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-/* jv.c
- *
- * Bessel function of noninteger order
- *
- *
- *
- * SYNOPSIS:
- *
- * double v, x, y, jv();
- *
- * y = jv( v, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of order v of the argument,
- * where v is real. Negative x is allowed if v is an integer.
- *
- * Several expansions are included: the ascending power
- * series, the Hankel expansion, and two transitional
- * expansions for large v. If v is not too large, it
- * is reduced by recurrence to a region of best accuracy.
- * The transitional expansions give 12D accuracy for v > 500.
- *
- *
- *
- * ACCURACY:
- * Results for integer v are indicated by *, where x and v
- * both vary from -125 to +125. Otherwise,
- * x ranges from 0 to 125, v ranges as indicated by "domain."
- * Error criterion is absolute, except relative when |jv()| > 1.
- *
- * arithmetic v domain x domain # trials peak rms
- * IEEE 0,125 0,125 100000 4.6e-15 2.2e-16
- * IEEE -125,0 0,125 40000 5.4e-11 3.7e-13
- * IEEE 0,500 0,500 20000 4.4e-15 4.0e-16
- * Integer v:
- * IEEE -125,125 -125,125 50000 3.5e-15* 1.9e-16*
- *
- */
-
-
-/*
-Cephes Math Library Release 2.8: June, 2000
-Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
-*/
-
-
-#include <math.h>
-#define DEBUG 0
-
-#ifdef DEC
-#define MAXGAM 34.84425627277176174
-#else
-#define MAXGAM 171.624376956302725
-#endif
-
-#ifdef ANSIPROT
-extern int airy ( double, double *, double *, double *, double * );
-extern double fabs ( double );
-extern double floor ( double );
-extern double frexp ( double, int * );
-extern double polevl ( double, void *, int );
-extern double j0 ( double );
-extern double j1 ( double );
-extern double sqrt ( double );
-extern double cbrt ( double );
-extern double exp ( double );
-extern double log ( double );
-extern double sin ( double );
-extern double cos ( double );
-extern double acos ( double );
-extern double pow ( double, double );
-extern double gamma ( double );
-extern double lgam ( double );
-static double recur(double *, double, double *, int);
-static double jvs(double, double);
-static double hankel(double, double);
-static double jnx(double, double);
-static double jnt(double, double);
-#else
-int airy();
-double fabs(), floor(), frexp(), polevl(), j0(), j1(), sqrt(), cbrt();
-double exp(), log(), sin(), cos(), acos(), pow(), gamma(), lgam();
-static double recur(), jvs(), hankel(), jnx(), jnt();
-#endif
-
-extern double MAXNUM, MACHEP, MINLOG, MAXLOG;
-#define BIG 1.44115188075855872E+17
-
-double jv( n, x )
-double n, x;
-{
-double k, q, t, y, an;
-int i, sign, nint;
-
-nint = 0; /* Flag for integer n */
-sign = 1; /* Flag for sign inversion */
-an = fabs( n );
-y = floor( an );
-if( y == an )
- {
- nint = 1;
- i = an - 16384.0 * floor( an/16384.0 );
- if( n < 0.0 )
- {
- if( i & 1 )
- sign = -sign;
- n = an;
- }
- if( x < 0.0 )
- {
- if( i & 1 )
- sign = -sign;
- x = -x;
- }
- if( n == 0.0 )
- return( j0(x) );
- if( n == 1.0 )
- return( sign * j1(x) );
- }
-
-if( (x < 0.0) && (y != an) )
- {
- mtherr( "Jv", DOMAIN );
- y = 0.0;
- goto done;
- }
-
-y = fabs(x);
-
-if( y < MACHEP )
- goto underf;
-
-k = 3.6 * sqrt(y);
-t = 3.6 * sqrt(an);
-if( (y < t) && (an > 21.0) )
- return( sign * jvs(n,x) );
-if( (an < k) && (y > 21.0) )
- return( sign * hankel(n,x) );
-
-if( an < 500.0 )
- {
-/* Note: if x is too large, the continued
- * fraction will fail; but then the
- * Hankel expansion can be used.
