Algorithm 484: Evaluation of the modified Bessel functions K0(Z) and K1(Z)for complex arguments

Fort ran subroutines are presented for the purpose of computing and evaluating g-splines interpolating Hermite-Birkoff data. The subroutines are based on a factorization method for comput ing g-splines discussed by Munteanu and Schumaker (Math. Comp. 27 (1973), 317-325). Description 1. bltroduction. In the following we present subroutines for calculating polynomial spline functions solving Hermite-Birkhoff (HB) interpolation problems. The subroutines are based on algori thms described in [9]. We begin by reviewing the definition of an HB-interpolation problem. Let N >_ 2 and xz < x,. < . . . < x N be prescribed. Suppose for each j, 1 < j < N, that zi is a positive integer, I M 1 j < I M ~ j < • • • < l M ~ i j are positive integers, and y~.:, y~.j, . . . , y~ . j are prescribed real numbers. The HB-interpolation problem is to determine s such that S(tMii--I)(Xi) = Y i j , i = 1, 2, . . . , zj, j = 1, 2, . . . , N. (1) We see that z j describes the number of derivatives prescribed at x j while the vector (IMI.j, . . . , lMzs. j) describes which derivatives. If z) = 1, j = 1, . . . , N, we have a simple interpolation problem. We are concerned with solving: HB-interpolation problems with polynomial splines. Let M be an integer, M >_ IMzj . i , j = 1, 2, . . . , N. Then (cf. [4]) there exists a function s satisfying (1) and s(2U)(t) = O, xj < t < x~+x, j = 1, 2, . . . , N 1; (2) s(M)(t) = 0, t < X~. t > Xlv; (3) s C C(~-z~( ° : , o¢); (4) s~ -"u-~ (xs+) = s~ ~--~-~) (xi-) , (5) l C {l . . . . . M} \ { IM,5, . . . , IM~j. j} j = 1,2, . . . , N . The function s is called a g-spline. It is a polynomial spline o f degree 2M -1 ; i.e. it is piecewise a polynomial o f degree 2M 1. The way in which the pieces tie together is described by (4) and (5). If the only polynomial o f degree M 1 which solves the homogeneous HB-interpolation problem (i.e. satisfies (1) with zero right-hand side) is the identically zero polynomial, then we say the HB-problem is M-poised. In this case there is a unique g-spline of degree 2M-1 solving the HB-problem (1). We consider constructing g-splines only for M-poised HB-problems. Given an M-poised HB-interpolat ion problem, the unique g-spline interpolant s satisfying (1)-115) can be represented as ~p,(t), t < xl s(t) = ~pi(t) , .rj-z < t ~ x ~ , j = 2, 3, . . . , N, (6) [pN+l(t), t > XN, where f o r j = 1, 2, . . . , N, ps(t) is a polynomial of the form 2 M pj( t ) = ~ C t . s ( t xs) z-1 and (7) l=1