Numerics.hh

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#ifndef G_NUMERICS_H
#define G_NUMERICS_H
 
#include <array>
#include <functional>
#include <complex>
#include <vector>
 
#include "Garfield/FundamentalConstants.hh"
 
namespace Garfield {
 
/// Collection of numerical routines.
namespace Numerics {
 
constexpr std::array<double, 2> GaussLegendreNodes2() {
  return {-0.577350269189625765, 0.577350269189625765};
}
constexpr std::array<double, 2> GaussLegendreNodes2() {…}
constexpr std::array<double, 2> GaussLegendreWeights2() {
  return {1., 1.};
}
constexpr std::array<double, 2> GaussLegendreWeights2() {…}
constexpr std::array<double, 3> GaussLegendreNodes3() {
  return {-0.774596669241483377, 0., 0.774596669241483377};
}
constexpr std::array<double, 3> GaussLegendreNodes3() {…}
constexpr std::array<double, 3> GaussLegendreWeights3() {
  return {5. / 9., 8. / 9., 5. / 9.};
}
constexpr std::array<double, 3> GaussLegendreWeights3() {…}
constexpr std::array<double, 4> GaussLegendreNodes4() {
  return {-0.861136311594052575, -0.339981043584856265,
           0.339981043584856265,  0.861136311594052575};
}
constexpr std::array<double, 4> GaussLegendreNodes4() {…}
constexpr std::array<double, 4> GaussLegendreWeights4() {
  return {0.347854845137453857, 0.652145154862546143,
          0.652145154862546143, 0.347854845137453857}; 
}
constexpr std::array<double, 4> GaussLegendreWeights4() {…}
constexpr std::array<double, 5> GaussLegendreNodes5() {
  return {-0.906179845938663993, -0.538469310105683091, 0.,
           0.538469310105683091,  0.906179845938663993};
}
constexpr std::array<double, 5> GaussLegendreNodes5() {…}
constexpr std::array<double, 5> GaussLegendreWeights5() {
  return {0.236926885056189088, 0.478628670499366468, 128. / 225.,
          0.478628670499366468, 0.236926885056189088};
}
constexpr std::array<double, 5> GaussLegendreWeights5() {…}
constexpr std::array<double, 6> GaussLegendreNodes6() {
  return {-0.932469514203152028, -0.661209386466264514,
          -0.238619186083196909,  0.238619186083196909,
           0.661209386466264514,  0.932469514203152028};
}
constexpr std::array<double, 6> GaussLegendreNodes6() {…}
constexpr std::array<double, 6> GaussLegendreWeights6() {
  return {0.171324492379170345, 0.360761573048138608,
          0.467913934572691047, 0.467913934572691047,
          0.360761573048138608, 0.171324492379170345};
}
constexpr std::array<double, 6> GaussLegendreWeights6() {…}
 
/// Functions for performing numerical integration (quadrature). 
/// Reimplemented from %QUADPACK.
///  
/// R. Piessens, E. de Doncker-Kapenger, C. Ueberhuber, D. Kahaner,
/// %QUADPACK, a Subroutine Package for Automatic Integration,
/// Springer, 1983
namespace QUADPACK {
 
/// Estimates an integral over a semi-infinite or infinite interval.
/// Calculates an approximation to an integral
/// \f[ I = \int_{a}^{\infty} f\left(x\right) dx, \f] 
/// or 
/// \f[ I = \int_{-\infty}^{a} f\left(x\right) dx, \f] 
/// or 
/// \f[ I = \int_{-\infty}^{\infty} f\left(x\right) dx \f] 
/// hopefully satisfying
/// \f[
///     \left| I - \mathrm{RESULT} \right| \leq
///     \max(\varepsilon_{\mathrm{abs}}, \varepsilon_{\mathrm{rel}} \left|I\right|).
/// \f]
/// \param f function to be integrated.
/// \param bound value of the finite endpoint of the integration range (if any).
/// \param inf indicates the type of integration range.
///   -  1: (    bound, +Infinity)
///   - -1: (-Infinity,     bound)
///   -  2: (-Infinity, +Infinity)
/// \param epsabs requested absolute accuracy
/// \param epsrel requested relative accuracy
/// \param result the estimated value of the integral 
/// \param abserr estimated error
/// \param status error flag
///   -   0: normal and reliable termination, requested accuracy
///           has been achieved.
///   - > 0: abnormal termination, estimates for result and error
///           are less reliable.
///     - 1: maximum number of subdivisions reached.
///     - 2: occurance of roundoff error prevents the requested 
///          tolerance from being achieved. Error may be underestimated.
///     - 3: extremely bad integrand behaviour at some points of the 
///          integration interval.
///     - 4: algorithm does not converge, roundoff error is detected in
///          the extrapolation table. It is assumed that the requested
///          tolerance cannot be achieved and that the returned result
///          is the best that can be obtained.
///     - 5: integral is probably divergent, or slowly convergent.
///     - 6: invalid input.  
void qagi(std::function<double(double)> f, double bound, const int inf, 
          const double epsabs, const double epsrel, 
          double& result, double& abserr, unsigned int& status);
 
