A useful expansion of the exponential of the sum of two non-commuting matrices, one of which is diagonal

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 ^λbndelta;_n,m + ∑_q=1 ^∞ ∑_l1=0^N ⋯ ∑ _lq-1=0^N a_lq-1,mC_{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_nδ _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.