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 Sn,m = e λbndelta;n,m + ∑q=1 ∞ ∑l1=0N ⋯ ∑ lq-1=0N alq-1,mCn,l1,...,lq-1,m (q) (B, λ) with Cn,l1,...,lq-1,m(q) (B, λ) being an analytical expression in bN, bl1, bl2, ... blq-1, bm, and the scalar coefficient λ A is a general N × N matrix with elements an,m and B a diagonal matrix with elements bn,m = bnδ 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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)