fermi_dirac.c

该文件为c++的数学函数库!是一个非常有用的编程工具.它含有各种数学函数,为科学计算、工程应用等程序编写提供方便！

 -1.3333e-15,
  1.898e-16,
  2.72e-17,
 -3.9e-18,
 -6.e-19,
  1.e-19
};
static cheb_series fd_half_a_cs = {
  fd_half_a_data,
  22,
  -1, 1,
  11
};


/* Chebyshev fit for F_{1/2}(3/2(t+1) + 1);  -1 < t < 1, 1 < x < 4
 */
static double fd_half_b_data[20] = {
  7.651013792074984027,
  2.475545606866155737,
  0.218335982672476128,
 -0.007730591500584980,
 -0.000217443383867318,
  0.000147663980681359,
 -0.000021586361321527,
  8.07712735394e-7,
  3.28858050706e-7,
 -7.9474330632e-8,
  6.940207234e-9,
  6.75594681e-10,
 -3.10200490e-10,
  4.2677233e-11,
 -2.1696e-14,
 -1.170245e-12,
  2.34757e-13,
 -1.4139e-14,
 -3.864e-15,
  1.202e-15
};
static cheb_series fd_half_b_cs = {
  fd_half_b_data,
  19,
  -1, 1,
  12
};


/* Chebyshev fit for F_{1/2}(3(t+1) + 4);  -1 < t < 1, 4 < x < 10
 */
static double fd_half_c_data[23] = {
  29.584339348839816528,
  8.808344283250615592,
  0.503771641883577308,
 -0.021540694914550443,
  0.002143341709406890,
 -0.000257365680646579,
  0.000027933539372803,
 -1.678525030167e-6,
 -2.78100117693e-7,
  1.35218065147e-7,
 -3.3740425009e-8,
  6.474834942e-9,
 -1.009678978e-9,
  1.20057555e-10,
 -6.636314e-12,
 -1.710566e-12,
  7.75069e-13,
 -1.97973e-13,
  3.9414e-14,
 -6.374e-15,
  7.77e-16,
 -4.0e-17,
 -1.4e-17
};
static cheb_series fd_half_c_cs = {
  fd_half_c_data,
  22,
  -1, 1,
  13
};


/* Chebyshev fit for F_{1/2}(x) / x^(3/2)
 * 10 < x < 30 
 * -1 < t < 1
 * t = 1/10 (x-10) - 1 = x/10 - 2
 */
static double fd_half_d_data[30] = {
  1.5116909434145508537,
 -0.0036043405371630468,
  0.0014207743256393359,
 -0.0005045399052400260,
  0.0001690758006957347,
 -0.0000546305872688307,
  0.0000172223228484571,
 -5.3352603788706e-6,
  1.6315287543662e-6,
 -4.939021084898e-7,
  1.482515450316e-7,
 -4.41552276226e-8,
  1.30503160961e-8,
 -3.8262599802e-9,
  1.1123226976e-9,
 -3.204765534e-10,
  9.14870489e-11,
 -2.58778946e-11,
  7.2550731e-12,
 -2.0172226e-12,
  5.566891e-13,
 -1.526247e-13,
  4.16121e-14,
 -1.12933e-14,
  3.0537e-15,
 -8.234e-16,
  2.215e-16,
 -5.95e-17,
  1.59e-17,
 -4.0e-18
};
static cheb_series fd_half_d_cs = {
  fd_half_d_data,
  29,
  -1, 1,
  15
};



/* Chebyshev fit for F_{3/2}(t);  -1 < t < 1, -1 < x < 1
 */
static double fd_3half_a_data[20] = {
  2.0404775940601704976,
  0.8122168298093491444,
  0.1536371165644008069,
  0.0156174323847845125,
  0.0005943427879290297,
 -0.0000429609447738365,
 -3.8246452994606e-6,
  3.802306180287e-7,
  4.05746157593e-8,
 -4.5530360159e-9,
 -5.306873139e-10,
  6.37297268e-11,
  7.8403674e-12,
 -9.840241e-13,
 -1.255952e-13,
  1.62617e-14,
  2.1318e-15,
 -2.825e-16,
 -3.78e-17,
  5.1e-18
};
static cheb_series fd_3half_a_cs = {
  fd_3half_a_data,
  19,
  -1, 1,
  11
};