- */
- if( nint != 0 )
- {
- k = 0.0;
- q = recur( &n, x, &k, 1 );
- if( k == 0.0 )
- {
- y = j0(x)/q;
- goto done;
- }
- if( k == 1.0 )
- {
- y = j1(x)/q;
- goto done;
- }
- }
-
-if( an > 2.0 * y )
- goto rlarger;
-
- if( (n >= 0.0) && (n < 20.0)
- && (y > 6.0) && (y < 20.0) )
- {
-/* Recur backwards from a larger value of n
- */
-rlarger:
- k = n;
-
- y = y + an + 1.0;
- if( y < 30.0 )
- y = 30.0;
- y = n + floor(y-n);
- q = recur( &y, x, &k, 0 );
- y = jvs(y,x) * q;
- goto done;
- }
-
- if( k <= 30.0 )
- {
- k = 2.0;
- }
- else if( k < 90.0 )
- {
- k = (3*k)/4;
- }
- if( an > (k + 3.0) )
- {
- if( n < 0.0 )
- k = -k;
- q = n - floor(n);
- k = floor(k) + q;
- if( n > 0.0 )
- q = recur( &n, x, &k, 1 );
- else
- {
- t = k;
- k = n;
- q = recur( &t, x, &k, 1 );
- k = t;
- }
- if( q == 0.0 )
- {
-underf:
- y = 0.0;
- goto done;
- }
- }
- else
- {
- k = n;
- q = 1.0;
- }
-
-/* boundary between convergence of
- * power series and Hankel expansion
- */
- y = fabs(k);
- if( y < 26.0 )
- t = (0.0083*y + 0.09)*y + 12.9;
- else
- t = 0.9 * y;
-
- if( x > t )
- y = hankel(k,x);
- else
- y = jvs(k,x);
-#if DEBUG
-printf( "y = %.16e, recur q = %.16e\n", y, q );
-#endif
- if( n > 0.0 )
- y /= q;
- else
- y *= q;
- }
-
-else
- {
-/* For large n, use the uniform expansion
- * or the transitional expansion.
- * But if x is of the order of n**2,
- * these may blow up, whereas the
- * Hankel expansion will then work.
- */
- if( n < 0.0 )
- {
- mtherr( "Jv", TLOSS );
- y = 0.0;
- goto done;
- }
- t = x/n;
- t /= n;
- if( t > 0.3 )
- y = hankel(n,x);
- else
- y = jnx(n,x);
- }
-
-done: return( sign * y);
-}
-
-/* Reduce the order by backward recurrence.
- * AMS55 #9.1.27 and 9.1.73.
