, V. Noferini, and M. Sharify. Locating the eigenvalues of matrix polynomials. SIAM J. Matrix Anal. Appl., 34, 1708–1727 (2013). Zbl1291.15048MR3144796
- D. Bini and V. Pan. Polynomial and Matrix Computations. Birkhäuser, 1994. Zbl0809.65012MR1289412
- D.A. Bini, L. Robol, Solving secular and polynomial equations: A multiprecision algorithm. J. Comput. Appl. Math., 272, 276–292 (2014). Zbl1310.65052MR3227386
- D.A. Bini, L. Robol, On a class of matrix pencils and -ifications equivalent to a given matrix polynomial. Linear Algebra Appl., in Press, doi:10.1016/j.laa.2015.07.017, arXiv:1406.1025 (2014).
- R.R. Bitmead, B.D.O. Anderson, Asymptotically fast solution of Toeplitz and related systems of linear equations. Linear Algebra Appl., 34, 103–116 (1980). Zbl1290.65031MR3177956
- P. Boito, Y. Eidelman, L. Gemignani, Implicit QR for Rank-Structured Matrix Pencils, BIT Numerical Mathematics 54, 85–111 (2014). Zbl1266.65059MR2991921
- P. Boito, Y. Eidelman, L. Gemignani, I. Gohberg, Implicit QR with Compression, Indag. Math. 23, 733–761 (2012). Zbl0952.47025
- A. Böttcher, S.M. Grudsky, Toeplitz Matrices, Asymptotic Linear Algebra and Functional Analysis. Text and Readings in Mathematics 18, Hindustan Book Agency, New Delhi 2000. Zbl0916.15012MR1724795
- A. Böttcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, Springer, New York 1999. Zbl1280.47027MR3060410
- A. Böttcher, I. Spitkovsky, The factorization problem: Some known results and open questions, Operator Theory: Advances and App., 229, 101–122 (013). Zbl1136.65041MR2397851
- S. Chandrasekaran, M. Gu, J. Xia, J. Zhu, A fast QR algorithm for companion matrices, Recent Advances in Matrix and Operator Theory, Oper. Theory Adv. Appl., vol. 179, Birkhüser, Basel (2008). Zbl0646.65042MR945937
- R. Chan, Circulant preconditioners for Hermitian Toeplitz systems, SIAM J. Matrix Anal. Appl. 10, 542–550 (1989). Zbl0684.65035MR1016802
- R. H.-F. Chan and X.-Q. Jin. An introduction to iterative Toeplitz solvers, volume 5 of Fundamentals of Algorithms. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. Zbl0666.65030MR976165
- T. F. Chan. An optimal circulant preconditioner for Toeplitz systems. SIAM J. Sci. Statist. Comput., 9(4):766–771, 1988. Zbl0646.65042
- R. Chan, G. Strang, Toeplitz equations by conjugate gradients with circulant preconditioner, SIAM J. Sci. Statist. Comput. 10, 104–119 (1989). Zbl0666.65030
- R.M. Corless. Generalized companion matrices in the Lagrange basis. In L. Gonzalez-Vega and T. Recio, editors, Proceedings EACA, June 2004.