/* Chebyshev fit for F_{3/2}(3/2(t+1) + 1);  -1 < t < 1, 1 < x < 4
 */
static double fd_3half_b_data[22] = {
  13.403206654624176674,
  5.574508357051880924,
  0.931228574387527769,
  0.054638356514085862,
 -0.001477172902737439,
 -0.000029378553381869,
  0.000018357033493246,
 -2.348059218454e-6,
  8.3173787440e-8,
  2.6826486956e-8,
 -6.011244398e-9,
  4.94345981e-10,
  3.9557340e-11,
 -1.7894930e-11,
  2.348972e-12,
 -1.2823e-14,
 -5.4192e-14,
  1.0527e-14,
 -6.39e-16,
 -1.47e-16,
  4.5e-17,
 -5.e-18
};
static cheb_series fd_3half_b_cs = {
  fd_3half_b_data,
  21,
  -1, 1,
  12
};


/* Chebyshev fit for F_{3/2}(3(t+1) + 4);  -1 < t < 1, 4 < x < 10
 */
static double fd_3half_c_data[21] = {
  101.03685253378877642,
  43.62085156043435883,
  6.62241373362387453,
  0.25081415008708521,
 -0.00798124846271395,
  0.00063462245101023,
 -0.00006392178890410,
  6.04535131939e-6,
 -3.4007683037e-7,
 -4.072661545e-8,
  1.931148453e-8,
 -4.46328355e-9,
  7.9434717e-10,
 -1.1573569e-10,
  1.304658e-11,
 -7.4114e-13,
 -1.4181e-13,
  6.491e-14,
 -1.597e-14,
  3.05e-15,
 -4.8e-16
};
static cheb_series fd_3half_c_cs = {
  fd_3half_c_data,
  20,
  -1, 1,
  12
};


/* Chebyshev fit for F_{3/2}(x) / x^(5/2)
 * 10 < x < 30 
 * -1 < t < 1
 * t = 1/10 (x-10) - 1 = x/10 - 2
 */
static double fd_3half_d_data[25] = {
  0.6160645215171852381,
 -0.0071239478492671463,
  0.0027906866139659846,
 -0.0009829521424317718,
  0.0003260229808519545,
 -0.0001040160912910890,
  0.0000322931223232439,
 -9.8243506588102e-6,
  2.9420132351277e-6,
 -8.699154670418e-7,
  2.545460071999e-7,
 -7.38305056331e-8,
  2.12545670310e-8,
 -6.0796532462e-9,
  1.7294556741e-9,
 -4.896540687e-10,
  1.380786037e-10,
 -3.88057305e-11,
  1.08753212e-11,
 -3.0407308e-12,
  8.485626e-13,
 -2.364275e-13,
  6.57636e-14,
 -1.81807e-14,
  4.6884e-15
};
static cheb_series fd_3half_d_cs = {
  fd_3half_d_data,
  24,
  -1, 1,
  16
};


/* Goano's modification of the Levin-u implementation.
 * This is a simplification of the original WHIZ algorithm.
 * See [Fessler et al., ACM Toms 9, 346 (1983)].
 */
static
int
fd_whiz(const double term, const int iterm,
        double * qnum, double * qden,
        double * result, double * s)
{
  if(iterm == 0) *s = 0.0;

  *s += term;

  qden[iterm] = 1.0/(term*(iterm+1.0)*(iterm+1.0));
  qnum[iterm] = *s * qden[iterm];

  if(iterm > 0) {
    double factor = 1.0;
    double ratio  = iterm/(iterm+1.0);
    int j;
    for(j=iterm-1; j>=0; j--) {
      double c = factor * (j+1.0) / (iterm+1.0);
      factor *= ratio;
      qden[j] = qden[j+1] - c * qden[j];
      qnum[j] = qnum[j+1] - c * qnum[j];
    }
  }