- */
-
-static double recur( n, x, newn, cancel )
-double *n;
-double x;
-double *newn;
-int cancel;
-{
-double pkm2, pkm1, pk, qkm2, qkm1;
-/* double pkp1; */
-double k, ans, qk, xk, yk, r, t, kf;
-static double big = BIG;
-int nflag, ctr;
-
-/* continued fraction for Jn(x)/Jn-1(x) */
-if( *n < 0.0 )
- nflag = 1;
-else
- nflag = 0;
-
-fstart:
-
-#if DEBUG
-printf( "recur: n = %.6e, newn = %.6e, cfrac = ", *n, *newn );
-#endif
-
-pkm2 = 0.0;
-qkm2 = 1.0;
-pkm1 = x;
-qkm1 = *n + *n;
-xk = -x * x;
-yk = qkm1;
-ans = 1.0;
-ctr = 0;
-do
- {
- yk += 2.0;
- pk = pkm1 * yk + pkm2 * xk;
- qk = qkm1 * yk + qkm2 * xk;
- pkm2 = pkm1;
- pkm1 = pk;
- qkm2 = qkm1;
- qkm1 = qk;
- if( qk != 0 )
- r = pk/qk;
- else
- r = 0.0;
- if( r != 0 )
- {
- t = fabs( (ans - r)/r );
- ans = r;
- }
- else
- t = 1.0;
-
- if( ++ctr > 1000 )
- {
- mtherr( "jv", UNDERFLOW );
- goto done;
- }
- if( t < MACHEP )
- goto done;
-
- if( fabs(pk) > big )
- {
- pkm2 /= big;
- pkm1 /= big;
- qkm2 /= big;
- qkm1 /= big;
- }
- }
-while( t > MACHEP );
-
-done:
-
-#if DEBUG
-printf( "%.6e\n", ans );
-#endif
-
-/* Change n to n-1 if n < 0 and the continued fraction is small
- */
-if( nflag > 0 )
- {
- if( fabs(ans) < 0.125 )
- {
- nflag = -1;
- *n = *n - 1.0;
- goto fstart;
- }
- }
-
-
-kf = *newn;
-
-/* backward recurrence
- * 2k
- * J (x) = --- J (x) - J (x)
- * k-1 x k k+1
- */
-
-pk = 1.0;
-pkm1 = 1.0/ans;
-k = *n - 1.0;
-r = 2 * k;
-do
- {
- pkm2 = (pkm1 * r - pk * x) / x;
- /* pkp1 = pk; */
- pk = pkm1;
- pkm1 = pkm2;
- r -= 2.0;
-/*
- t = fabs(pkp1) + fabs(pk);
- if( (k > (kf + 2.5)) && (fabs(pkm1) < 0.25*t) )
- {
- k -= 1.0;
- t = x*x;
- pkm2 = ( (r*(r+2.0)-t)*pk - r*x*pkp1 )/t;
- pkp1 = pk;
- pk = pkm1;
- pkm1 = pkm2;
- r -= 2.0;
- }
-*/
- k -= 1.0;
- }
-while( k > (kf + 0.5) );
-
-/* Take the larger of the last two iterates
- * on the theory that it may have less cancellation error.
- */
-
-if( cancel )
- {
- if( (kf >= 0.0) && (fabs(pk) > fabs(pkm1)) )
- {
- k += 1.0;
- pkm2 = pk;
- }
- }
-*newn = k;
-#if DEBUG
-printf( "newn %.6e rans %.6e\n", k, pkm2 );
-#endif
-return( pkm2 );
-}
-
-
-
-/* Ascending power series for Jv(x).
- * AMS55 #9.1.10.
- */
-
-extern double PI;
-extern int sgngam;
-
-static double jvs( n, x )
-double n, x;
-{
-double t, u, y, z, k;
-int ex;
-
-z = -x * x / 4.0;
-u = 1.0;
-y = u;
-k = 1.0;
-t = 1.0;
-
-while( t > MACHEP )
- {
- u *= z / (k * (n+k));
- y += u;
- k += 1.0;
- if( y != 0 )
- t = fabs( u/y );
- }
-#if DEBUG
-printf( "power series=%.5e ", y );
-#endif
-t = frexp( 0.5*x, &ex );
-ex = ex * n;
-if( (ex > -1023)
- && (ex < 1023)
- && (n > 0.0)
- && (n < (MAXGAM-1.0)) )
- {
- t = pow( 0.5*x, n ) / gamma( n + 1.0 );
-#if DEBUG
-printf( "pow(.5*x, %.4e)/gamma(n+1)=%.5e\n", n, t );
-#endif
- y *= t;
- }
-else
- {
-#if DEBUG
- z = n * log(0.5*x);
- k = lgam( n+1.0 );
- t = z - k;
- printf( "log pow=%.5e, lgam(%.4e)=%.5e\n", z, n+1.0, k );
-#else
- t = n * log(0.5*x) - lgam(n + 1.0);
-#endif
- if( y < 0 )
- {
- sgngam = -sgngam;
- y = -y;
- }
- t += log(y);
-#if DEBUG
-printf( "log y=%.5e\n", log(y) );
-#endif
- if( t < -MAXLOG )
- {
- return( 0.0 );
- }
- if( t > MAXLOG )
- {
- mtherr( "Jv", OVERFLOW );
- return( MAXNUM );
- }
- y = sgngam * exp( t );
- }
-return(y);
-}
-
-/* Hankel's asymptotic expansion
- * for large x.