- R.M. Corless and G. Litt, Generalized Companion Matrices for Polynomials not expressed in Monomial Bases, Ontario Research Centre for Computer Algebra, Department of Applied Mathematics University of Western Ontario http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/PABV/CMP.pdf. Zbl1259.15031MR2921748
- F. De Terán, F.M. Dopico, J. Pérez, Backward stability of polynomial root-finding using Fiedler companion matrices, IMA J. Numer. Anal. (2014). Zbl1198.65068MR2338469
- F. De Terán, F.M. Dopico, D.S. Mackey, Fiedler companion linearizations, for rectangular matrix polynomials. Linear Algebra Appl., 437, 957–991 (2012). Zbl1195.65035MR2530262
- S. Delvaux, M. Van Barel, A Hessenberg reduction algorithm for rank structured matrices, SIAM J. on Matrix Analysis and App., 29, 895–926 (2007). Zbl1139.65026MR2385737
- F. Di Benedetto, Gram matrices of fast algebras have a rank structure, SIAM J. Matrix Anal. Appl. 31, 526–545 (2009). Zbl1280.65027MR3137083
- Y. Eidelman, L. Gemignani, and I. Gohberg. Efficient eigenvalue computation for quasiseparable Hermitian matrices under low rank perturbations. Numer. Algorithms, 47, 253–273 (2008). Zbl1280.65034MR3136431
- Y. Eidelman, I. Gohberg, and I. Haimovici. Separable type representations of matrices and fast algorithms. Vol. 1, volume 234 of Operator Theory: Advances and Applications. Birkhäuser/Springer, Basel, 2014. Basics. Completion problems. Multiplication and inversion algorithms. Zbl1031.15014MR1999154
- Y. Eidelman, I. Gohberg, and I. Haimovici. Separable type representations of matrices and fast algorithms. Vol. 2, volume 235 of Operator Theory: Advances and Applications. Birkhäuser/Springer Basel AG, Basel, 2014. Eigenvalue method. Zbl1260.65024MR3011390
- M. Fiedler, A note on companion matrices. Linear Algebra and Its Applications, 372:325–331, 2003. Zbl0017.00102
- K. Frederix, S. Delvaux, M. Van Barel, An algorithm for computing the eigenvalues of block companion matrices, Numerical Algorithms, volume 62, 261–287 (2013). Zbl1186.15007MR2596623
- F.R. Gantmacher, M.G. Krein, Sur les matrices complètement non négatives et oscillatoires. Compositio Mathematica 4, 445–476 (1937). Zbl0848.65010MR1312096
- S. Gaubert and M. Sharify, Tropical scaling of polynomial matrices. In Positive systems, volume 389 of Lecture Notes in Control and Inform. Sci., 291–303. Springer, Berlin, 2009. Zbl1170.15300MR662418
- I. Gohberg, T. Kailath, V. Olshevsky (1995). “Fast Gaussian elimination with partial pivoting for matrices with displacement structure”. Mathematics of Computation 64 (212): 1557–1576. Zbl0288.15004MR353038
- I. Gohberg, P. Lancaster, and L. Rodman. Matrix polynomials. Academic Press Inc., New York, 1982. Computer Science and Applied Mathematics. Zbl0254.65027MR329227
- I. Gohberg and A. Semencul, On the inversion of finite Toeplitz matrices and their continuous analogs (in Russian), Mat. Issled 7, 201–223 (1972). Zbl0103.01003MR132755
- G.H. Golub, Some modified matrix eigenvalue problems, SIAM Review, 15, 318–334 (1973). Zbl0886.65013MR1470294
- I.J. Good, The colleague matrix, a Chebyshev analogue of the companion matrix Quart. J. Math., Oxf. Ser., 12, 61–68 (1961). Zbl0611.47018MR890515
- T.NT Goodman, C. A. Micchelli, G. Rodriguez, S. Seatzu, Spectral factorization of Laurent polynomials, Advances in Computational Mathematics, 7 429–454 (1997). Zbl1167.15001MR2396439
- U. Grenander, G. Szegő. Toeplitz Forms and Their Applications. Second Edition, Chelsea, New York, 1984. Zbl0839.65028MR1355506
- N. J. Higham. Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. Zbl0931.65018MR1715813
- T. Kailath, A H. Sayed, Displacement Structure: theory and applications, SIAM Review, 297–386, 1995. Zbl0433.15001MR533501
- T. Kailath and A. H. Sayed, eds., Fast Reliable Algorithms for Matrices with Structure, SIAM Press, 1999. Zbl1103.65310MR2073063
- T. Kailath, S. Y. Kung and M. Morf, Displacement ranks of matrices and linear equations, J. Math. Appl. 68, 395-407 (1979). Zbl0695.60088MR1010040
- F.-R. Lin, W.-K. Ching, M.K. Ng, Fast inversion of triangular Toeplitz matrices, Theoretical Computer Science, 315, 511–523 (2004). Zbl0469.60002MR1313503
- M.F. Neuts. Structured stochastic matrices of type and their applications, volume 5 of Probability: Pure and Applied. Marcel Dekker, Inc., New York, 1989. MR4222
