📄 tzerojp_cpp.txt
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/*************************************************************
* CALCULATE THE Kth ZERO OF THE DERIVATIVE OF BESSEL *
* FUNCTION OF ORDER N, J(N,X) *
* ---------------------------------------------------------- *
* SAMPLE RUN: *
* (Calculate the 10th zero of the derivative of J(1,X) ). *
* *
* X= 30.60192297 *
* *
* BESSJP(X)= -1.743397e-016 *
* *
* ---------------------------------------------------------- *
* Reference: From Numath Library By Tuan Dang Trong in *
* Fortran 77. *
* *
* C++ Release By J-P Moreau, Paris *
*************************************************************/
#include <stdio.h>
#include <math.h>
#define TRUE 1
double RES; int K, N;
double Sign(double a, double b) {
if (b<0) return (-fabs(a));
else return (fabs(a));
}
double BESSJ (int N, double X);
double BESSJ0(double X);
double BESSJ1(double X);
double BESSJP(int N, double X);
double ZEROJP(int N, int K) {
/*-------------------------------------------------------------------
! CALCULATE THE Kth ZERO OF THE DERIVATIVE OF BESSEL FUNCTION
! OF ORDER N, J(N,X)
!--------------------------------------------------------------------
! CALLING MODE:
! RES = ZEROJP(N,K)
!
! INPUTS:
! N ORDER OF BESSEL FUNCTION J (INTEGER >= 0) I*4
! K RANK OF ZERO (INTEGER > 0) I*4
! OUTPUT:
! ZEROJP R*8
! REFERENCE:
! ABRAMOWITZ M. & STEGUN IRENE A.
! HANDBOOK OF MATHEMATICAL FUNCTIONS
!-------------------------------------------------------------------*/
// Labels: e10, e20, e25
double B0,B1,B2,B3,B5,B7,T0,T1,T3,T5,T7,FN,FK;
double C1,C2,C3,C4,F1,F2,P,DP,P0,P1,Q0,Q1,TOL;
double PI, ZJP;
int IER,IT,NEV,NITMX;
bool IMPROV;
PI = 4.0*atan(1.0);
TOL=1e-7; NITMX=15;
C1=0.8086165; C2=0.072490; C3=0.05097; C4=0.0094;
IMPROV=TRUE;
FN = 1.0*N;
FK = 1.0*K;
if (K > 1) goto e10;
// if N = 0 ET K = 1
if (N == 0)
return 0.0;
// TCHEBYCHEV'S SERIES FOR K <= N
else {
F1 = pow(FN,1.0/3.0);
F2 = F1*F1*FN;
ZJP = FN+C1*F1+(C2/F1)-(C3/FN)+(C4/F2);
goto e20;
}
// MAC MAHON'S SERIES FOR K >> N
e10: B0 = (FK+0.5*FN-0.75)*PI;
B1 = 8.0*B0;
B2 = B1*B1;
B3 = 3.0*B1*B2;
B5 = 5.0*B3*B2;
B7 = 7.0*B5*B2;
T0 = 4.0*FN*FN;
T1 = T0 + 3.0;
T3 = 4.0*((7.0*T0+82.0)*T0-9.0);
T5 = 32.0*(((83.0*T0+2075.0)*T0-3039.0)*T0+3537.0);
T7 = 64.0*((((6949.0*T0+296492.0)*T0-1248002.0)*T0
+7414380.0)*T0-5853627.0);
ZJP = B0-(T1/B1)-(T3/B3)-(T5/B5)-(T7/B7);
e20: if (IMPROV) {
// IMPROVE SOLUTION BY SECANT METHOD WHEN K > N
// AND IMPROV = TRUE
P0 = 0.9*ZJP;
P1 = ZJP;
IER = 0;
NEV = 2;
Q0 = BESSJP(N,P0);
Q1 = BESSJP(N,P1);
for (IT=1; IT<=NITMX; IT++) {
P = P1-Q1*(P1-P0)/(Q1-Q0);
DP = P-P1;
if (IT == 1) goto e25;
if (fabs(DP) < TOL) return P;
e25: NEV = NEV+1;
P0 = P1;
Q0 = Q1;
P1 = P ;
Q1 = BESSJP(N,P1);
}
IER = 1;
printf(" ** ZEROJP ** NITMX EXCEEDED.\n");
return ZJP;
}
return ZJP;
}
double BESSJP(int N, double X) {
/* ----------------------------------------------------------------------
! NAME : BESSJP
! DATE : 06/01/1982
! IV : 1
! IE : 1
! AUTHOR: DANG TRONG TUAN
! ......................................................................