- * AMS55 #9.2.5.
- */
-
-static double hankel( n, x )
-double n, x;
-{
-double t, u, z, k, sign, conv;
-double p, q, j, m, pp, qq;
-int flag;
-
-m = 4.0*n*n;
-j = 1.0;
-z = 8.0 * x;
-k = 1.0;
-p = 1.0;
-u = (m - 1.0)/z;
-q = u;
-sign = 1.0;
-conv = 1.0;
-flag = 0;
-t = 1.0;
-pp = 1.0e38;
-qq = 1.0e38;
-
-while( t > MACHEP )
- {
- k += 2.0;
- j += 1.0;
- sign = -sign;
- u *= (m - k * k)/(j * z);
- p += sign * u;
- k += 2.0;
- j += 1.0;
- u *= (m - k * k)/(j * z);
- q += sign * u;
- t = fabs(u/p);
- if( t < conv )
- {
- conv = t;
- qq = q;
- pp = p;
- flag = 1;
- }
-/* stop if the terms start getting larger */
- if( (flag != 0) && (t > conv) )
- {
-#if DEBUG
- printf( "Hankel: convergence to %.4E\n", conv );
-#endif
- goto hank1;
- }
- }
-
-hank1:
-u = x - (0.5*n + 0.25) * PI;
-t = sqrt( 2.0/(PI*x) ) * ( pp * cos(u) - qq * sin(u) );
-#if DEBUG
-printf( "hank: %.6e\n", t );
-#endif
-return( t );
-}
-
-
-/* Asymptotic expansion for large n.
- * AMS55 #9.3.35.
- */
-
-static double lambda[] = {
- 1.0,
- 1.041666666666666666666667E-1,
- 8.355034722222222222222222E-2,
- 1.282265745563271604938272E-1,
- 2.918490264641404642489712E-1,
- 8.816272674437576524187671E-1,
- 3.321408281862767544702647E+0,
- 1.499576298686255465867237E+1,
- 7.892301301158651813848139E+1,
- 4.744515388682643231611949E+2,
- 3.207490090890661934704328E+3
-};
-static double mu[] = {
- 1.0,
- -1.458333333333333333333333E-1,
- -9.874131944444444444444444E-2,
- -1.433120539158950617283951E-1,
- -3.172272026784135480967078E-1,
- -9.424291479571202491373028E-1,
- -3.511203040826354261542798E+0,
- -1.572726362036804512982712E+1,
- -8.228143909718594444224656E+1,
- -4.923553705236705240352022E+2,
- -3.316218568547972508762102E+3
-};
-static double P1[] = {
- -2.083333333333333333333333E-1,
- 1.250000000000000000000000E-1
-};
-static double P2[] = {
- 3.342013888888888888888889E-1,
- -4.010416666666666666666667E-1,
- 7.031250000000000000000000E-2
-};
-static double P3[] = {
- -1.025812596450617283950617E+0,
- 1.846462673611111111111111E+0,
- -8.912109375000000000000000E-1,
- 7.324218750000000000000000E-2
-};
-static double P4[] = {
- 4.669584423426247427983539E+0,
- -1.120700261622299382716049E+1,
- 8.789123535156250000000000E+0,
- -2.364086914062500000000000E+0,
- 1.121520996093750000000000E-1
-};
-static double P5[] = {
- -2.8212072558200244877E1,
- 8.4636217674600734632E1,
- -9.1818241543240017361E1,
- 4.2534998745388454861E1,
- -7.3687943594796316964E0,
- 2.27108001708984375E-1
-};
-static double P6[] = {
- 2.1257013003921712286E2,
- -7.6525246814118164230E2,
- 1.0599904525279998779E3,
- -6.9957962737613254123E2,
- 2.1819051174421159048E2,
- -2.6491430486951555525E1,
- 5.7250142097473144531E-1
-};
-static double P7[] = {
- -1.9194576623184069963E3,
- 8.0617221817373093845E3,
- -1.3586550006434137439E4,
- 1.1655393336864533248E4,
- -5.3056469786134031084E3,
- 1.2009029132163524628E3,
- -1.0809091978839465550E2,
- 1.7277275025844573975E0
-};
-
-
-static double jnx( n, x )
-double n, x;
-{
-double zeta, sqz, zz, zp, np;
-double cbn, n23, t, z, sz;
-double pp, qq, z32i, zzi;
-double ak, bk, akl, bkl;
-int sign, doa, dob, nflg, k, s, tk, tkp1, m;
-static double u[8];
-static double ai, aip, bi, bip;
-
-/* Test for x very close to n.
