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Some applications of the Hermite matrix polynomials series expansions. (English) Zbl 0929.33006

Evaluation of matrix functions using Hermite matrix polynomials was proposed to avoid computational difficulties. The results were applied to construct approximations (with a prefixed accuracy) of problems \(Y'= AY\), \(Y(0)=y_0\) (\(A\) is a matrix, \(y_0\) a vector), \[ Y''+A^2Y= 0,\;Y(0)=P,\;Y'(0) =Q \] \((A\) is a matrix, \(P\) and \(Q\) are vectors), and the Sylvester matrix differential equation. The solution of the problems can be expressed in terms of \(\exp(At)\), \(\cos(At)\), \(\sin(At)\), and \(\exp(Bt)\). Some new properties of Hermite matrix polynomials were established. Hermite matrix polynomial series expansion of \(e^{At}\), \(\sin(At)\), and \(\cos(At)\) of any matrix, also with their finite series truncation, with a prefixed accuracy in a bounded domain was dealt with. Approximations were derived so that the error with respect to the exact solution is uniformly upper bounded.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
34A30 Linear ordinary differential equations and systems
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
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[1] Apostol, T., Explicit formulas for solutions of the second-order matrix differential equation Y″ = AY, Amer. math. monthly, 8, 159-162, (1975) · Zbl 0302.34001
[2] Bellman, R., Introduction to matrix analysis, (1965), McGraw-Hill New York · Zbl 0124.01001
[3] Dunford, N.; Schwartz, J., Linear operators, part I, (1957), Interscience New York
[4] Gajić, Z.; Qureshi, M., Lyapunov matrix equationin systems stability and control, (1995), Academic Press New York
[5] Golub, G.; Van Loan, C.F., Matrix computations, (1989), Johns Hopkins Univ. Press Baltimore, M.A · Zbl 0733.65016
[6] Jódar, L., Explicit solutions for second order operator differential equations with two boundary value conditions, Linear algebra appl., 103, 73-86, (1988) · Zbl 0651.34062
[7] Jódar, L.; Company, R., Hermite matrix polynomials and second order matrix differential equations, J. approx. theory appl., 12, 2, 20-30, (1996) · Zbl 0858.15014
[8] Jódar, L.; Defez, E., On Hermite matrix polynomials and Hermite matrix functions, J. approx. theory appl., 14, 1, 36-48, (1998) · Zbl 0911.15015
[9] Keller, H.B., Numerical solution of two point boundary value problems, (), Philadelphia · Zbl 0172.19503
[10] Law, A.G.; Zhang, C.N.; Rezazadeh, A.; Jódar, L., Evaluation of a rational function, Numer. algorithms, 3, 265-272, (1992) · Zbl 0788.65016
[11] Moler, C.B.; Van Loan, C.F., Nineteen dubious ways to compute the exponential of a matrix, SIAM rev., 20, 801-836, (1978) · Zbl 0395.65012
[12] Rainville, E.D., Special functions, (1960), Chelsea New York · Zbl 0050.07401
[13] Saks, S.; Zygmund, A., Analytic functions, (1971), Elsevier Amsterdam · JFM 60.0243.01
[14] Serbin, S.M., Rational approximations of trigonometric matrices with applications to second-order systems of differential equations, Appl. math. comput., 5, 75-92, (1979) · Zbl 0408.65047
[15] Serbin, S.M.; Blalock, S.A., An algorithm for computing the matrix cosine, SIAM J. sci. statist. comput., 1, 2, 198-204, (1980) · Zbl 0445.65023
[16] Serbin, S.; Serbin, C., A time-stepping procedure for X′ = A1X + XA2 + D,X(0) = C, IEEE trans. autom. control., 25, 1138-1141, (1980)
[17] Varga, R.S., On higher order stable implicit methods for solving parabolic partial differential equations, J. math. phys., 40, 220-231, (1961) · Zbl 0106.10805
[18] Ward, R.C., Numerical computation of the matrix exponential with accuracy estimate, SIAM J. numer. anal., 14, 600-619, (1977) · Zbl 0363.65031
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