' -------------------------------------------------------------------------- ' Program to calculate the first zeroes (root abscissas) of the first ' kind Bessel function of integer order N using the subroutine ROOTJ. ' -------------------------------------------------------------------------- ' SAMPLE RUN: ' ' (Calculate the first 10 zeroes of 1st kind Bessel function of order 2). ' ' Zeroes of Bessel Function of order: 2 ' ' Number of calculated zeroes: 10 ' ' Table of root abcissas (5 items per line) ' 5.135622 8.417244 11.619841 14.795952 17.959819 ' 21.116997 24.270112 27.420574 30.569204 33.716520 ' ' Table of error codes (5 items per line) ' 0 0 0 0 0 ' 0 0 0 0 0 ' ' -------------------------------------------------------------------------- ' Reference: From Numath Library By Tuan Dang Trong in Fortran 77 ' [BIBLI 18]. ' ' Basic Release 1.0 By J-P Moreau, Paris ' (www.jpmoreau.fr) ' -------------------------------------------------------------------------- 'PROGRAM TROOTJ DEFDBL A-H, O-Z DEFINT I-N DIM JZERO(10) AS DOUBLE DIM IER(10) N = 2 NK = 10 CLS GOSUB 1000 'CALL ROOTJ(N,NR,JZERO,IER) PRINT PRINT " Zeroes of Bessel Function of order: "; N PRINT PRINT " Number of calculated zeroes: "; NK PRINT PRINT " Table of root abcissas (5 items per line)" J = 1 FOR i = 1 TO NK PRINT USING "####.######"; JZERO(i); J = J + 1 IF INT(J / 6) = J / 6 THEN PRINT : J = 1 END IF NEXT i PRINT PRINT " Table of error codes (5 items per line)" J = 1 FOR i = 1 TO NK PRINT " "; IER(i); J = J + 1 IF INT(J / 6) = J / 6 THEN PRINT : J = 1 END IF NEXT i PRINT END 1000 'SUBROUTINE ROOTJ(N,NK,JZERO,IER) ' ---------------------------------------------------------------------- ' CALCULATE THE FIRST NK ZEROES OF BESSEL FUNCTION J(N,X) ' ' INPUTS: ' N ORDER OF FUNCTION J (INTEGER >= 0) I*4 ' NK NUMBER OF FIRST ZEROES (INTEGER > 0) I*4 ' OUTPUTS: ' JZERO(NK) TABLE OF FIRST ZEROES (ABCISSAS) R*8 ' IER(NK) TABLE OF ERROR CODES (MUST BE ZEROES) I*4 ' ' REFERENCE : ' ABRAMOWITZ M. & STEGUN IRENE A. ' HANDBOOK OF MATHEMATICAL FUNCTIONS ' ---------------------------------------------------------------------- PI = 4# * ATN(1#) TOL = 1E-08 NITMX = 10 C1 = 1.8557571# C2 = 1.03315# C3 = .00397# C4 = 9.080000000000001D-02 C5 = .043# XFN = 1# * N ' FIRST ZERO IF N = 0 THEN ZEROJ = C1 + C2 - C3 - C4 + C5 GOSUB 2000 'CALL SECANT(N,NITMX,TOL,ZEROJ,IERROR) IER(1) = IERROR JZERO(1) = ZEROJ ELSE F1 = XFN ^ (1# / 3#) F2 = F1 * F1 * XFN F3 = F1 * XFN * XFN ZEROJ = XFN + C1 * F1 + (C2 / F1) - (C3 / XFN) - (C4 / F2) + (C5 / F3) GOSUB 2000 'CALL SECANT(N,NITMX,TOL,ZEROJ,IERROR) IER(1) = IERROR JZERO(1) = ZEROJ END IF T0 = 4# * XFN * XFN T1 = T0 - 1# T3 = 4# * T1 * (7# * T0 - 31#) T5 = 32# * T1 * ((83# * T0 - 982#) * T0 + 3779#) T7 = 64# * T1 * (((6949# * T0 - 153855#) * T0 + 1585743#) * T0 - 6277237#) ' OTHER ZEROES FOR K = 2 TO NK