jv.cpp

math software

//
// jv.cpp
//
// jv.cpp,v 1.10 2003/12/12 09:54:54 tonyottosson Exp
//
 
#include <cmath>

#include "base/itassert.h"
#include "base/scalfunc.h"

#include "../src/base/bessel/bessel_internal.h"

using namespace itpp;

// This is slightly modified routine from the Cephes library, see http://www.netlib.org/cephes/
//  
// According to licence agreement this software can be used freely.
//

/*
 *	Bessel function of noninteger order
 *
 * 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
*/


#define MAXGAM 171.624376956302725


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);

#define MAXNUM 1.79769313486231570815E308    /* 2**1024*(1-MACHEP) */
#define MACHEP 1.11022302462515654042E-16   /* 2**-53 */
#define MAXLOG 7.08396418532264106224E2     /* log 2**1022 */
#define MINLOG -7.08396418532264106224E2    /* log 2**-1022 */
#define PI 3.14159265358979323846       /* pi */

#define BIG  1.44115188075855872E+17



// ---------------------------- jv() -------------------------------------------------------
double jv(double n, double 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 = int(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 ) // use 0th order bessel function
	return( j0(x) );
      if( n == 1.0 ) // use 1th order bessel function
	return( sign * j1(x) );
    }

  it_error_if( (x < 0.0) && (y != an), "besselj:: negative values only allowed for integer orders.");

  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( 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 )
	{
	  it_warning("besselj:: partial loss of precision");
	  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(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:
  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 )
	{
	  it_warning("besselj:: Underflow");
	  //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:

  /* 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;

  return( pkm2 );
}



/* Ascending power series for Jv(x).
 * AMS55 #9.1.10.
 */


static double jvs(double n, double 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 );
    }

  t = frexp( 0.5*x, &ex );
  ex = int(ex * n);
  if(  (ex > -1023)
       && (ex < 1023) 
       && (n > 0.0)
       && (n < (MAXGAM-1.0)) )
    {
      t = pow( 0.5*x, n ) / itpp::gamma( n + 1.0 );

      y *= t;
    }
  else
    {
      t = n * log(0.5*x) - lgamma(n + 1.0);
      if( y < 0 )
	{
#ifdef _MSC_VER
		it_error("lgamma only defined for positive values for Visual C++");
#else
	  signgam = -signgam;
	  y = -y;
#endif
	}
      t += log(y);

      if( t < -MAXLOG )
	{
	  return( 0.0 );
	}

      if( t > MAXLOG )
	{
	  it_warning("besselj:: Overflow");
	  //mtherr( "Jv", OVERFLOW );
	  return( MAXNUM );
	}
#ifdef _MSC_VER
      y = exp( t );
#else
      y = signgam * exp( t );
#endif
    }
  return(y);
}

/* Hankel's asymptotic expansion
 * for large x.
 * AMS55 #9.2.5.
 */
static double hankel(double n, double 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) )
	{
	  goto hank1;
	}
    }	

 hank1:
  u = x - (0.5*n + 0.25) * PI;
  t = sqrt( 2.0/(PI*x) ) * ( pp * cos(u) - qq * sin(u) );

  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(double n, double 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;

  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);

  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( 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(double n, double 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;

  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;
	}

      nk /= n23;
    }

  fk = cbtwo * ai * pp/cbn  +  cbrt(4.0) * aip * qq/n;
  return(fk);
}