- * Use expansion for transition region if so.
- */
-cbn = cbrt(n);
-z = (x - n)/cbn;
-if( fabs(z) <= 0.7 )
- return( jnt(n,x) );
-
-z = x/n;
-zz = 1.0 - z*z;
-if( zz == 0.0 )
- return(0.0);
-
-if( zz > 0.0 )
- {
- sz = sqrt( zz );
- t = 1.5 * (log( (1.0+sz)/z ) - sz ); /* zeta ** 3/2 */
- zeta = cbrt( t * t );
- nflg = 1;
- }
-else
- {
- sz = sqrt(-zz);
- t = 1.5 * (sz - acos(1.0/z));
- zeta = -cbrt( t * t );
- nflg = -1;
- }
-z32i = fabs(1.0/t);
-sqz = cbrt(t);
-
-/* Airy function */
-n23 = cbrt( n * n );
-t = n23 * zeta;
-
-#if DEBUG
-printf("zeta %.5E, Airy(%.5E)\n", zeta, t );
-#endif
-airy( t, &ai, &aip, &bi, &bip );
-
-/* polynomials in expansion */
-u[0] = 1.0;
-zzi = 1.0/zz;
-u[1] = polevl( zzi, P1, 1 )/sz;
-u[2] = polevl( zzi, P2, 2 )/zz;
-u[3] = polevl( zzi, P3, 3 )/(sz*zz);
-pp = zz*zz;
-u[4] = polevl( zzi, P4, 4 )/pp;
-u[5] = polevl( zzi, P5, 5 )/(pp*sz);
-pp *= zz;
-u[6] = polevl( zzi, P6, 6 )/pp;
-u[7] = polevl( zzi, P7, 7 )/(pp*sz);
-
-#if DEBUG
-for( k=0; k<=7; k++ )
- printf( "u[%d] = %.5E\n", k, u[k] );
-#endif
-
-pp = 0.0;
-qq = 0.0;
-np = 1.0;
-/* flags to stop when terms get larger */
-doa = 1;
-dob = 1;
-akl = MAXNUM;
-bkl = MAXNUM;
-
-for( k=0; k<=3; k++ )
- {
- tk = 2 * k;
- tkp1 = tk + 1;
- zp = 1.0;
- ak = 0.0;
- bk = 0.0;
- for( s=0; s<=tk; s++ )
- {
- if( doa )
- {
- if( (s & 3) > 1 )
- sign = nflg;
- else
- sign = 1;
- ak += sign * mu[s] * zp * u[tk-s];
- }
-
- if( dob )
- {
- m = tkp1 - s;
- if( ((m+1) & 3) > 1 )
- sign = nflg;
- else
- sign = 1;
- bk += sign * lambda[s] * zp * u[m];
- }
- zp *= z32i;
- }
-
- if( doa )
- {
- ak *= np;
- t = fabs(ak);
- if( t < akl )
- {
- akl = t;
- pp += ak;
- }
- else
- doa = 0;
- }
-
- if( dob )
- {
- bk += lambda[tkp1] * zp * u[0];
- bk *= -np/sqz;
- t = fabs(bk);
- if( t < bkl )
- {
- bkl = t;
- qq += bk;
- }
- else
- dob = 0;
- }
-#if DEBUG
- printf("a[%d] %.5E, b[%d] %.5E\n", k, ak, k, bk );
-#endif
- if( np < MACHEP )
- break;
- np /= n*n;
- }
-
-/* normalizing factor ( 4*zeta/(1 - z**2) )**1/4 */
-t = 4.0 * zeta/zz;
-t = sqrt( sqrt(t) );
-
-t *= ai*pp/cbrt(n) + aip*qq/(n23*n);
-return(t);
-}
-
-/* Asymptotic expansion for transition region,
- * n large and x close to n.
