legendre_con.c

开放gsl矩阵运算

    double im_tmp = im_multiplier*re_term + re_multiplier*im_term;
    double asum = fabs(re_sum) + fabs(im_sum);
    re_term = y/k * re_tmp;
    im_term = y/k * im_tmp;
    if(fabs(re_term/asum) < GSL_DBL_EPSILON && fabs(im_term/asum) < GSL_DBL_EPSILON) break;
    re_sum += re_term;
    im_sum += im_term;
  }

  *reF = re_sum;
  *imF = im_sum;

  if(k == kmax)
    GSL_ERROR ("error", GSL_EMAXITER);
  else  
    return GSL_SUCCESS;
}


/* P^{mu}_{-1/2 + I tau}
 * x->Inf
 */
int
gsl_sf_conicalP_large_x_e(const double mu, const double tau, const double x,
                             gsl_sf_result * result, double * ln_multiplier)
{
  /* 2F1 term
   */
  double y = ( x < 0.5*GSL_SQRT_DBL_MAX ? 1.0/(x*x) : 0.0 );
  double reF, imF;
  int stat_F = conicalP_hyperg_large_x(mu, tau, y, &reF, &imF);

  /* f = Gamma(+i tau)/Gamma(1/2 - mu + i tau)
   * FIXME: shift so it's better for tau-> 0
   */
  gsl_sf_result lgr_num, lgth_num;
  gsl_sf_result lgr_den, lgth_den;
  int stat_gn = gsl_sf_lngamma_complex_e(0.0,tau,&lgr_num,&lgth_num);
  int stat_gd = gsl_sf_lngamma_complex_e(0.5-mu,tau,&lgr_den,&lgth_den);

  double angle = lgth_num.val - lgth_den.val + atan2(imF,reF);

  double lnx   = log(x);
  double lnxp1 = log(x+1.0);
  double lnxm1 = log(x-1.0);
  double lnpre_const = 0.5*M_LN2 - 0.5*M_LNPI;
  double lnpre_comm = (mu-0.5)*lnx - 0.5*mu*(lnxp1 + lnxm1);
  double lnpre_err  =   GSL_DBL_EPSILON * (0.5*M_LN2 + 0.5*M_LNPI)
                      + GSL_DBL_EPSILON * fabs((mu-0.5)*lnx)
		      + GSL_DBL_EPSILON * fabs(0.5*mu)*(fabs(lnxp1)+fabs(lnxm1));

  /*  result = pre*|F|*|f| * cos(angle - tau * (log(x)+M_LN2))
   */
  gsl_sf_result cos_result;
  int stat_cos = gsl_sf_cos_e(angle + tau*(log(x) + M_LN2), &cos_result);
  int status = GSL_ERROR_SELECT_4(stat_cos, stat_gd, stat_gn, stat_F);
  if(cos_result.val == 0.0) {
    result->val = 0.0;
    result->err = 0.0;
    return status;
  }
  else {
    double lnFf_val = 0.5*log(reF*reF+imF*imF) + lgr_num.val - lgr_den.val;
    double lnFf_err = lgr_num.err + lgr_den.err + GSL_DBL_EPSILON * fabs(lnFf_val);
    double lnnoc_val = lnpre_const + lnpre_comm + lnFf_val;
    double lnnoc_err = lnpre_err + lnFf_err + GSL_DBL_EPSILON * fabs(lnnoc_val);
    int stat_e = gsl_sf_exp_mult_err_e(lnnoc_val, lnnoc_err,
                                          cos_result.val, cos_result.err,
                                          result);
    if(stat_e == GSL_SUCCESS) {
      *ln_multiplier = 0.0;
    }
    else {
      result->val  = cos_result.val;
      result->err  = cos_result.err;
      result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
      *ln_multiplier = lnnoc_val;
    }
    return status;
  }
}


