Rational Chebyshev Approximations for the Bickley Functions Kin(x)

Transcription

1 MATHEMATICS OF COMPUTATION, VOLUME, NUMBER JULY, PAGES - Rational Chebyshev Approximations for the Bickley Functions Kin(x) By J. M. Blair, C. A. Edwards and J. H. Johnson Abstract. This report presents near-minimax rational approximations for the Bickley functions Kix(x) for x > 0, KU(x) and Ki(x) for 0 < x <, and Ki$(x), Ki^(x) and KíXq(x) for x >, with relative errors ranging down to. The approximations, combined with the recurrence relation, yield a stable method of computing Kin(x), n =,,...,, for the complete range of the argument.. Introduction. The Bessel function integrals defined by () Kin(x) = rkin_x(t)dt, n=,,,..., J X with Ki0(x) = KQ(x), were first introduced by Bickley [] in connection with the solution of heat convection problems. They arise in neutron transport calculations, and are widely used in nuclear reactor computer codes. Taylor series and asymptotic expansions for Kin(x) are developed in [], [] and [], and [] contains a discussion of the numerical stability of the four-term recurrence relation. Chebyshev series and rational approximations to the Kin(x) have been published in a number of reports. [] gives S rational approximations to Ki and Ki^; [] gives S, S and S rational approximations to Kix, Ki and Ki, respectively; [] gives S rational approximations to Kix ; [] gives S rational approximations to Kix Ki; [] gives 0D Chebyshev series approximations to Kix; and [] gives S rational approximations to Kiv Ki and Ki for 0 < x <, and to Ki, Kix and Kix for x >. A number of the approximations in [] [] suffer from significant digit cancellation. This report gives rational minimax approximations to Kix (x) for x > 0, to Kix, Ki and Ki for 0 < x <, and to Ki, Ki and Kix0 for x >, with relative errors ranging down to -. The approximations, combined with the recurrence relation, yield a stable method of computing Kin(x), n =,,...,, for the complete range of the argument. The results in [] may be used to determine the accuracy of the Kin when the recurrence relation is extended to higher orders.. Functional Properties. Most of the results of this section are given in more general terms in []. Alternative definitions of Kin(x) axe Received February,. AMS (MOS) subject classifications (0). Primary D0; Secondary A0, AS0. Key words and phrases. Rational Chebyshev approximations, Bickley functions, Bessel function integrals. Copyright, American Mathematical Society

2 CHEBYSHEV APPROXIMATIONS FOR THE BICKLEY FUNCTIONS () Kin(x) = f g xcosh t 0 cosh" t dt o) = r e du J l u"iu - I)* () = fr/v*mc,'cos,,-üdi;. J 0 The derivative of Kin is given by Ki' ix) =-Kin_xix), and higher derivatives may be computed recursively from the formulae Ki^ix) = -Kifjj >(*), k =,,.n = 0,,,..., () Ki<* (x) = - «L*f (*) - * o*-!)(*) -^-AT/(0k->(x), fc =,,,... The latter equation is obtained by repeated differentiation of the formula K'xix) = ~Kxix)-K0ix), where K0 and ATt are the modified Bessel functions, and Ki0 = K0, Ki_x = Kx. By integrating () by parts we can derive the recurrence relation () (- lakijx) = x[kin_ix) -Kin_xix)] + (- )Kin_ix). From () and () it follows that -/, =, () ^ (0)= ' n = ' Ascending series for the Kin can be developed by repeated integration of the ascending series for KQix). The resulting formula is Kinix)=pnix) + i-iy X,+- + '*'H-Y-ln-! \ ', wr (+U-q+: *=i(*!)^.(*+/) fc Â:+- -L + _i_ H-, In * k + ' where is Euler's constant, and Pnix) is defined recursively, starting with P0ix) = 0, by

