Bounding error of calculating the matrix functions

Dehghani-Madiseh, Marzieh

doi:10.22034/cmde.2020.38964.1708

فهرست نشریات دارای اعتبار وزارت علوم، تحقیقات و فناوری

تعداد نشریات	43
تعداد شماره‌ها	1,224
تعداد مقالات	15,034
تعداد مشاهده مقاله	50,540,285
تعداد دریافت فایل اصل مقاله	13,636,783

	Bounding error of calculating the matrix functions
Computational Methods for Differential Equations
مقاله 19، دوره 10، شماره 1، فروردین 2022، صفحه 259-273 اصل مقاله (3.72 M)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2020.38964.1708
نویسنده
Marzieh Dehghani-Madiseh^*
Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
چکیده
Matrix functions play important roles in various branches of science and engineering. In numerical computations and physical measurements there are several sources of error which significantly affect the main results obtained from solving the problems. This effect also influences the matrix computations. In this paper, we propose some approaches to enclose the matrix functions. We then present some analytical arguments to ensure that the obtained enclosures contain the exact result. Numerical experiments are given to illustrate the performance and effectiveness of the proposed approaches.
کلیدواژه‌ها
Matrix function؛ Floating point arithmetic؛ Interval arithmetic

مراجع
[1] B. Adcock and D. Huybrechs, Multivariate modified Fourier expansions, Proceedings of the International Confer- ence on Spectral and High Order Methods, (2009), 85–92. [2] G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, New York, 1983. [3] S. Barnett, Matrices: Methods and Applications, 2nd ed., Oxford University Press, 1994. [4] R. Bellman, Stability Theory of Diﬀerential Equations, McGraw-Hill, 1953. [5] J. C. Burkill, Functions of intervals, Proceedings of the London Mathematical Society, 22 (1924), 375–446. [6] D. Cheu and L. Longpre, Towards the possibility of objective interval uncertainty in physics, Reliab. Comput., 15 (2011) 43–46. [7] G. W. Cross and P. Lancaster, Square roots of complex matrices, Linear Multilinear Algebra, 1 (1974), 289–293. [8] M. Dehghan and M. Hajarian, Determination of a matrix function using the divided diﬀerence method of Newton and the interpolation technique of Hermite, J. Comput. Appl. Math., 231 (2009), 67–81. [9] M. Dehghan and M. Hajarian, Computing matrix functions using mixed interpolation methods, Math. Comput. Modelling, 52 (2010), 826–836. [10] L. Dymova and M. Pilarek, Organizing Calculations in Algorithms for Solving Systems of Interval Linear Equa- tions Using the Interval Extended Zero Method, Parallel Processing and Applied Mathematics, 7204 (2012), 439–446. [11] L. Dymova, P. Sevastjanov, and M. Pilarek, A method for solving systems of linear interval equations applied to the Leontief input output model of economics, Expert Systems with Applications, 40 (2013), 222–230. [12] S. S. Ganji, A. Barari, and D. D. Ganji, Approximate analysis of two-mass spring systems and buckling of a column, Comput. Math. Appl., 61 (2011), 1088–1095. [13] J. Garloﬀ, Bibliography on interval mathematics, continuation, Freiburger Intervall-Berichte, 2 (1987), 1–50. [14] J. Garloﬀ and K. P. Schwierz, A bibliography on interval mathematics, Comput. Appl. Math., 6 (1980), 67–79. [15] M. Ghanbari, T. Allahviranloo, and E. Haghi, Estimation of algebraic solution by limiting the solution set of an interval linear system, Soft Comput., 16 (2012), 2135–2142. [16] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996. [17] A. Goldsztejn, Modal intervals revisited, Part 1: A generalized interval natural extension, Reliab. Comput., 16 (2012), 130–183. [18] A. Goldsztejn, Modal intervals revisited, Part 2: A generalized interval mean value extension, Reliab. Comput., 16 (2012), 184–209. [19] E. R. Hansen, Interval arithmetic in matrix computations, Part I, SIAM J. Numer. Anal., 2 (1965), 308–320. [20] E. R. Hansen, Global Optimization Using Interval Analysis, Dekker, Inc., New York, 1992. [21] G. I. Hargreaves and N. J. Higham, Eﬃcient algorithms for the matrix cosine and sine, Numer. Algorithms, 40 (2005), 383–400. [22] N. J. Higham, The scaling and squaring method for the matrix exponential revisited, SIAM J. Matrix Anal. Appl., 26 (2005), 1179–1193. [23] N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, 2008. [24] N. J. Higham and M. I. Smith, Computing the matrix cosine, Numer. Algorithms 34 (2003), 13–26. [25] M. Hladik, Solution sets of complex linear interval systems of equations, Reliab. Comput., 14 (2010), 78–87. [26] M. Hladik, Weak and strong solvability of interval linear systems of equations and inequalities, Linear Algebra Appl., 438 (2013), 4156–4165. [27] R. B. Kearfott and V. Kreinovich, editors. Applications of Interval Computations, Applied Optimization, Dor- drecht, Netherlands, Kluwer, 1996. [28] R. B. Kearfott, Interval computations: Introduction, uses, and resources, Euromath Bulletin, (1996), 95–112. [29] B. J. Kubica and A. Wozniak, Interval methods for computing the Pareto-front of a multicriterial problem, Parallel Processing and Applied Mathematics, 4967 (2008), 1382–1391. [30] B. J. Kubica and A. Wozniak, An Interval Method for Seeking the Nash Equilibria of Non-cooperative Games, Parallel Processing and Applied Mathematics, 6068 (2010), 446–455. [31] Y. Y. Lu, A Pade approximation method for square roots of symmetric positive definite matrices, SIAM J. Matrix Anal. Appl. 19 (1998), 833–845. [32] R. E. Moore, Interval Arithmetic and Automatic Error Analysis in Digital Computing, Thesis (Ph.D.)–Stanford University, 1962 [33] R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM, 2009. [34] A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, England, 1990. [35] M. Pilarek and R. Wyrzykowski, Solving Systems of Interval Linear Equations in Parallel Using Multithreaded Model and Interval Extended Zero Method, Parallel Processing and Applied Mathematics, 7203 (2012), 206–214. [36] H. Richter, Uber Matrixfunktion, Math. Ann., 122 (1950), 16–34. [37] R. F. Rinehart, The equivalence of definitions of a matrix function, Amer. Math. Monthly, 62 (1955), 395–414. [38] S. M. Rump, Fast and parallel interval arithmetic, BIT Numer. Math., 39 (1999), 534–554. [39] A. Sasane, Diﬀerential Equations, Department of Mathematics, London School of Economics, Lecture notes. [40] S. P. Shary, On nonnegative interval linear systems and their solution, Reliab. Comput., 15 (2011), 358–369. [41] A. Shehata and M. Abul-Dahab, On Humbert matrix functions, J. Egyptian Math. Soc., 20 (2012), 167–171. [42] T. Sunaga, Theory of an Interval Algebra and its Application to Numerical Analysis, Gaukutsu Bunken Fukeyu- kai, Tokyo, 1958. [43] J. J. Sylvester, On the equation to the secular inequalities in the planetary theory, Phill. Mag., 16 (1883), 267–269. [44] R. C. Young, The algebra of many-valued quantities, Math. Ann., 104 (1931), 260–290.
آمار تعداد مشاهده مقاله: 480 تعداد دریافت فایل اصل مقاله: 440

سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب

پیوندهای مفید

آمار

Bounding error of calculating the matrix functions