!
! FIRST DERIVATIVE OF FIRST KIND BESSEL FUNCTION OF ORDER N, FOR REAL X
!
! MODULE BESSJP
! ......................................................................
!
! THIS SUBROUTINE CALCULATES THE FIRST DERIVATIVE OF FIRST KIND BESSEL
! FUNCTION OF ORDER N, FOR REAL X.
!
! ......................................................................
!
! I VARIABLE DIMENSION/TYPE DESCRIPTION (INPUTS)
! N I*4 ORDER OF FUNCTION
! X R*8 ABSCISSA OF FUNCTION BESSJP(N,X)
!
! O VARIABLE,DIMENSION/TYPE DESCRIPTION (OUTPUT)
!
! BESSJP R*8 FUNCTION EVALUATION AT X
!.......................................................................
! CALLED SUBROUTINE
!
! BESSJ FIRST KIND BESSEL FUNCTION
!
! ----------------------------------------------------------------------*/
if (N == 0)
return (-BESSJ(1,X));
else if (X == 0.0)
return 0.0;
else
return (BESSJ(N-1,X)-(1.0*N/X)*BESSJ(N,X));
}
double BESSJ(int N, double X) {
/*----------------------------------------------------------------------
! THIS SUBROUTINE CALCULATES THE 1ST KIND BESSEL FUNCTION OF ORDER
! N, INTEGER FOR ANY REAL X. THE CLASSICAL RECURRENCE FORMULA IS USED
! HERE, WHEN X IS > N, FOR X < N, THE MILLER'S ALGORITHM ALLOWS
! AVOIDING OVERFLOWS.
! REFERENCE:
! C.W.CLENSHAW, CHEBYSHEV SERIES FOR MATHEMATICAL FUNCTIONS,
! MATHEMATICAL TABLES, VOL.5, 1962.
!---------------------------------------------------------------------*/
int IACC=40; double BIGNO = 1e10, BIGNI = 1e-10;
double TOX,BJM,BJ,BJP,BSJ,SUM;
int J,JSUM,M;
if (N == 0) return BESSJ0(X);
if (N == 1) return BESSJ1(X);
if (X == 0.0) return 0.0;
TOX = 2.0/X;
if (X > 1.0*N) {
BJM = BESSJ0(X);
BJ = BESSJ1(X);
for (J=1; J<N; J++) {
BJP = J*TOX*BJ-BJM;
BJM = BJ;
BJ = BJP;
}
return BJ;
}
else {
M = (int) floor(2*((N+floor(sqrt(IACC*N)))/2.0));
BSJ = 0.0;
JSUM = 0;
SUM = 0.0;
BJP = 0.0;
BJ = 1.0;
for (J=M; J>0; J--) {
BJM = J*TOX*BJ-BJP;
BJP = BJ;
BJ = BJM;
if (fabs(BJ) > BIGNO) {
BJ = BJ*BIGNI;
BJP = BJP*BIGNI;
BSJ = BSJ*BIGNI;
SUM = SUM*BIGNI;
}
if (JSUM != 0) SUM = SUM + BJ;
JSUM = 1 - JSUM;
if (J == N) BSJ = BJP;
}
SUM = 2.0*SUM-BJ;
return (BSJ/SUM);
}
}
double BESSJ0(double X) {
/*---------------------------------------------------------------------
! THIS SUBROUTINE CALCULATES THE 1ST KIND BESSEL FUNCTION OF ORDER
! ZERO FOR ANY REAL X. THE CHEBYSHEV'S POLYNOMIAL SERIES IS USED
! FOR 0<X<8 AND 0<8/X<1.
! REFERENCE:
! M.ABRAMOWITZ,I.A.STEGUN, HANDBOOK OF MATHEMATICAL FUNCTIONS, 1965.
! C.W.CLENSHAW, NATIONAL PHYSICAL LABORATORY MATHEMATICAL TABLES,
! VOL.5, 1962.
