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Numerical computation of exponential matrices using the Cayley-Hamilton theoremA method for computing exponential matrices, which often arise naturally in the solution of systems of linear differential equations, is developed. An exponential matrix is generated as a linear combination of a finite number (equal to the matrix order) of matrices, the coefficients of which are scalar infinite sums. The method can be generalized to apply to any formal power series of matrices. Attention is focused upon the exponential function, and the matrix exponent is assumed tri-diagonal in form. In such cases, the terms in the coefficient infinite sums can be extracted, as recursion relations, from the characteristic polynomial of the matrix exponent. Two numerical examples are presented in some detail: (1) the three dimensional infinitesimal rotation rate matrix, which is skew symmetric, and (2) an N-dimensional tri-diagonal and symmetric finite difference matrix which arises in the numerical solution of the heat conduction partial differential equation. In the second example, the known eigenvalues and eigenvectors of the finite difference matrix permit an analytical solution for the exponential matrix, through the theory of diagonalization and similarity transformations, which is used for independent verification. The convergence properties of the scalar infinite summations are investigated for finite difference matrices of various orders up to ten, and it is found that the number of terms required for convergence increases slowly with the order of the matrix.
Document ID
19830007841
Acquisition Source
Legacy CDMS
Document Type
Technical Memorandum (TM)
Authors
Walden, H.
(NASA Goddard Space Flight Center Greenbelt, MD, United States)
Roelof, E. C.
(APL Laurel, Md., United States)
Date Acquired
September 4, 2013
Publication Date
July 1, 1982
Subject Category
Theoretical Mathematics
Report/Patent Number
NAS 1.15:83982
NASA-TM-83982
Report Number: NAS 1.15:83982
Report Number: NASA-TM-83982
Accession Number
83N16112
Distribution Limits
Public
Copyright
Work of the US Gov. Public Use Permitted.
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