  *result = qnum[0] / qden[0];
  return GSL_SUCCESS;
}


/* Handle case of integer j <= -2.
 */
static
int
fd_nint(const int j, const double x, gsl_sf_result * result)
{
/*    const int nsize = 100 + 1; */
  enum {
    nsize = 100+1
  };
  double qcoeff[nsize];

  if(j >= -1) {
    result->val = 0.0;
    result->err = 0.0;
    GSL_ERROR ("error", GSL_ESANITY);
  }
  else if(j < -(nsize)) {
    result->val = 0.0;
    result->err = 0.0;
    GSL_ERROR ("error", GSL_EUNIMPL);
  }
  else {
    double a, p, f;
    int i, k;
    int n = -(j+1);

    qcoeff[1] = 1.0;

    for(k=2; k<=n; k++) {
      qcoeff[k] = -qcoeff[k-1];
      for(i=k-1; i>=2; i--) {
        qcoeff[i] = i*qcoeff[i] - (k-(i-1))*qcoeff[i-1];
      }
    }

    if(x >= 0.0) {
      a = exp(-x);
      f = qcoeff[1];
      for(i=2; i<=n; i++) {
        f = f*a + qcoeff[i];
      }
    }
    else {
      a = exp(x);
      f = qcoeff[n];
      for(i=n-1; i>=1; i--) {
        f = f*a + qcoeff[i];
      }
    }

    p = gsl_sf_pow_int(1.0+a, j);
    result->val = f*a*p;
    result->err = 3.0 * GSL_DBL_EPSILON * fabs(f*a*p);
    return GSL_SUCCESS;
  }
}


/* x < 0
 */
static
int
fd_neg(const double j, const double x, gsl_sf_result * result)
{
  enum {
    itmax = 100,
    qsize = 100+1
  };
/*    const int itmax = 100; */
/*    const int qsize = 100 + 1; */
  double qnum[qsize], qden[qsize];

  if(x < GSL_LOG_DBL_MIN) {
    result->val = 0.0;
    result->err = 0.0;
    return GSL_SUCCESS;
  }
  else if(x < -1.0 && x < -fabs(j+1.0)) {
    /* Simple series implementation. Avoid the
     * complexity and extra work of the series
     * acceleration method below.
     */
    double ex   = exp(x);
    double term = ex;
    double sum  = term;
    int n;
    for(n=2; n<100; n++) {
      double rat = (n-1.0)/n;
      double p   = pow(rat, j+1.0);
      term *= -ex * p;
      sum  += term;
      if(fabs(term/sum) < GSL_DBL_EPSILON) break;
    }
    result->val = sum;
    result->err = 2.0 * GSL_DBL_EPSILON * fabs(sum);
    return GSL_SUCCESS;
  }
  else {
    double s;
    double xn = x;
    double ex  = -exp(x);
    double enx = -ex;
    double f = 0.0;
    double f_previous;
    int jterm;
    for(jterm=0; jterm<=itmax; jterm++) {
      double p = pow(jterm+1.0, j+1.0);
      double term = enx/p;
      f_previous = f;
      fd_whiz(term, jterm, qnum, qden, &f, &s);
      xn += x;
      if(fabs(f-f_previous) < fabs(f)*2.0*GSL_DBL_EPSILON || xn < GSL_LOG_DBL_MIN) break;
      enx *= ex;
    }

    result->val  = f;
    result->err  = fabs(f-f_previous);
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(f);

    if(jterm == itmax)
      GSL_ERROR ("error", GSL_EMAXITER);
    else
      return GSL_SUCCESS;
  }
}


