BESSIK cribed by Gautschi [1]. For arguments of large modulus, I~(z)is determined from its asymptotic expansion for large Catalogue number: AAQZ argument and backward recurrence. K~(z)and K~. 1(z)for vt ~1/2 are determined from Program obtaintable from: CPC Program Library, Queen's Neumann series when real part of z is small [2] and from University of Belfast, N. Ireland (see application form in Temme's algorithm [3] and some modifications [4) for this issue) arguments of moderately large or large modulus. For real arguments, Temme's algorithm for small arguments is also Computer: IBM 3033;Installation: National Research Coun-used. K~(z)for large order is obtained from the forward dil, Ottawa recurrence. Operating system: TSS/370 Restrictions on the complexity of the problem The functions are determined only for non-negative order. Programming language used: FORTRAN N K~(z)is determined only for values ofz with non-negative real part. The subroutines are inefficient when both order High speed storage required: 14700 words and modulus of argument are very large. No. of bits in a word: 32 Typical running time For order and modulus of argument not both large, the Overlay structure: none determination of a single function requires less than 1.6 ms. The determination of a sequence of functions with the orders No. of magnetic tapes required: none of two successive members of the sequence differing by 1 requires, in addition, 15 ~isfor each member of the sequence. Other peripherals used: card reader, line printer The running times of subroutines RBESI and RBESK for real argument are comparable to those of RBESJ and RBESY No. of cords in combinedprogram and test deck: 1955 [5J and are about 1/2 to 1/3 of the running times of subroutines CBESI and CBESK for complex argument. Running Card punching code: EBCDIC times on the IBM 3033 are about 1.9 times faster than times on the IBM 3032 which was the computer used for the Keywords: general purpose, Bessel, cylindrical, backward results reported in