/* P^{mu}_{-1/2 + I tau}  first hypergeometric representation
 * -1 < x < 1
 * This is more effective for |x| small, however it will work w/o
 * reservation for any x < 0 because everything is positive
 * definite in that case.
 *
 * [Kolbig,   (3)] (note typo in args of gamma functions)
 * [Bateman, (22)] (correct form)
 */
static
int
conicalP_xlt1_hyperg_A(double mu, double tau, double x, gsl_sf_result * result)
{
  double x2 = x*x;
  double err_amp = 1.0 + 1.0/(GSL_DBL_EPSILON + fabs(1.0-fabs(x)));
  double pre_val = M_SQRTPI / pow(0.5*sqrt(1-x2), mu);
  double pre_err = err_amp * GSL_DBL_EPSILON * (fabs(mu)+1.0) * fabs(pre_val) ;
  gsl_sf_result ln_g1, ln_g2, arg_g1, arg_g2;
  gsl_sf_result F1, F2;
  gsl_sf_result pre1, pre2;
  double t1_val, t1_err;
  double t2_val, t2_err;

  int stat_F1 = gsl_sf_hyperg_2F1_conj_e(0.25 - 0.5*mu, 0.5*tau, 0.5, x2, &F1);
  int stat_F2 = gsl_sf_hyperg_2F1_conj_e(0.75 - 0.5*mu, 0.5*tau, 1.5, x2, &F2);
  int status = GSL_ERROR_SELECT_2(stat_F1, stat_F2);

  gsl_sf_lngamma_complex_e(0.75 - 0.5*mu, -0.5*tau, &ln_g1, &arg_g1);
  gsl_sf_lngamma_complex_e(0.25 - 0.5*mu, -0.5*tau, &ln_g2, &arg_g2);

  gsl_sf_exp_err_e(-2.0*ln_g1.val, 2.0*ln_g1.err, &pre1);
  gsl_sf_exp_err_e(-2.0*ln_g2.val, 2.0*ln_g2.err, &pre2);
  pre2.val *= -2.0*x;
  pre2.err *=  2.0*fabs(x);
  pre2.err +=  GSL_DBL_EPSILON * fabs(pre2.val);

  t1_val = pre1.val * F1.val;
  t1_err = fabs(pre1.val) * F1.err + pre1.err * fabs(F1.val);
  t2_val = pre2.val * F2.val;
  t2_err = fabs(pre2.val) * F2.err + pre2.err * fabs(F2.val);

  result->val  = pre_val * (t1_val + t2_val);
  result->err  = pre_val * (t1_err + t2_err);
  result->err += pre_err * fabs(t1_val + t2_val);
  result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

  return status;
}


/* P^{mu}_{-1/2 + I tau}
 * defining hypergeometric representation
 * [Abramowitz+Stegun, 8.1.2]
 * 1 < x < 3
 * effective for x near 1
 *
 */
#if 0
static
int
conicalP_def_hyperg(double mu, double tau, double x, double * result)
{
  double F;
  int stat_F = gsl_sf_hyperg_2F1_conj_renorm_e(0.5, tau, 1.0-mu, 0.5*(1.0-x), &F);
  *result = pow((x+1.0)/(x-1.0), 0.5*mu) * F;
  return stat_F;
}
#endif /* 0 */


/* P^{mu}_{-1/2 + I tau}  second hypergeometric representation
 * [Zhurina+Karmazina, (3.1)] 
 * -1 < x < 3
 * effective for x near 1
 *
 */
#if 0
static
int
conicalP_xnear1_hyperg_C(double mu, double tau, double x, double * result)
{
  double ln_pre, arg_pre;
  double ln_g1, arg_g1;
  double ln_g2, arg_g2;
  double F;

  int stat_F = gsl_sf_hyperg_2F1_conj_renorm_e(0.5+mu, tau, 1.0+mu, 0.5*(1.0-x), &F);

  gsl_sf_lngamma_complex_e(0.5+mu, tau, &ln_g1, &arg_g1);
  gsl_sf_lngamma_complex_e(0.5-mu, tau, &ln_g2, &arg_g2);

  ln_pre  = mu*M_LN2 - 0.5*mu*log(fabs(x*x-1.0)) + ln_g1 - ln_g2;
  arg_pre = arg_g1 - arg_g2;

  *result = exp(ln_pre) * F;
  return stat_F;
}
#endif /* 0 */


/* V0, V1 from Kolbig, m = 0
 */
static
int
conicalP_0_V(const double t, const double f, const double tau, const double sgn,
             double * V0, double * V1)
{
  double C[8];
  double T[8];
  double H[8];
  double V[12];
  int i;
  T[0] = 1.0;
  H[0] = 1.0;
  V[0] = 1.0;
  for(i=1; i<=7; i++) {
    T[i] = T[i-1] * t;
    H[i] = H[i-1] * (t*f);
  }
  for(i=1; i<=11; i++) {
    V[i] = V[i-1] * tau;
  }