- M.F. Neuts. Matrix-geometric solutions in stochastic models. Dover Publications, Inc., New York, 1994. An algorithmic approach, Corrected reprint of the 1981 original. Zbl0028.19901MR12681
- A. Ostrowski, Recherches sur la méthode de Graeffe et les zéros des polynomes et des séries de Laurent. Chapitres III et IV. Acta Math. 72, 157–257 (1940) Zbl0099.32403MR120492
- A. Ostrowski, Addition à notre mémoire: ”Recherches sur la méthode de Graeffe et les zéros des polynomes et des séries de Laurent.” Acta Math. 75, 183–186 (1943) Zbl0028.19901
- S.V. Parter, Extreme eigenvalues of Toeplitz forms and applications to elliptic difference equations, Trans. Amer. Math. Soc., 99 (1966), pp. 153–192. Zbl1195.65057MR2679754
- M.A. Pellet. Sur un mode de séparation des racines des équations et la formule de Lagrange. Bull. Sci. Math., 5:393–395, 1881. Zbl1031.65046MR1990645
- F. Poloni, A note on the -storage implementation of the GKO algorithm and its adaptation to Trummer-like matrices, Numer. Algorithms, 55, 115–139 (2010). Zbl1195.65057
- Y. Saad, Iterative methods for sparse linear systems, second Edition, SIAM, Philadelphia 2003. Zbl0877.65025MR1410714
- S. Serra Capizzano, Toeplitz matrices: spectral properties and preconditioning in the CG method, NLA courses at FMB in Uppsala, TR 23 (2014), Information Technology Dept, Uppsala University. Zbl0864.65019MR1401945
- S. Serra Capizzano, Preconditioning strategies for Hermitian Toeplitz systems with nondefinite generating functions, SIAM J. Matrix Anal. Appl., 17, 1007–1019 (1996). Zbl1021.15005MR1972737
- S. Serra Capizzano. Optimal, quasi–optimal and superlinear preconditioners for asymptotically ill–conditioned positive definite Toeplitz systems, Math. Comput. 66, 651–665 (1997). Zbl0215.27305MR279998
- S. Serra Capizzano, E. Tyrtyshnikov, How to prove that a preconditioner cannot be superlinear, Math. Comput. 1305-1316, (2003). Zbl0621.65025
- B.T. Smith, Error bounds for zeros of a polynomial based upon Gerschgorin’s theorems, J. Assoc. Comput. Math. 17, 661–674 (1970). Zbl0131.36002MR173681
- G. Strang, A proposal for Toeplitz matrix calculation, Stnd. Appl. Math. 74, 171–176 (1986). Zbl0841.15006MR1366576
- W.F. Trench, An algorithm for the inversion of finite Toeplitz matrices, SIAM J. Appl. Math. 12, 515–522 (1964). Zbl0890.15006MR1484073
- E.E. Tyrtyshnikov, A Unifying Approach to Some Old and New Theorems on Distribution and Clustering, Linear Algebra Appl., 232 1–43 (1996). Zbl1196.65076MR2672262
- E.E. Tyrtyshnikov, N. Zamarashkin. Spectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationships, Linear Algebra Appl., 270, 15–27 (1998). Zbl1216.65045MR2678097
- M. Van Barel, R. Vandebril, P. Van Dooren, K. Frederix, Implicit double shift QR-algorithm for companion matrices, Numerische Mathematik, 116, 177–212 (2010). Zbl1168.15312MR2191201
- R. Vandebril and G.M. Del Corso. An implicit multishift qr-algorithm for hermitian plus low rank matrices. SIAM Journal on Scientific Computing, 32, 2190–2212 (2010). Zbl1175.65045MR2378139
- R. Vandebril, M. Van Barel, G. Golub, N. Mastronardi, A bibliography on semiseparable matrices, Calcolo 42, 249–270 (2005) Zbl1141.65019MR2460593
- R. Vandebril, M. Van Barel, N. Mastronardi, Matrix Computations and Semiseparable Matrices, Vol. 1 Linear Systems. Johns Hopkins, Baltimore, Maryland, 2008. Zbl1258.65030MR3023454
- R. Vandebril, M. Van Barel, N. Mastronardi, Matrix Computations and Semiseparable Matrices, Vol. 2 Eigenvalue and Singular Value Methods. Johns Hopkins, Baltimore, Maryland, 2008. Zbl0522.15013MR719861
- J. Xia, Y. Xi, M. Gu, A superfast structured solver for toeplitz linear systems via randomized sampling, SIAM J. Matrix Anal. Appl., 33, 837–858 (2012). Zbl1263.15010MR3023468
- W. Werner,A generalized companion matrix of a polynomial and some applications, Linear Algebra Appl., 55, 19–36 (1983). Zbl0194.18102MR247762
- P. Zhlobich, Differential qd algorithm with shifts for rank-structured matrices. SIAM J. Matrix Anal. Appl. 33, 1153–1171 (2012). Zbl1263.15010
- S. Zohar, Toeplitz matrix inversion: The algorithm of W.F. Trench, J. Assoc. Comput. Mach. 16, 592–601 (1967). Zbl0194.18102
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