/// 15-point Gauss-Kronrod integration with (semi-)infinite integration range.
/// \param f function to be integrated.
/// \param bound finite bound of original integration range (0 if inf = 2).
/// \param inf indicates the type of integration range.
/// \param a lower limit for integration over subrange of (0, 1).
/// \param b upper limit for integration over subrange of (0, 1).
/// \param result approximation to the integral.
/// \param abserr estimate of the modulus of the absolute error.
/// \param resabs approximation to the integral over \f$\left|f\right|\f$.
/// \param resasc approximation to the integral of 
///               \f$\left|f - I / (b-a)\right|\f$ over \f$(a,b)\f$.
void qk15i(std::function<double(double)> f, double bound, const int inf,
           const double a, const double b, double& result, double& abserr,
           double& resabs, double& resasc);
 
/// 15-point Gauss-Kronrod integration with finite integration range.
void qk15(std::function<double(double)> f, const double a, const double b,
          double& result, double& abserr, double& resabs, double& resasc);
 
}
namespace QUADPACK {…}
 
/// Linear algebra routines from CERNLIB.
namespace CERNLIB {
 
/// Replaces b by the solution x of Ax = b, after which A is undefined.
/// \param n order of the square matrix A.
/// \param a n by n matrix.
/// \param b right-hand side vector.
/// \returns 0: normal exit, -1: singular matrix.
int deqn(const int n, std::vector<std::vector<double> >& a,
         std::vector<double>& b);
/// Replaces b by the solution x of Ax = b, and replace A by its inverse.
int deqinv(const int n, std::vector<std::vector<double> >& a, 
           std::vector<double>& b);
 
void dfact(const int n, std::vector<std::vector<double> >& a,
           std::vector<int>& ir, int& ifail, double& det, int& jfail);
void dfeqn(const int n, std::vector<std::vector<double> >& a,
           std::vector<int>& ir, std::vector<double>& b);
void dfinv(const int n, std::vector<std::vector<double> >& a,
           std::vector<int>& ir);
/// Replace square matrix A by its inverse.
int dinv(const int n, std::vector<std::vector<double> >& a);
 
void cfact(const int n, std::vector<std::vector<std::complex<double> > >& a,
           std::vector<int>& ir, int& ifail, std::complex<double>& det,
           int& jfail);
void cfinv(const int n, std::vector<std::vector<std::complex<double> > >& a,
           std::vector<int>& ir);
 
/// Replace square matrix A by its inverse.
int cinv(const int n, std::vector<std::vector<std::complex<double> > >& a);
 
void cfft(std::vector<std::complex<double> >& a, const int msign);
}
namespace CERNLIB {…}
 
/// Legendre polynomials.
inline double Legendre(const unsigned int n, const double x) {
 
  if (std::abs(x) > 1.) return 0.;
  double p0 = 1.;
  double p1 = x;
  if (n == 0) return p0;
  if (n == 1) return p1;
  for (unsigned int k = 1; k < n; ++k) {
    p0 = ((2 * k + 1) * x * p1 - k * p0) / (k + 1);
    std::swap(p0, p1);
  }
  return p1;
}
inline double Legendre(const unsigned int n, const double x) {…}
 