3 J. M. BLAIR, C. A. EDWARDS AND J. H. JOHNSON P ix) = KiniO)-fxo P _xit)dt, =,,,... Asymptotic expansions for large arguments can be developed by writing () in the form and expanding the integrand binomially. The resulting asymptotic formula is () Kinix)~e-*ffi io(-»m n'm Y m where, = 0, (") tt- ~~ j (m - ) «jk)\jn + m - k - l)\ The alternative formula min - l)\ k=o ^ Skikl)im nk(ku(m - k)\,"' () ( + l)am + = (m + H){(w + Vi + n)am - im - Yi)(m - tt + n)am_x} is derived in []. The change of variable» = v in () gives the formula () «L,«= (/*)*e-* f~ e- (l + )"" (l + g )"* dv, which proves to be useful for computations.. Stability of Recurrence Relation. If () is used for forward recursion, the growth of the absolute error in Kinix) is determined by the factor x/(t - ). Since Kin is a slowly decreasing function of, the relative error is comparable to the absolute error, and forward recursion over a short range is stable provided x/(«- ) is not large. As a check of this result, Ki, Ki,...,KiX0 were computed by forward recursion on Kix,Ki and for x = 0(0.0).0. The greatest loss of accuracy noted was one digit, which occurred near x =. In general, the accuracy loss was less than one digit. If () is used for backward recursion, the main factor determining the growth of the absolute error is (tj - l)/x. Since the relative error grows more slowly than the absolute error, backward recursion is stable provided ( - l)/x is not large. A numerical test consisted of computing Kin, Ki,..., Kix by backward recursion on Ki%, Kig and Kï, for x>. The greatest loss of accuracy noted was less than one digit, and occurred near x =. In general, the accuracy loss was very small. [] contains a more detailed discussion of the stability of the recurrence relation, and tables of values of the relative error from forward recursion for < 0 and x < 00.

4 CHEBYSHEV APPROXIMATIONS FOR THE BICKLEY FUNCTIONS. Generation of Approximations. Rational minimax approximations to Kinix) were computed in decimal arithmetic on a CDC 00 using a version of the second algorithm of Remes due to Ralston []. was levelled to three digits. The approximation forms and intervals are The relative error of the approximations Ki ix) - Rlmix) + x" In xs/m(x), 0 <x <, n -,., -x-*e-*/?/m(l/x), x>,«=, <x-*e-xrlm(llx), Kx<, = l,,, ^ x-'ae-xrlmillx), x>, =,,, where Rlmix) and Slm(x) axe rational functions of degree / in the numerator and in the denominator. For the range 0 < x <, Rlm is positive for n =,,, while Slm is positive for =, and negative for =, and so a loss of accuracy occurs for =, by subtraction. However, the amount of cancellation is small, being less than bit for =, and considerably less for =. The master routine for the range 0 < x < uses the power series expansions in (). For sufficiently large values of x, for which terms in the asymptotic series in () become less than -0 in magnitude, the master routine uses the series in (). For the intermediate range Kinix) is computed by a local Taylor series expansion of the form () Kinix0 +h) = Kinix0) + Z S *'"m)(*o)> m=l where Aïn(x0) is the closest of a set of reference values, and where the derivatives Kinm\x0) are computed by (). The table of reference values is constructed by using () at an appropriate large value, and then using () repeatedly with negative values of A. As a check of the master routine, the results of () and () were compared for a range of values of x between 0. and and for n = 0,,,...,. The relative difference was less than x - in every case. The values of KiAx) were compared to those in [], and showed agreement to at least digits. An independent check is provided by (), which was evaluated by a -point Gauss-Hermite integration formula. Agreement to digits was obtained for =,,...,, and x > x, where x increases slowly with n. Typical values of xn are Xj = and x =. The quadrature formula becomes progressively less accurate as x decreases. As a result of the tests, we conclude that the master routine is accurate to at least digits.. Results. The details of the approximations are given in Tables -, in a format similar to that used in []. Tables - summarize the best approximations in the ^ Walsh arrays of the functions, and Tables - give the coefficients of selected approximations. issue. Tables are included in the microfiche section of this

5 0 j. m. blair, c. a. edwards and j. h. johnson Table Kijfxj «PÄ(x)/Qm(xj txtnx S(x ) [0,] Table Ki(x) «R(x) + x Un x P^ (x)/qm(x) [0,] Table Ki(x)» P Cxj/Qm(Xj + x Un x S(xz) [0,]

6 CHEBYSHEV APPROXIMATIONS FOR THE BICKLEY FUNCTIONS Table Ki(x) «R(x) x Jin x PJl(x)/Qm(x) [0,] Table Ki(x) «PJl(x)/QniCx) + x S.n x S(x) [0,]., Table Ki(x) «R(x) + x Jin x P (x )/Qm(x ) [0,]

7 J. M. BLAIR, C. A. EDWARDS AND J. H. JOHNSON Table Kij(x) «x'h ex Pi(z)/Qm(z), z = /x [0,], S II 0 0 Table Kijfx) «x* e"x PJl(i/x)/Qm(l/x) [,]

