Computation of Exponential Integrals
Donald E. Amos · ACM Transactions on Mathematical Software · 1980
Exponential IntegralsFormulas leading to the c o m p u t a t m n of M m e m b e r sequences of exponential integrals EN+k(x),x _> 0, N -> 1, k = 0, 1 . . . .M -1, are presented here and n n p l e m e n t e d in Fortran subroutine E X P I N T .Sequences of exponential integrals can be generated in a numerically stable fashion if recurrence is carrmd forward or backward away from the integer closest to x.In keeping with this requirement, we select n, the integer closest to x within the constraint N -n -~ N + M -1, and use E,(x) to start the recursion E,(x) Is computed by m e a n s of the power series on 0 ~ x ": 2 and the confluent h y p e r g e o m e t n c function U(n, n, x) on 2 < x < o0.U(n, n, x) is, in turn, computed from the backward recurswe Miller algorithm for U(n + k, n, x), k = 0, 1, . . ., with a normalizing relatmn derived from the two-term recursmn relation satisfied by E,(x) and E,+l(x).Truncation error bounds are derived and used m error tests m E X P I N T Exponential scaling is also provided as a subroutine optmn.