Computation of modified Bessel functions and their ratios

An efficient algorithm for calculating ratios r,(x) = I,+i(x)/1I(x), v > 0, x > 0, is presented. This algorithm in conjunction with the recursion relation for r (x) gives an alternative to other recursive methods for Ih(x) when approximations for low-order Bessel functions are available. Sharp bounds on r,(x) and l(x) are also established in addition to some monotonicity properties of r,(x) and r,'(x). r,+k(X) = I.+k+1(X)/I.+k(X), a = V - (v), k = 0, 1, * , (v), where (v) is the integer part of v and 0 ? v - (v) < 1. The identity El, (3) I^(x) = Ia(x) II r^_k(x), a = v- (v), k=l