8 CHEBYSHEV APPROXIMATIONS FOR THE BICKLEY FUNCTIONS Table Ki(x) «x"* e"x Pjl(l/x)/Qm(l/x) [,] Table Ki(x) «x"" e"x PÄ(l/x)/Qm(l/x) [,]

9 j. m. blair, c. a. edwards and j. h. johnson Table Kig(x) «x'h e'x PjL(z)/Qm(z), z = /x [0,/], Table Kiq(x) «x'h e"x P (z)/q fz), z = /x [0,/],

10 CHEBYSHEV APPROXIMATIONS FOR THE BICKLEY FUNCTIONS Table Ki(x) «x'h e~x PJ,(z)/Qm(z), z = /x [0,/], The precision is defined as log max f(x)-rlm(x) f(x) where f(x) is the function being approximated and the maximum is taken over the appropriate interval. For the range 0 < x < the first auxiliary function Rlm(x) is ill-conditioned, and loses up to two significant digits by cancellation. To eliminate the cancellation the numerator and denominator were converted to minimal Newton form [], and the resulting coefficients rounded off by an algorithm similar to that described in []. For the range 0 < x < the first auxiliary function R x for Ki(x) is almost degenerate, and the rational function could only be found by Cody's method of artificial poles []. The approximations in Tables - were verified by comparing them with the master routine for 000 pseudorandom values of the argument in each interval.. Use of Coefficients. The coefficients may be used to construct a subroutine to compute Kin(x) for =,,...,, and for all values of x. For x <, approximations to Kix, Ki and Ki, obtained from Tables - and -, may be extended to fl, Ki,..., Kix0 by forward recursion in () with the loss of at most one digit of accuracy. For x >, Tables - give Ki&, Ki and Kixo, and these may be extended to Kin, Ki,..., Kix by backward recursion in () with less than one digit loss of accuracy.

11 J. M. BLAIR, C. A. EDWARDS AND J. H. JOHNSON Full range approximations to Kix(x) axe given in Tables - and, since it is the most commonly occurring function.. Acknowledgement. We wish to thank the referee for a suggestion which reduced the size of the tables. Atomic Energy of Canada Limited Chalk River Nuclear Laboratories Mathematics and Computation Branch Chalk River, Ontario KOJ J0, Canada Atomic Energy of Canada Limited Chalk River Nuclear Laboratories Mathematics and Computation Branch Chalk River, Ontario KOJ J0, Canada Statistics Canada Business Survey Methods Division Coats Building Tunney's Pasture Ottawa, Ontario KA 0T, Canada. W. G. BICKLEY, "Some solutions of the problem of forced convection," Philos. Mag., v. 0, S, pp. -.. W. G. BICKLEY & J. NAYLER, "A short table of the functions Ki (x), from n = to n =," Philos. Mag., v. 0,, pp. -.. Y. L. LUKE, Integrals of Bessel Functions, McGraw-Hill, New York,.. D. DAVIERWALLA, Tschebyscheff Rational Approximations to Bickley Functions, EIR Report No.,.. A. J. CLAYTON, "Some rational approximations for the Kij and JC functions," /. Nuclear Energy, Parts A/B, v.,, pp. -.. I. GARGANTINI & T. POMENTALE, "Rational Chebyshev approximations to the Bessel function integrals Ki (x)," Comm. ACM, v.,, pp I. GARGANTINI, "On the application of the process of equalization of maxima to obtain rational approximations to certain modified Bessel functions," Comm. ACM, v.,, pp. -.. K. MAKINO, "Some rational approximations for the Bickley functions Kin (n =,,,, )," Nukleonik, v.,, pp. -.. Y. L. LUKE, Mathematical Functions and Their Approximations, Academic Press, New York,.. J. H. JOHNSON & J. M. BLAIR, REMES-A FORTRAN Program to Calculate Rational Minimax Approximations to a Given Function, Report AECL-, Atomic Energy of Canada Limited, Chalk River, Ontario,.. B. S. BERGER & H. McALLISTER, "A table of the modified Bessel functions Kn(x) and In(x) to at least 0S for n = 0, and x =,,..., 0," Math. Comp., v., 0, p., RMT.. J. F. HART, et al., Computer Approximations, Wiley, New York,.. C. MESZTENYI & C. WITZGALL, "Stable evaluation of polynomials," /. Res. Nat. Bur. Standards Sect. B, v. B,, pp. -.. W. J. CODY, "A survey of practical rational and polynomial approximation of functions," SIAM Rev., v., 0, pp. 00-.