bessel/bessel.js

var M = Math;
function _horner(arr, v) { return arr.reduce(function(z,w){return v * z + w;},0); };
function _bessel_iter(x, n, f0, f1, sign) {
  if(!sign) sign = -1;
  var tdx = 2 / x, f2;
  if(n === 0) return f0;
  if(n === 1) return f1;
  for(var o = 1; o != n; ++o) {
    f2 = f1 * o * tdx + sign * f0;
    f0 = f1; f1 = f2;
  }
  return f1;
}
function _bessel_wrap(bessel0, bessel1, name, nonzero, sign) {
  return function bessel(x,n) {
    if(n === 0) return bessel0(x);
    if(n === 1) return bessel1(x);
    if(n < 0) throw name + ': Order (' + n + ') must be nonnegative';
    if(nonzero == 1 && x === 0) throw name + ': Undefined when x == 0';
    if(nonzero == 2 && x <= 0) throw name + ': Undefined when x <= 0';
    var b0 = bessel0(x), b1 = bessel1(x);
    return _bessel_iter(x, n, b0, b1, sign);
  };
}
var besselj = (function() {
  var b0_a1a = [57568490574.0,-13362590354.0,651619640.7,-11214424.18,77392.33017,-184.9052456].reverse();
  var b0_a2a = [57568490411.0,1029532985.0,9494680.718,59272.64853,267.8532712,1.0].reverse();
  var b0_a1b = [1.0, -0.1098628627e-2, 0.2734510407e-4, -0.2073370639e-5, 0.2093887211e-6].reverse();
  var b0_a2b = [-0.1562499995e-1, 0.1430488765e-3, -0.6911147651e-5, 0.7621095161e-6, -0.934935152e-7].reverse();
  var W = 0.636619772; // 2 / Math.PI

  function bessel0(x) {
    var a, a1, a2, y = x * x, xx = M.abs(x) - 0.785398164;
    if(M.abs(x) < 8) {
      a1 = _horner(b0_a1a, y);
      a2 = _horner(b0_a2a, y);
      a = a1/a2;
    }
    else {
      y = 64 / y;
      a1 = _horner(b0_a1b, y);
      a2 = _horner(b0_a2b, y);
      a = M.sqrt(W/M.abs(x))*(M.cos(xx)*a1-M.sin(xx)*a2*8/M.abs(x));
    }
    return a;
  }
  var b1_a1a = [72362614232.0,-7895059235.0,242396853.1,-2972611.439, 15704.48260, -30.16036606].reverse();
  var b1_a2a = [144725228442.0, 2300535178.0, 18583304.74, 99447.43394, 376.9991397, 1.0].reverse();
  var b1_a1b = [1.0, 0.183105e-2, -0.3516396496e-4, 0.2457520174e-5, -0.240337019e-6].reverse();
  var b1_a2b = [0.04687499995, -0.2002690873e-3, 0.8449199096e-5, -0.88228987e-6, 0.105787412e-6].reverse();
  function bessel1(x) {
    var a, a1, a2, y = x*x, xx = M.abs(x) - 2.356194491;
    if(Math.abs(x)< 8) {
      a1 = x*_horner(b1_a1a, y);
      a2 = _horner(b1_a2a, y);
      a = a1 / a2;
    } else {
      y = 64 / y;
      a1=_horner(b1_a1b, y);
      a2=_horner(b1_a2b, y);
      a=M.sqrt(W/M.abs(x))*(M.cos(xx)*a1-M.sin(xx)*a2*8/M.abs(x));
      if(x < 0) a = -a;
    }
    return a;
  }
  return function besselj(x, n) {
    n = Math.round(n);
    if(n === 0) return bessel0(M.abs(x));
    if(n === 1) return bessel1(M.abs(x));
    if(n < 0) throw 'BESSELJ: Order (' + n + ') must be nonnegative';
    if(M.abs(x) === 0) return 0;

    var ret, j, tox = 2 / M.abs(x), m, jsum, sum, bjp, bj, bjm;
    if(M.abs(x) > n) {
      ret = _bessel_iter(x, n, bessel0(M.abs(x)), bessel1(M.abs(x)),-1);
    } else {
      m=2*M.floor((n+M.floor(M.sqrt(40*n)))/2);
      jsum=0;
      bjp=ret=sum=0.0;
      bj=1.0;
      for (j=m;j>0;j--) {
        bjm=j*tox*bj-bjp;
        bjp=bj;
        bj=bjm;
        if (M.abs(bj) > 1E10) {
          bj *= 1E-10;
          bjp *= 1E-10;
          ret *= 1E-10;
          sum *= 1E-10;
        }
        if (jsum) sum += bj;
        jsum=!jsum;
        if (j == n) ret=bjp;
      }
      sum=2.0*sum-bj;
      ret /= sum;
    }
    return x < 0 && (n%2) ? -ret : ret;
  };
})();
var bessely = (function() {
  var b0_a1a = [-2957821389.0, 7062834065.0, -512359803.6, 10879881.29, -86327.92757, 228.4622733].reverse();
  var b0_a2a = [40076544269.0, 745249964.8, 7189466.438, 47447.26470, 226.1030244, 1.0].reverse();
  var b0_a1b = [1.0, -0.1098628627e-2, 0.2734510407e-4, -0.2073370639e-5, 0.2093887211e-6].reverse();
  var b0_a2b = [-0.1562499995e-1, 0.1430488765e-3, -0.6911147651e-5, 0.7621095161e-6, -0.934945152e-7].reverse();