/* asymptotic expansion
 * j + 2.0 > 0.0
 */
static
int
fd_asymp(const double j, const double x, gsl_sf_result * result)
{
  const int j_integer = ( fabs(j - floor(j+0.5)) < 100.0*GSL_DBL_EPSILON );
  const int itmax = 200;
  gsl_sf_result lg;
  int stat_lg = gsl_sf_lngamma_e(j + 2.0, &lg);
  double seqn_val = 0.5;
  double seqn_err = 0.0;
  double xm2  = (1.0/x)/x;
  double xgam = 1.0;
  double add = GSL_DBL_MAX;
  double cos_term;
  double ln_x;
  double ex_term_1;
  double ex_term_2;
  gsl_sf_result fneg;
  gsl_sf_result ex_arg;
  gsl_sf_result ex;
  int stat_fneg;
  int stat_e;
  int n;
  for(n=1; n<=itmax; n++) {
    double add_previous = add;
    gsl_sf_result eta;
    gsl_sf_eta_int_e(2*n, &eta);
    xgam = xgam * xm2 * (j + 1.0 - (2*n-2)) * (j + 1.0 - (2*n-1));
    add  = eta.val * xgam;
    if(!j_integer && fabs(add) > fabs(add_previous)) break;
    if(fabs(add/seqn_val) < GSL_DBL_EPSILON) break;
    seqn_val += add;
    seqn_err += 2.0 * GSL_DBL_EPSILON * fabs(add);
  }
  seqn_err += fabs(add);

  stat_fneg = fd_neg(j, -x, &fneg);
  ln_x = log(x);
  ex_term_1 = (j+1.0)*ln_x;
  ex_term_2 = lg.val;
  ex_arg.val = ex_term_1 - ex_term_2; /*(j+1.0)*ln_x - lg.val; */
  ex_arg.err = GSL_DBL_EPSILON*(fabs(ex_term_1) + fabs(ex_term_2)) + lg.err;
  stat_e    = gsl_sf_exp_err_e(ex_arg.val, ex_arg.err, &ex);
  cos_term  = cos(j*M_PI);
  result->val  = cos_term * fneg.val + 2.0 * seqn_val * ex.val;
  result->err  = fabs(2.0 * ex.err * seqn_val);
  result->err += fabs(2.0 * ex.val * seqn_err);
  result->err += fabs(cos_term) * fneg.err;
  result->err += 4.0 * GSL_DBL_EPSILON * fabs(result->val);
  return GSL_ERROR_SELECT_3(stat_e, stat_fneg, stat_lg);
}


/* Series evaluation for small x, generic j.
 * [Goano (8)]
 */
#if 0
static
int
fd_series(const double j, const double x, double * result)
{
  const int nmax = 1000;
  int n;
  double sum = 0.0;
  double prev;
  double pow_factor = 1.0;
  double eta_factor;
  gsl_sf_eta_e(j + 1.0, &eta_factor);
  prev = pow_factor * eta_factor;
  sum += prev;
  for(n=1; n<nmax; n++) {
    double term;
    gsl_sf_eta_e(j+1.0-n, &eta_factor);
    pow_factor *= x/n;
    term = pow_factor * eta_factor;
    sum += term;
    if(fabs(term/sum) < GSL_DBL_EPSILON && fabs(prev/sum) < GSL_DBL_EPSILON) break;
    prev = term;
  }

  *result = sum;
  return GSL_SUCCESS;
}
#endif /* 0 */


/* Series evaluation for small x > 0, integer j > 0; x < Pi.
 * [Goano (8)]
 */
static
int
fd_series_int(const int j, const double x, gsl_sf_result * result)
{
  int n;
  double sum = 0.0;
  double del;
  double pow_factor = 1.0;
  gsl_sf_result eta_factor;
  gsl_sf_eta_int_e(j + 1, &eta_factor);
  del  = pow_factor * eta_factor.val;
  sum += del;

  /* Sum terms where the argument
   * of eta() is positive.
   */
  for(n=1; n<=j+2; n++) {
    gsl_sf_eta_int_e(j+1-n, &eta_factor);
    pow_factor *= x/n;
    del  = pow_factor * eta_factor.val;
    sum += del;
    if(fabs(del/sum) < GSL_DBL_EPSILON) break;
  }