### Abstract

The matrix exponential plays an important role in solving systems of linear differential equations. We will give a general expansion of the matrix exponential S = exp[λ(A + B)] as S_{n,m} = e ^{λbn}delta;_{n,m} + ∑_{q=1} ^{∞} ∑_{l1=0}^{N} ⋯ ∑ _{lq-1=0}^{N} a_{lq-1,m}C_{n,l1,...,lq-1,m} ^{(q)} (B, λ) with C_{n,l1,...,lq-1,m}^{(q)} (B, λ) being an analytical expression in b_{N}, b_{l1}, b_{l2}, ... b_{lq-1}, b_{m}, and the scalar coefficient λ A is a general N × N matrix with elements a_{n,m} and B a diagonal matrix with elements b_{n,m} = b_{n}δ _{n,m} along its diagonal. The convergence of this expansion is shown to be superior to the Taylor expansion in terms of (λ[A + B]), especially if elements of B are larger than the elements of A. The convergence and possibility of solving the phase problem through multiple scattering is demonstrated by using this expansion for the computation of large-angle convergent beam electron diffraction pattern intensities.

Original language | English (US) |
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Pages (from-to) | 803-816 |

Number of pages | 14 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 36 |

Issue number | 3 |

DOIs | |

State | Published - Jan 24 2003 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)