  C[0] = 1.0;
  C[1] = (H[1]-1.0)/(8.0*T[1]);
  C[2] = (9.0*H[2] + 6.0*H[1] - 15.0 - sgn*8.0*T[2])/(128.0*T[2]);
  C[3] = 5.0*(15.0*H[3] + 27.0*H[2] + 21.0*H[1] - 63.0 - sgn*T[2]*(16.0*H[1]+24.0))/(1024.0*T[3]);
  C[4] = 7.0*(525.0*H[4] + 1500.0*H[3] + 2430.0*H[2] + 1980.0*H[1] - 6435.0
              + 192.0*T[4] - sgn*T[2]*(720.0*H[2]+1600.0*H[1]+2160.0)
              ) / (32768.0*T[4]);
  C[5] = 21.0*(2835.0*H[5] + 11025.0*H[4] + 24750.0*H[3] + 38610.0*H[2]
               + 32175.0*H[1] - 109395.0 + T[4]*(1984.0*H[1]+4032.0)
               - sgn*T[2]*(4800.0*H[3]+15120.0*H[2]+26400.0*H[1]+34320.0)
	       ) / (262144.0*T[5]);
  C[6] = 11.0*(218295.0*H[6] + 1071630.0*H[5] + 3009825.0*H[4] + 6142500.0*H[3]
               + 9398025.0*H[2] + 7936110.0*H[1] - 27776385.0
	       + T[4]*(254016.0*H[2]+749952.0*H[1]+1100736.0)
	       - sgn*T[2]*(441000.0*H[4] + 1814400.0*H[3] + 4127760.0*H[2]
	                 + 6552000.0*H[1] + 8353800.0 + 31232.0*T[4]
			 )
               ) / (4194304.0*T[6]);

  *V0 = C[0] + (-4.0*C[3]/T[1]+C[4])/V[4]
             + (-192.0*C[5]/T[3]+144.0*C[6]/T[2])/V[8]
             + sgn * (-C[2]/V[2]
	              + (-24.0*C[4]/T[2]+12.0*C[5]/T[1]-C[6])/V[6] 
	              + (-1920.0*C[6]/T[4])/V[10]
		      );
  *V1 = C[1]/V[1] + (8.0*(C[3]/T[2]-C[4]/T[1])+C[5])/V[5]
                  + (384.0*C[5]/T[4] - 768.0*C[6]/T[3])/V[9]
                  + sgn * ((2.0*C[2]/T[1]-C[3])/V[3]
		           + (48.0*C[4]/T[3]-72.0*C[5]/T[2] + 18.0*C[6]/T[1])/V[7]
		           + (3840.0*C[6]/T[5])/V[11]
		           );

  return GSL_SUCCESS;
}


/* V0, V1 from Kolbig, m = 1
 */
static
int
conicalP_1_V(const double t, const double f, const double tau, const double sgn,
             double * V0, double * V1)
{
  double Cm1;
  double C[8];
  double T[8];
  double H[8];
  double V[12];
  int i;
  T[0] = 1.0;
  H[0] = 1.0;
  V[0] = 1.0;
  for(i=1; i<=7; i++) {
    T[i] = T[i-1] * t;
    H[i] = H[i-1] * (t*f);
  }
  for(i=1; i<=11; i++) {
    V[i] = V[i-1] * tau;
  }

  Cm1  = -1.0;
  C[0] = 3.0*(1.0-H[1])/(8.0*T[1]);
  C[1] = (-15.0*H[2]+6.0*H[1]+9.0+sgn*8.0*T[2])/(128.0*T[2]);
  C[2] = 3.0*(-35.0*H[3] - 15.0*H[2] + 15.0*H[1] + 35.0 + sgn*T[2]*(32.0*H[1]+8.0))/(1024.0*T[3]);
  C[3] = (-4725.0*H[4] - 6300.0*H[3] - 3150.0*H[2] + 3780.0*H[1] + 10395.0
          -1216.0*T[4] + sgn*T[2]*(6000.0*H[2]+5760.0*H[1]+1680.0)) / (32768.0*T[4]);
  C[4] = 7.0*(-10395.0*H[5] - 23625.0*H[4] - 28350.0*H[3] - 14850.0*H[2]
              +19305.0*H[1] + 57915.0 - T[4]*(6336.0*H[1]+6080.0)
	      + sgn*T[2]*(16800.0*H[3] + 30000.0*H[2] + 25920.0*H[1] + 7920.0)
	      ) / (262144.0*T[5]);
  C[5] = (-2837835.0*H[6] - 9168390.0*H[5] - 16372125.0*H[4] - 18918900*H[3]
          -10135125.0*H[2] + 13783770.0*H[1] + 43648605.0
	  -T[4]*(3044160.0*H[2] + 5588352.0*H[1] + 4213440.0)
	  +sgn*T[2]*(5556600.0*H[4] + 14817600.0*H[3] + 20790000.0*H[2]
	             + 17297280.0*H[1] + 5405400.0 + 323072.0*T[4]
		     )
          ) / (4194304.0*T[6]);
  C[6] = 0.0;

