'********************************************************************* '* Compute the zeros of Bessel functions Jn(x), Yn(x), and their * '* derivatives using subroutine JYZO * '* ----------------------------------------------------------------- * '* SAMPLE RUN: * '* (Compute 10 zeroes for n=1). * '* * '* Please enter order n and number of zeroes: 1,10 * '* * '* Zeros of Bessel functions Jn(x), Yn(x) and their derivatives * '* ( n = 1 ) * '* m jnm j'nm ynm y'nm * '* ----------------------------------------------------------- * '* 1 3.8317060 1.8411838 2.1971413 3.6830229 * '* 2 7.0155867 5.3314428 5.4296810 6.9415000 * '* 3 10.1734681 8.5363164 8.5960059 10.1234047 * '* 4 13.3236919 11.7060049 11.7491548 13.2857582 * '* 5 16.4706301 14.8635886 14.8974421 16.4400580 * '* 6 19.6158585 18.0155279 18.0434023 19.5902418 * '* 7 22.7600844 21.1643699 21.1880689 22.7380347 * '* 8 25.9036721 24.3113269 24.3319426 25.8843146 * '* 9 29.0468285 27.4570506 27.4752950 29.0295758 * '* 10 32.1896799 30.6019230 30.6182865 32.1741182 * '* ----------------------------------------------------------- * '* * '* ----------------------------------------------------------------- * '* Ref.: www.esrf.fr/computing/expg/libraries/smf/PROGRAMS/MJYZO.FOR * '* * '* Basic Release 1.0 By J-P Moreau, Paris. * '* (www.jpmoreau.fr) * '********************************************************************* 'PROGRAM MJYZO DEFDBL A-H, O-Z DEFINT I-N DIM RJ0(30), RJ1(30), RY0(30), RY1(30) DIM BJ(31), BY(31) CLS PRINT INPUT " Please enter order n and number of zeroes: ", N, NT PRINT GOSUB 1000 'call JYZO(N,NT,RJ0,RJ1,RY0,RY1) FF$ = "######.#######" FF1$ = "####" PRINT " Zeros of Bessel functions Jn(x), Yn(x), and their derivatives" PRINT " ( n = "; N; " )" PRINT " m jnm j'nm ynm y'nm" PRINT " -----------------------------------------------------------" FOR M = 1 TO NT PRINT USING FF1$; M; PRINT USING FF$; RJ0(M); PRINT USING FF$; RJ1(M); PRINT USING FF$; RY0(M); PRINT USING FF$; RY1(M) NEXT M PRINT " -----------------------------------------------------------" END 1000 'Subroutine JYZO(N,NT,RJ0,RJ1,RY0,RY1) ' ======================================================== ' Purpose: Compute the zeros of Bessel functions Jn(x), ' Yn(x), and their derivatives ' Input : n --- Order of Bessel functions (0 to 100) ' NT --- Number of zeros (roots) ' Output: RJ0(L) --- L-th zero of Jn(x), L=1,2,...,NT ' RJ1(L) --- L-th zero of Jn'(x), L=1,2,...,NT ' RY0(L) --- L-th zero of Yn(x), L=1,2,...,NT ' RY1(L) --- L-th zero of Yn'(x), L=1,2,...,NT ' Routine called: JYNDD for computing Jn(x), Yn(x), and ' their first and second derivatives ' ======================================================== ' Labels: 10, 15, 20, 25 IF N <= 20 THEN X = 2.82141# + 1.15859# * N ELSE X = N + 1.85576# * N ^ .333333# + 1.03315# / N ^ .333333# END IF L = 0 