  var W = 0.636619772;
  function bessel0(x) {
    var a, a1, a2, y = x * x, xx = x - 0.785398164;
    if(x < 8) {
      a1 = _horner(b0_a1a, y);
      a2 = _horner(b0_a2a, y);
      a = a1/a2 + W * besselj(x,0) * M.log(x);
    } else {
      y = 64 / y;
      a1 = _horner(b0_a1b, y);
      a2 = _horner(b0_a2b, y);
      a = M.sqrt(W/x)*(M.sin(xx)*a1+M.cos(xx)*a2*8/x);
    }
    return a;
  }

  var b1_a1a = [-0.4900604943e13, 0.1275274390e13, -0.5153438139e11, 0.7349264551e9, -0.4237922726e7, 0.8511937935e4].reverse();
  var b1_a2a = [0.2499580570e14, 0.4244419664e12, 0.3733650367e10, 0.2245904002e8, 0.1020426050e6, 0.3549632885e3, 1].reverse();
  var b1_a1b = [1.0, 0.183105e-2, -0.3516396496e-4, 0.2457520174e-5, -0.240337019e-6].reverse();
  var b1_a2b = [0.04687499995, -0.2002690873e-3, 0.8449199096e-5, -0.88228987e-6, 0.105787412e-6].reverse();
  function bessel1(x) {
    var a, a1, a2, y = x*x, xx = x - 2.356194491;
    if(x < 8) {
      a1 = x*_horner(b1_a1a, y);
      a2 = _horner(b1_a2a, y);
      a = a1/a2 + W * (besselj(x,1) * M.log(x) - 1 / x);
    } else {
      y = 64 / y;
      a1=_horner(b1_a1b, y);
      a2=_horner(b1_a2b, y);
      a=M.sqrt(W/x)*(M.sin(xx)*a1+M.cos(xx)*a2*8/x);
    }
    return a;
  }

  return _bessel_wrap(bessel0, bessel1, 'BESSELY', 1, -1);
})();
var besseli = (function() {
  var b0_a = [1.0, 3.5156229, 3.0899424, 1.2067492, 0.2659732, 0.360768e-1, 0.45813e-2].reverse();
  var b0_b = [0.39894228, 0.1328592e-1, 0.225319e-2, -0.157565e-2, 0.916281e-2, -0.2057706e-1, 0.2635537e-1, -0.1647633e-1, 0.392377e-2].reverse();
  function bessel0(x) {
    if(x <= 3.75) return _horner(b0_a, x*x/(3.75*3.75));
    return M.exp(M.abs(x))/M.sqrt(M.abs(x))*_horner(b0_b, 3.75/M.abs(x));
  }

  var b1_a = [0.5, 0.87890594, 0.51498869, 0.15084934, 0.2658733e-1, 0.301532e-2, 0.32411e-3].reverse();
  var b1_b = [0.39894228, -0.3988024e-1, -0.362018e-2, 0.163801e-2, -0.1031555e-1, 0.2282967e-1, -0.2895312e-1, 0.1787654e-1, -0.420059e-2].reverse();
  function bessel1(x) {
    if(x < 3.75) return x * _horner(b1_a, x*x/(3.75*3.75));
    return (x < 0 ? -1 : 1) * M.exp(M.abs(x))/M.sqrt(M.abs(x))*_horner(b1_b, 3.75/M.abs(x));
  }

  return function besseli(x, n) {
    n = Math.round(n);
    if(n === 0) return bessel0(x);
    if(n == 1) return bessel1(x);
    if(n < 0) throw 'BESSELI Order (' + n + ') must be nonnegative';
    if(M.abs(x) === 0) return 0;

    var ret, j, tox = 2 / M.abs(x), m, bip, bi, bim;
    m=2*M.round((n+M.round(M.sqrt(40*n)))/2);
    bip=ret=0.0;
    bi=1.0;
    for (j=m;j>0;j--) {
      bim=j*tox*bi + bip;
      bip=bi; bi=bim;
      if (M.abs(bi) > 1E10) {
        bi *= 1E-10;
        bip *= 1E-10;
        ret *= 1E-10;
      }
      if(j == n) ret = bip;
    }
    ret *= besseli(x, 0) / bi;
    return x < 0 && (n%2) ? -ret : ret;
  };

})();

var besselk = (function() {
  var b0_a = [-0.57721566, 0.42278420, 0.23069756, 0.3488590e-1, 0.262698e-2, 0.10750e-3, 0.74e-5].reverse();
  var b0_b = [1.25331414, -0.7832358e-1, 0.2189568e-1, -0.1062446e-1, 0.587872e-2, -0.251540e-2, 0.53208e-3].reverse();
  function bessel0(x) {
    if(x <= 2) return -M.log(x/2)*besseli(x,0) + _horner(b0_a, x*x/4);
    return M.exp(-x)/M.sqrt(x)*_horner(b0_b, 2/x);
  }

  var b1_a = [1.0, 0.15443144, -0.67278579, -0.18156897, -0.1919402e-1, -0.110404e-2, -0.4686e-4].reverse();
  var b1_b = [1.25331414, 0.23498619, -0.3655620e-1, 0.1504268e-1, -0.780353e-2, 0.325614e-2, -0.68245e-3].reverse();
  function bessel1(x) {
    if(x <= 2) return M.log(x/2)*besseli(x,1) + (1/x)*_horner(b1_a, x*x/4);
    return M.exp(-x)/M.sqrt(x)*_horner(b1_b, 2/x);
  }

  return _bessel_wrap(bessel0, bessel1, 'BESSELK', 2, 1);
})();
if(typeof exports !== "undefined") {
  exports.besselj = besselj;
  exports.bessely = bessely;
  exports.besseli = besseli;
  exports.besselk = besselk;
}