  *V0 = C[0] + (-4.0*C[3]/T[1]+C[4])/V[4]
             + (-192.0*C[5]/T[3]+144.0*C[6]/T[2])/V[8]
             + sgn * (-C[2]/V[2]
	              + (-24.0*C[4]/T[2]+12.0*C[5]/T[1]-C[6])/V[6] 
		      );
  *V1 = C[1]/V[1] + (8.0*(C[3]/T[2]-C[4]/T[1])+C[5])/V[5]
                  + (384.0*C[5]/T[4] - 768.0*C[6]/T[3])/V[9]
                  + sgn * (Cm1*V[1] + (2.0*C[2]/T[1]-C[3])/V[3]
		           + (48.0*C[4]/T[3]-72.0*C[5]/T[2] + 18.0*C[6]/T[1])/V[7]
		           );

  return GSL_SUCCESS;
}



/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/

/* P^0_{-1/2 + I lambda}
 */
int
gsl_sf_conicalP_0_e(const double lambda, const double x, gsl_sf_result * result)
{
  /* CHECK_POINTER(result) */

  if(x <= -1.0) {
    DOMAIN_ERROR(result);
  }
  else if(x == 1.0) {
    result->val = 1.0;
    result->err = 0.0;
    return GSL_SUCCESS;
  }
  else if(lambda == 0.0) {
    gsl_sf_result K;
    int stat_K;
    if(x < 1.0) {
      const double th = acos(x);
      const double s  = sin(0.5*th);
      stat_K = gsl_sf_ellint_Kcomp_e(s, GSL_MODE_DEFAULT, &K);
      result->val  = 2.0/M_PI * K.val;
      result->err  = 2.0/M_PI * K.err;
      result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
      return stat_K;
    }
    else {
      const double xi = acosh(x);
      const double c  = cosh(0.5*xi);
      const double t  = tanh(0.5*xi);
      stat_K = gsl_sf_ellint_Kcomp_e(t, GSL_MODE_DEFAULT, &K);
      result->val  = 2.0/M_PI / c * K.val;
      result->err  = 2.0/M_PI / c * K.err;
      result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
      return stat_K;
    }
  }
  else if(   (x <= 0.0 && lambda < 1000.0)
          || (x <  0.1 && lambda < 17.0)
	  || (x <  0.2 && lambda < 5.0 )
    ) {
    return conicalP_xlt1_hyperg_A(0.0, lambda, x, result);
  }
  else if(   (x <= 0.2 && lambda < 17.0)
          || (x <= 1.5 && lambda < 20.0)
    ) {
    return gsl_sf_hyperg_2F1_conj_e(0.5, lambda, 1.0, (1.0-x)/2, result);
  }
  else if(1.5 < x && lambda < GSL_MAX(x,20.0)) {
    gsl_sf_result P;
    double lm;
    int stat_P = gsl_sf_conicalP_large_x_e(0.0, lambda, x,
                                              &P, &lm
                                              );
    int stat_e = gsl_sf_exp_mult_err_e(lm, 2.0*GSL_DBL_EPSILON * fabs(lm),
    					  P.val, P.err,
    					  result);
    return GSL_ERROR_SELECT_2(stat_e, stat_P);
  }
  else {
    double V0, V1;
    if(x < 1.0) {
      double th  = acos(x);
      double sth = sqrt(1.0-x*x);  /* sin(th) */
      gsl_sf_result I0, I1;
      int stat_I0 = gsl_sf_bessel_I0_scaled_e(th * lambda, &I0);
      int stat_I1 = gsl_sf_bessel_I1_scaled_e(th * lambda, &I1);
      int stat_I  = GSL_ERROR_SELECT_2(stat_I0, stat_I1);
      int stat_V  = conicalP_0_V(th, x/sth, lambda, -1.0, &V0, &V1);
      double bessterm = V0 * I0.val + V1 * I1.val;
      double besserr  = fabs(V0) * I0.err + fabs(V1) * I1.err;
      double arg1 = th*lambda;
      double sqts = sqrt(th/sth);
      int stat_e = gsl_sf_exp_mult_err_e(arg1, 4.0 * GSL_DBL_EPSILON * fabs(arg1),
                                            sqts * bessterm, sqts * besserr,
					    result);
      return GSL_ERROR_SELECT_3(stat_e, stat_V, stat_I);
    }
    else {
      double sh = sqrt(x-1.0)*sqrt(x+1.0);  /* sinh(xi)      */
      double xi = log(x + sh);              /* xi = acosh(x) */
      gsl_sf_result J0, J1;
      int stat_J0 = gsl_sf_bessel_J0_e(xi * lambda, &J0);
      int stat_J1 = gsl_sf_bessel_J1_e(xi * lambda, &J1);
      int stat_J  = GSL_ERROR_SELECT_2(stat_J0, stat_J1);
      int stat_V  = conicalP_0_V(xi, x/sh, lambda, 1.0, &V0, &V1);
      double bessterm = V0 * J0.val + V1 * J1.val;
      double besserr  = fabs(V0) * J0.err + fabs(V1) * J1.err;
      double pre_val = sqrt(xi/sh);
      double pre_err = 2.0 * fabs(pre_val);
      result->val  = pre_val * bessterm;
      result->err  = pre_val * besserr;
      result->err += pre_err * fabs(bessterm);
      result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
      return GSL_ERROR_SELECT_2(stat_V, stat_J);
    }
  }
}