10 X0 = X GOSUB 2000 'call JYNDD(N,X,BJN,DJN,FJN,BYN,DYN,FYN) X = X - BJN / DJN IF ABS(X - X0) > .000000001# THEN GOTO 10 L = L + 1 RJ0(L) = X X = X + 3.1416# + (9.719999999999999D-02 + .0679# * N - .000354# * N * N) / L IF L < NT THEN GOTO 10 IF N <= 20 THEN X = .961587# + 1.07703# * N ELSE X = N + .8086100000000001# * N ^ .333333# + .07249# / N ^ .333333# END IF IF N = 0 THEN X = 3.8317 L = 0 15 X0 = X GOSUB 2000 'call JYNDD(N,X,BJN,DJN,FJN,BYN,DYN,FYN) X = X - DJN / FJN IF ABS(X - X0) > .000000001# THEN GOTO 15 L = L + 1 RJ1(L) = X X = X + 3.1416# + (.4955# + .0915# * N - .000435# * N * N) / L IF L < NT THEN GOTO 15 IF N <= 20 THEN X = 1.19477# + 1.08933# * N ELSE X = N + .93158# * N ^ .333333# + .26035# / N ^ .333333# END IF L = 0 20 X0 = X GOSUB 2000 'call JYNDD(N,X,BJN,DJN,FJN,BYN,DYN,FYN) X = X - BYN / DYN IF ABS(X - X0) > .000000001# THEN GOTO 20 L = L + 1 RY0(L) = X X = X + 3.1416# + (.312# + .0852# * N - .000403# * N * N) / L IF L < NT THEN GOTO 20 IF N <= 20 THEN X = 2.67257# + 1.16099# * N ELSE X = N + 1.8211# * N ^ .333333# + .94001# / N ^ .333333# END IF L = 0 25 X0 = X GOSUB 2000 'call JYNDD(N,X,BJN,DJN,FJN,BYN,DYN,FYN) X = X - DYN / FYN IF ABS(X - X0) > .000000001# THEN GOTO 25 L = L + 1 RY1(L) = X X = X + 3.1416# + (.197# + .0643# * N - .000286# * N * N) / L IF L < NT THEN GOTO 25 RETURN 2000 'Subroutine JYNDD(N, X, BJN, DJN, FJN, BYN, DYN, FYN) ' ========================================================= ' Purpose: Compute Bessel functions Jn(x) and Yn(x), and ' their first and second derivatives ' Input: x --- Argument of Jn(x) and Yn(x) ( x > 0 ) ' n --- Order of Jn(x) and Yn(x) ' Output: BJN --- Jn(x) ' DJN --- Jn'(x) ' FJN --- Jn"(x) ' BYN --- Yn(x) ' DYN --- Yn'(x) ' FYN --- Yn"(x) ' ========================================================= ' Label: 50 FOR N1 = 1 TO 900 XN1 = 1# * N1 MT = INT(.5# * (LOG(6.28# * XN1) / LOG(10#)) - XN1 * LOG(1.36# * ABS(X) / XN1) / LOG(10#) + .5#) IF MT > 20 THEN GOTO 50 NEXT N1 50 M = N1 BS = 0# F0 = 0# F1 = 1D-35 SU = 0# FOR K = M TO 0 STEP -1 F = 2# * (K + 1#) * F1 / X - F0 IF K <= N + 1 THEN BJ(K + 1) = F IF INT(K / 2) = K / 2 THEN BS = BS + 2# * F IF K <> 0 THEN IF INT(K / 4) = K / 4 THEN SU = SU + F / K ELSE SU = SU - F / K END IF END IF END IF F0 = F1 F1 = F NEXT K FOR K = 0 TO N + 1 BJ(K + 1) = BJ(K + 1) / (BS - F) NEXT K BJN = BJ(N + 1) EC = .5772156649015329# E0 = .3183098861837907# S1 = 2# * E0 * (LOG(X / 2#) + EC) * BJ(1) F0 = S1 - 8# * E0 * SU / (BS - F) F1 = (BJ(2) * F0 - 2# * E0 / X) / BJ(1) BY(1) = F0 BY(2) = F1 FOR K = 2 TO N + 1 F = 2# * (K - 1#) * F1 / X - F0 BY(K + 1) = F F0 = F1 F1 = F NEXT K BYN = BY(N + 1) DJN = -BJ(N + 2) + N * BJ(N + 1) / X DYN = -BY(N + 2) + N * BY(N + 1) / X FJN = (N * N / (X * X) - 1#) * BJN - DJN / X FYN = (N * N / (X * X) - 1#) * BYN - DYN / X RETURN ' End of file mjyzo.bas