/// Modified Bessel functions.
/// Series expansions from Abramowitz and Stegun.
inline double BesselI0S(const double xx) {
  const double y = xx / 3.75;
  const double y2 = y * y;
  return 1. + 3.5156229 * y2 + 3.0899424 * y2 * y2 +
         1.2067492 * pow(y2, 3) + 0.2659732 * pow(y2, 4) +
         0.0360768 * pow(y2, 5) + 0.0045813 * pow(y2, 6);
}
inline double BesselI0S(const double xx) {…}
 
inline double BesselI1S(const double xx) {
  const double y = xx / 3.75;
  const double y2 = y * y;
  return xx *
         (0.5 + 0.87890594 * y2 + 0.51498869 * y2 * y2 + 
          0.15084934 * pow(y2, 3) + 0.02658733 * pow(y2, 4) + 
          0.00301532 * pow(y2, 5) + 0.00032411 * pow(y2, 6));
}
inline double BesselI1S(const double xx) {…}
 
inline double BesselK0S(const double xx) {
  const double y = xx / 2.;
  const double y2 = y * y;
  return -log(y) * BesselI0S(xx) - Gamma +
         0.42278420 * y2 + 0.23069756 * y2 * y2 +
         0.03488590 * pow(y2, 3) + 0.00262698 * pow(y2, 4) +
         0.00010750 * pow(y2, 5) + 0.00000740 * pow(y2, 6);
}
inline double BesselK0S(const double xx) {…}
 
inline double BesselK0L(const double xx) {
  const double y = 2. / xx;
  return (exp(-xx) / sqrt(xx)) *
         (1.25331414 - 0.07832358 * y + 0.02189568 * y * y -
          0.01062446 * pow(y, 3) + 0.00587872 * pow(y, 4) -
          0.00251540 * pow(y, 5) + 0.00053208 * pow(y, 6));
}
inline double BesselK0L(const double xx) {…}
 
inline double BesselK1S(const double xx) {
  const double y = xx / 2.;
  const double y2 = y * y;
  return log(y) * BesselI1S(xx) +
         (1. / xx) *
             (1. + 0.15443144 * y2 - 0.67278579 * y2 * y2 -
              0.18156897 * pow(y2, 3) - 0.01919402 * pow(y2, 4) -
              0.00110404 * pow(y2, 5) - 0.00004686 * pow(y2, 6));
}
inline double BesselK1S(const double xx) {…}
 
inline double BesselK1L(const double xx) {
  const double y = 2. / xx;
  return (exp(-xx) / sqrt(xx)) *
         (1.25331414 + 0.23498619 * y - 0.03655620 * y * y +
          0.01504268 * pow(y, 3) - 0.00780353 * pow(y, 4) +
          0.00325614 * pow(y, 5) - 0.00068245 * pow(y, 6));
}
inline double BesselK1L(const double xx) {…}
 
/// C++ version of DIVDIF (CERN program library E105) which performs
/// tabular interpolation using symmetrically placed argument points.
double Divdif(const std::vector<double>& f, const std::vector<double>& a,
              int nn, double x, int mm);
 
/// Interpolation of order 1 and 2 in an irregular rectangular
/// two-dimensional grid.
bool Boxin2(const std::vector<std::vector<double> >& value,
            const std::vector<double>& xAxis, const std::vector<double>& yAxis,
            const int nx, const int ny, const double xx, const double yy,
            double& f, const int iOrder);
/// Interpolation of order 1 and 2 in an irregular rectangular 
/// three-dimensional grid.
bool Boxin3(const std::vector<std::vector<std::vector<double> > >& value,
            const std::vector<double>& xAxis, const std::vector<double>& yAxis,
            const std::vector<double>& zAxis, const int nx, const int ny,
            const int nz, const double xx, const double yy, const double zz,
            double& f, const int iOrder);
 
/// Least-squares minimisation.
bool LeastSquaresFit(
    std::function<double(double, const std::vector<double>&)> f, 
    std::vector<double>& par, std::vector<double>& epar,
    const std::vector<double>& x, const std::vector<double>& y,
    const std::vector<double>& ey, const unsigned int nMaxIter,
    const double diff, double& chi2, const double eps, 
    const bool debug, const bool verbose);
 
}
namespace Numerics {…}
}
 
#endif