/* P^1_{-1/2 + I lambda}
 */
int
gsl_sf_conicalP_1_e(const double lambda, const double x, gsl_sf_result * result)
{
  /* CHECK_POINTER(result) */

  if(x <= -1.0) {
    DOMAIN_ERROR(result);
  }
  else if(lambda == 0.0) {
    gsl_sf_result K, E;
    int stat_K, stat_E;
    if(x == 1.0) {
      result->val = 0.0;
      result->err = 0.0;
      return GSL_SUCCESS;
    }
    else if(x < 1.0) {
      if(1.0-x < GSL_SQRT_DBL_EPSILON) {
        double err_amp = GSL_MAX_DBL(1.0, 1.0/(GSL_DBL_EPSILON + fabs(1.0-x)));
        result->val = 0.25/M_SQRT2 * sqrt(1.0-x) * (1.0 + 5.0/16.0 * (1.0-x));
	result->err = err_amp * 3.0 * GSL_DBL_EPSILON * fabs(result->val);
	return GSL_SUCCESS;
      }
      else {
        const double th = acos(x);
        const double s  = sin(0.5*th);
        const double c2 = 1.0 - s*s;
        const double sth = sin(th);
	const double pre = 2.0/(M_PI*sth);
        stat_K = gsl_sf_ellint_Kcomp_e(s, GSL_MODE_DEFAULT, &K);
        stat_E = gsl_sf_ellint_Ecomp_e(s, GSL_MODE_DEFAULT, &E);
        result->val  = pre * (E.val - c2 * K.val);
	result->err  = pre * (E.err + fabs(c2) * K.err);
	result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
        return stat_K;
      }
    }
    else {
      if(x-1.0 < GSL_SQRT_DBL_EPSILON) {
        double err_amp = GSL_MAX_DBL(1.0, 1.0/(GSL_DBL_EPSILON + fabs(1.0-x)));
        result->val = -0.25/M_SQRT2 * sqrt(x-1.0) * (1.0 - 5.0/16.0 * (x-1.0));
	result->err = err_amp * 3.0 * GSL_DBL_EPSILON * fabs(result->val);
	return GSL_SUCCESS;
      }
      else {
        const double xi = acosh(x);
        const double c  = cosh(0.5*xi);
        const double t  = tanh(0.5*xi);
        const double sxi = sinh(xi);
	const double pre = 2.0/(M_PI*sxi) * c;
        stat_K = gsl_sf_ellint_Kcomp_e(t, GSL_MODE_DEFAULT, &K);
        stat_E = gsl_sf_ellint_Ecomp_e(t, GSL_MODE_DEFAULT, &E);