!---------------------------------------------------------------------*/
// Labels: e1, e2
double AX,FR,FS,Z,FP,FQ,XX;
double Y,P1,P2,P3,P4,P5,R1,R2,R3,R4,R5,R6;
double Q1,Q2,Q3,Q4,Q5,S1,S2,S3,S4,S5,S6;
P1=1.0; P2=-0.1098628627e-02; P3=0.2734510407e-04;
P4=-0.2073370639e-05; P5=0.2093887211e-06;
Q1=-0.1562499995e-01; Q2=0.1430488765e-03; Q3=-0.6911147651e-05;
Q4=0.7621095161e-06; Q5=-0.9349451520e-07;
R1=57568490574.0; R2=-13362590354.0; R3=651619640.7;
R4=-11214424.18; R5=77392.33017; R6=-184.9052456;
S1=57568490411.0; S2=1029532985.0; S3=9494680.718;
S4=59272.64853; S5=267.8532712; S6=1.0;
if (X == 0.0) goto e1;
AX = fabs(X);
if (AX < 8.0) {
Y = X*X;
FR = R1+Y*(R2+Y*(R3+Y*(R4+Y*(R5+Y*R6))));
FS = S1+Y*(S2+Y*(S3+Y*(S4+Y*(S5+Y*S6))));
return (FR/FS);
}
else {
Z = 8.0/AX;
Y = Z*Z;
XX = AX-0.785398164;
FP = P1+Y*(P2+Y*(P3+Y*(P4+Y*P5)));
FQ = Q1+Y*(Q2+Y*(Q3+Y*(Q4+Y*Q5)));
return (sqrt(0.636619772/AX)*(FP*cos(XX)-Z*FQ*sin(XX)));
}
e1: return (1.0);
}
double BESSJ1(double X) {
/*---------------------------------------------------------------------
! THIS SUBROUTINE CALCULATES THE 1ST KIND BESSEL FUNCTION OF ORDER
! ONE FOR ANY REAL X. THE CHEBYSHEV'S POLYNOMIAL SERIES IS USED
! FOR 0<X<8 AND 0<8/X<1.
! REFERENCE:
! M.ABRAMOWITZ,I.A.STEGUN, HANDBOOK OF MATHEMATICAL FUNCTIONS, 1965.
! C.W.CLENSHAW, NATIONAL PHYSICAL LABORATORY MATHEMATICAL TABLES,
! VOL.5, 1962.
!---------------------------------------------------------------------*/
double AX,FR,FS,Z,FP,FQ,XX;
double Y,P1,P2,P3,P4,P5,P6,R1,R2,R3,R4,R5,R6;
double Q1,Q2,Q3,Q4,Q5,S1,S2,S3,S4,S5,S6;
P1=1.0; P2=0.183105e-02; P3=-0.3516396496e-04;
P4=0.2457520174e-05; P5=-0.240337019e-06; P6=0.636619772;
Q1=0.04687499995; Q2=-0.2002690873e-03; Q3=0.8449199096e-05;
Q4=-0.88228987e-06; Q5=0.105787412e-06;
R1=72362614232.0; R2=-7895059235.0; R3=242396853.1;
R4=-2972611.439; R5=15704.48260; R6=-30.16036606;
S1=144725228442.0; S2=2300535178.0; S3=18583304.74;
S4=99447.43394; S5=376.9991397; S6=1.0;
AX = fabs(X);
if (AX < 8.0) {
Y = X*X;
FR = R1+Y*(R2+Y*(R3+Y*(R4+Y*(R5+Y*R6))));
FS = S1+Y*(S2+Y*(S3+Y*(S4+Y*(S5+Y*S6))));
return (X*(FR/FS));
}
else {
Z = 8.0/AX;
Y = Z*Z;
XX = AX-2.35619491;
FP = P1+Y*(P2+Y*(P3+Y*(P4+Y*P5)));
FQ = Q1+Y*(Q2+Y*(Q3+Y*(Q4+Y*Q5)));
return (sqrt(P6/AX)*(cos(XX)*FP-Z*sin(XX)*FQ)*Sign(S6,X));
}
}
void main() {
N = 1;
K = 10;
RES = ZEROJP(N,K);
printf("\n X=%12.8f\n", RES);
printf("\n BESSJP(X)= %e\n\n", BESSJP(N,RES));
}
// end of file tzerojp.cpp
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