Certification of an algorithm for Bessel functions of real argument

Abstract

using multiprecis ion ari thm etic [2).2 Th e 60·bit mantissa of each part of the test value was sub· tracted from the correspo ndin g manti ssa of the check value , and the res ult expressed as a multiple, m , say, of the last bit. For most e rrors mentioned in this certification . 1 m 1 ~ 7. Bit comparison was used in preference to calc ulation of relative error in order to simplify co mputa tion s. An error of m bits in the mantissa corresponds to a relative error between 171 .2 60 and 111 • 259• This tes t is too stri ct , however, near a zero of the real or im agin ary part of th e fun ction be ing tested. For co mpl ex values of z, it is more reali stic to compare the bit error with the greater of the real and im aginary parts , since the relative error in a complex number ~+ i'Y/ is (o~+ iO'Y/) /I ~+ i'Y/ I, o~ and o'Y/ de notin g th e respective errors in the real and imaginary parts. For arguments in th e neighborhoods of zeroes of Re JII (Z), 1m J,, (z), Re l ,,(z) , or 1m J,,(z), an assess me nt of absolute errors was made to 60 bin ary places by right·shifting the mantissa of both test and c heck values before subtraction. Thi s was done whenever the order n was less than 1 Z 1 and the modulus of the 1 check value less than 2' These cases are distinguished by asterisks in the computer printout;