- * AMS55 #9.3.23.
- */
-
-static double PF2[] = {
- -9.0000000000000000000e-2,
- 8.5714285714285714286e-2
-};
-static double PF3[] = {
- 1.3671428571428571429e-1,
- -5.4920634920634920635e-2,
- -4.4444444444444444444e-3
-};
-static double PF4[] = {
- 1.3500000000000000000e-3,
- -1.6036054421768707483e-1,
- 4.2590187590187590188e-2,
- 2.7330447330447330447e-3
-};
-static double PG1[] = {
- -2.4285714285714285714e-1,
- 1.4285714285714285714e-2
-};
-static double PG2[] = {
- -9.0000000000000000000e-3,
- 1.9396825396825396825e-1,
- -1.1746031746031746032e-2
-};
-static double PG3[] = {
- 1.9607142857142857143e-2,
- -1.5983694083694083694e-1,
- 6.3838383838383838384e-3
-};
-
-
-static double jnt( n, x )
-double n, x;
-{
-double z, zz, z3;
-double cbn, n23, cbtwo;
-double ai, aip, bi, bip; /* Airy functions */
-double nk, fk, gk, pp, qq;
-double F[5], G[4];
-int k;
-
-cbn = cbrt(n);
-z = (x - n)/cbn;
-cbtwo = cbrt( 2.0 );
-
-/* Airy function */
-zz = -cbtwo * z;
-airy( zz, &ai, &aip, &bi, &bip );
-
-/* polynomials in expansion */
-zz = z * z;
-z3 = zz * z;
-F[0] = 1.0;
-F[1] = -z/5.0;
-F[2] = polevl( z3, PF2, 1 ) * zz;
-F[3] = polevl( z3, PF3, 2 );
-F[4] = polevl( z3, PF4, 3 ) * z;
-G[0] = 0.3 * zz;
-G[1] = polevl( z3, PG1, 1 );
-G[2] = polevl( z3, PG2, 2 ) * z;
-G[3] = polevl( z3, PG3, 2 ) * zz;
-#if DEBUG
-for( k=0; k<=4; k++ )
- printf( "F[%d] = %.5E\n", k, F[k] );
-for( k=0; k<=3; k++ )
- printf( "G[%d] = %.5E\n", k, G[k] );
-#endif
-pp = 0.0;
-qq = 0.0;
-nk = 1.0;
-n23 = cbrt( n * n );
-
-for( k=0; k<=4; k++ )
- {
- fk = F[k]*nk;
- pp += fk;
- if( k != 4 )
- {
- gk = G[k]*nk;
- qq += gk;
- }
-#if DEBUG
- printf("fk[%d] %.5E, gk[%d] %.5E\n", k, fk, k, gk );
-#endif
- nk /= n23;
- }
-
-fk = cbtwo * ai * pp/cbn + cbrt(4.0) * aip * qq/n;
-return(fk);
-}
generated by cgit v1.2.3 (git 2.25.1) at 2024-11-23 03:11:54 +0000