JZERO(K) = 0# FK = 1# * K ' MAC MAHON'S SERIES FOR K>>N B0 = (FK + .5# * XFN - .25#) * PI B1 = 8# * B0 B2 = B1 * B1 B3 = 3# * B1 * B2 B5 = 5# * B3 * B2 B7 = 7# * B5 * B2 ZEROJ = B0 - (T1 / B1) - (T3 / B3) - (T5 / B5) - (T7 / B7) 'ERRJ=ABS(BESSJ(N,ZEROJ)) X = ZEROJ: GOSUB 500: ERRJ = BESSJ ' IMPROVE SOLUTION USING SUBROUTINE SECANT IF ERRJ > TOL THEN GOSUB 2000 'CALL SECANT(N,NITMX,TOL,ZEROJ,IERROR) JZERO(K) = ZEROJ IER(K) = IERROR NEXT K RETURN ' ---------------------------------------------------------------------- 2000 'SUBROUTINE SECANT(N,NITMX,TOL,ZEROJ,IERROR) DIM C(2) C(1) = .95# C(2) = .999# NTRY = 1 IERROR = 0 5 P0 = C(NTRY) * ZEROJ XP1 = ZEROJ 'Caution - P1, Q1 modified by 600 and 700 NEV = 2 'Q0 = BESSJ(N,P0) X = P0: GOSUB 500: Q0 = BESSJ 'Q1 = BESSJ(N,P1) X = XP1: GOSUB 500: XQ1 = BESSJ FOR IT = 1 TO NITMX IF XQ1 = Q0 THEN GOTO 15 P = XP1 - XQ1 * (XP1 - P0) / (XQ1 - Q0) DP = P - XP1 IF IT = 1 THEN GOTO 10 IF ABS(DP) < TOL THEN GOTO 20 'NORMAL EXIT 10 NEV = NEV + 1 P0 = XP1 Q0 = XQ1 XP1 = P 'Q1 = BESSJ(N,P1) X = XP1: GOSUB 500: XQ1 = BESSJ NEXT IT 15 NTRY = NTRY + 1 IF NTRY <= 2 THEN GOTO 5 IERROR = NTRY 20 ZEROJ = P RETURN ' ---------------------------------------------------------------------- 500 'FUNCTION BESSJ (N,X) ' THIS FUNCTION RETURNS THE VALUE OF THE FIRST KIND BESSEL FUNCTION ' OF ORDER N, INTEGER FOR ANY REAL X. WE USE HERE THE CLASSICAL ' RECURRENT FORMULA, WHEN X > N. FOR X < N, THE MILLER'S ALGORITHM ' IS USED TO AVOID OVERFLOWS. ' REFERENCE : ' C.W.CLENSHAW, CHEBYSHEV SERIES FOR MATHEMATICAL FUNCTIONS, ' MATHEMATICAL TABLES, VOL.5, 1962. IACC = 40 BIGNO = 1E+10 BIGNI = 1E-10 IF N = 0 THEN 'BESSJ = BESSJ0(X) GOSUB 600: BESSJ = BESSJ0 RETURN END IF IF N = 1 THEN 'BESSJ = BESSJ1(X) GOSUB 700: BESSJ = BESSJ1 RETURN END IF IF X = 0# THEN BESSJ = 0# RETURN END IF TOX = 2# / X IF X > 1# * N THEN 'BJM = BESSJ0(X) GOSUB 600: BJM = BESSJ0 'BJ = BESSJ1(X) GOSUB 700: BJ = BESSJ1 FOR J = 1 TO N - 1 BJP = J * TOX * BJ - BJM BJM = BJ BJ = BJP NEXT J BESSJ = BJ ELSE M = 2 * ((N + INT(SQR(1# * IACC * N))) / 2) BESSJ = 0# JSUM = 0 SUM = 0# BJP = 0# BJ = 1# FOR J = M TO 1 STEP -1 BJM = J * TOX * BJ - BJP BJP = BJ BJ = BJM IF ABS(BJ) > BIGNO THEN BJ = BJ * BIGNI BJP = BJP * BIGNI BESSJ = BESSJ * BIGNI SUM = SUM * BIGNI END IF IF JSUM <> 0 THEN SUM = SUM + BJ JSUM = 1 - JSUM IF J = N THEN BESSJ = BJP NEXT J SUM = 2# * SUM - BJ BESSJ = BESSJ / SUM END IF RETURN ' ---------------------------------------------------------------------- 600 'FUNCTION BESSJ0 (X) ' THIS FUNCTION RETURNS THE VALUE OF THE FIRST KIND BESSEL FUNCTION ' OF ORDER 0 FOR ANY REAL X. WE USE HERE THE POLYNOMIAL APPROXIMATION ' BY SERIES OF CHEBYSHEV POLYNOMIALS FOR 0= 0# THEN SIGN = ABS(S6) ELSE SIGN = -ABS(S6) END IF BESSJ1 = SQR(P6 / AX) * (COS(XX) * FP - Z * SIN(XX) * FQ) * SIGN END IF RETURN ' end of file trootj.bas