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A practical method for computing the exponential of a matrix and its integral. (English) Zbl 0704.65028

Consider a time-invariant state-space system \(\dot x=Ax\) where A is a constant \(n\times n\) matrix. This paper describes a practical algorithm for the computation of the transition matrix \(\Phi (T)=e^{AT}=\exp (AT)=\sum^{\infty}_{i=1}\frac{A^ iT^ i}{i!}\) and its integral \(\Theta (T)=\int^{T}_{0}e^{A(T-\tau)}d\tau\) for some sampling period T. \(\Theta\) (T) can be approximated by summing \(\Theta (N,T)=T\sum^{N}_{i=1}(AT)^ i/(i+1)!,\) where N is chosen large enough. After computing \(\Theta\) (N,T), \(\Phi\) (T) is computed by using \(\Theta\) (N,T). \(\Theta\) (N,T) is summed by a modified Horner nested computing rule [cf. G. H. Golub and C. F. Van Loan, Matrix Computations, 392-393 (1983; Zbl 0559.65011)]. The remaining part gives estimation and reduction of the number of terms to be summed and numerical tests.
Reviewer: M.Kono

MSC:

65F30 Other matrix algorithms (MSC2010)
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A30 Linear ordinary differential equations and systems

Citations:

Zbl 0559.65011
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References:

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