Approximating the exponential from a Lie algebra to a Lie group

Consider a differential equation y ′ = A ( t , y ) y , y ( 0 ) = y 0 y’=A(t,y)y,\ y(0)=y_0 with y 0 ∈ G y_ 0∈ G and A : R + × G → g A:R⁺× G→ g , where g g is a Lie algebra of the matricial Lie group G G . Every B ∈ g B∈ g can be mapped to G G by the matrix exponential map exp ⁡ ( t B ) exp{(tB)} with t ∈ R t∈ R . Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation <inline-formula content-type="math/ma