/******************************************************************** * 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 * * * * C++ Release 1.0 By J-P Moreau, Paris. * * (www.jpmoreau.fr) * ********************************************************************/ #include #include #define NMAX 101 double RJ0[NMAX], RJ1[NMAX], RY0[NMAX], RY1[NMAX]; int M, N, NT; char res[2]; void JYNDD(int N, double X, double *BJN, double *DJN, double *FJN, double *BYN, double *DYN, double *FYN); void JYZO(int N, int NT, double *RJ0, double *RJ1, double *RY0, double *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: e10, e15, e20, e25 double BJN,DJN,FJN,BYN,DYN,FYN; double X, X0; int L; if (N <= 20) X = 2.82141+1.15859*N; else X = N + pow(1.85576*N,0.33333) + pow(1.03315/N,0.33333); L=0; e10: X0=X; JYNDD(N,X,&BJN,&DJN,&FJN,&BYN,&DYN,&FYN); X -= BJN/DJN; if (fabs(X-X0) > 1e-9) goto e10; L++; RJ0[L]=X; X=X+3.1416+(0.0972+0.0679*N-0.000354*N*N)/L; if (L < NT) goto e10; if (N <= 20) X=0.961587+1.07703*N; else X=N + pow(0.80861*N,0.33333) + pow(0.07249/N,0.33333); if (N == 0) X=3.8317; L=0; e15: X0=X; JYNDD(N,X,&BJN,&DJN,&FJN,&BYN,&DYN,&FYN); X=X-DJN/FJN; if (fabs(X-X0) > 1e-9) goto e15; L++; RJ1[L]=X; X=X+3.1416+(0.4955+0.0915*N-0.000435*N*N)/L; if (L < NT) goto e15; if (N <= 20) X=1.19477+1.08933*N; else X=N + pow(0.93158*N,0.33333) + pow(0.26035/N,0.33333); L=0; e20: X0=X; JYNDD(N,X,&BJN,&DJN,&FJN,&BYN,&DYN,&FYN); X=X-BYN/DYN; if (fabs(X-X0) > 1e-9) goto e20; L++; RY0[L]=X; X=X+3.1416+(0.312+0.0852*N-0.000403*N*N)/L; if (L < NT) goto e20; if (N <= 20) X=2.67257+1.16099*N; else X=N + pow(1.8211*N,0.33333) + pow(0.94001/N,0.33333); L=0; e25: X0=X; JYNDD(N,X,&BJN,&DJN,&FJN,&BYN,&DYN,&FYN); X=X-DYN/FYN; if (fabs(X-X0) > 1e-9) goto e25; L++; RY1[L]=X; X=X+3.1416+(0.197+0.0643*N-0.000286*N*N)/L; if (L < NT) goto e25; } void JYNDD(int N, double X, double *BJN, double *DJN, double *FJN, double *BYN, double *DYN, double *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 e15 double BJ[NMAX], BY[NMAX]; int K,M, MT, NT; double F,F0,F1,BS,SU; double E0,EC,S1; for (NT=1; NT<=900; NT++) { MT=(int) floor(0.5*log10(6.28*NT)-NT*log10(1.36*fabs(X)/NT)); if (MT > 20) goto e15; } e15: M=NT; BS=0.0; F0=0.0; F1=1e-35; SU=0.0; for (K=M; K>=0; K--) { F=2.0*(K+1.0)*F1/X-F0; if (K <= N+1) BJ[K+1]=F; if ((K % 2) == 0) { BS=BS+2.0*F; if (K != 0) if (((K / 2) % 2) == 0) SU += F/K; else SU -= F/K; } F0=F1; F1=F; } for (K=0; K<=N+1; K++) BJ[K+1] /= BS-F; *BJN=BJ[N+1]; EC=0.5772156649015329; E0=0.3183098861837907; S1=2.0*E0*(log(X/2.0)+EC)*BJ[1]; F0=S1-8.0*E0*SU/(BS-F); F1=(BJ[2]*F0-2.0*E0/X)/BJ[1]; BY[1]=F0; BY[2]=F1; for (K=2; K<=N+1; K++) { F=2.0*(K-1.0)*F1/X-F0; BY[K+1]=F; F0=F1; F1=F; } *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)-1.0)*(*BJN)-*DJN/X; *FYN=(N*N/(X*X)-1.0)*(*BYN)-*DYN/X; } void main() { printf("\n Please enter order n and number of zeroes: "); scanf("%d %d", &N, &NT); JYZO(N,NT,RJ0,RJ1,RY0,RY1); printf("\n Zeros of Bessel functions Jn(x), Yn(x), and their derivatives"); printf("\n ( n = %d )", N); printf("\n m jnm j'nm ynm y'nm"); printf("\n -----------------------------------------------------------\n"); for (M=1; M<=NT; M++) printf(" %3d%14.7f%14.7f%14.7f%14.7f\n", M, RJ0[M], RJ1[M], RY0[M], RY1[M]); printf(" -----------------------------------------------------------\n\n"); } // End of file mjyzo.cpp