math_funcs.cc

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/* Copyright (C) 2002-2012
   Patrick Eriksson <Patrick.Eriksson@chalmers.se>
   Stefan Buehler   <sbuehler@ltu.se>
 
   This program is free software; you can redistribute it and/or modify it
   under the terms of the GNU General Public License as published by the
   Free Software Foundation; either version 2, or (at your option) any
   later version.
 
   This program is distributed in the hope that it will be useful,
   but WITHOUT ANY WARRANTY; without even the implied warranty of
   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
   GNU General Public License for more details.
 
   You should have received a copy of the GNU General Public License
   along with this program; if not, write to the Free Software
   Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307,
   USA. */
 
/*****************************************************************************
 ***  File description 
 *****************************************************************************/
 
/*****************************************************************************
 *** External declarations
 *****************************************************************************/
 
#include "math_funcs.h"
#include <cmath>
#include <iostream>
#include <stdexcept>
#include "array.h"
#include "arts_constants.h"
#include "arts_conversions.h"
#include "logic.h"
#include "mystring.h"
 
inline constexpr Numeric DEG2RAD=Conversion::deg2rad(1);
inline constexpr Numeric PI=Constant::pi;
 
/*****************************************************************************
 *** The functions (in alphabetical order)
 *****************************************************************************/
 
 
Numeric fac(const Index n) {
  Numeric sum;
 
  if (n == 0) return (1.0);
 
  sum = 1.0;
  for (Index i = 1; i <= n; i++) sum *= Numeric(i);
 
  return (sum);
}
 
 
Index integer_div(const Index& x, const Index& y) {
  ARTS_ASSERT(is_multiple(x, y));
  return x / y;
}
 
 
Numeric LagrangeInterpol4(ConstVectorView x,
                          ConstVectorView y,
                          const Numeric a) {
  // lowermost grid spacing on x-axis
  const Numeric Dlimit = 1.00000e-15;
 
  // Check that dimensions of x and y vector agree
  const Index n_x = x.nelem();
  const Index n_y = y.nelem();
  ARTS_USER_ERROR_IF ((n_x != 4) || (n_y != 4),
      "The vectors x and y must all have the same length of 4 elements!\n"
      "Actual lengths:\n"
      "x:", n_x, ", "
      "y:", n_y, ".")
 
  // assure that x1 =< a < x2 holds
  ARTS_USER_ERROR_IF ((a < x[1]) || (a > x[2]),
    "LagrangeInterpol4: the relation x[1] =< a < x[2] is not satisfied. "
    "No interpolation can be calculated.\n")
 
  // calculate the Lagrange polynomial coefficients for a polynomial of the order of 3
  Numeric b[4];
  for (Index i = 0; i < 4; ++i) {
    b[i] = 1.000e0;
    for (Index k = 0; k < 4; ++k) {
      if ((k != i) && (fabs(x[i] - x[k]) > Dlimit))
        b[i] = b[i] * ((a - x[k]) / (x[i] - x[k]));
    };
  };
 
  Numeric ya = 0.000e0;
  for (Index i = 0; i < n_x; ++i) ya = ya + b[i] * y[i];
 
  return ya;
}
 
 
Numeric last(ConstVectorView x) {
  ARTS_ASSERT(x.nelem() > 0);
  return x[x.nelem() - 1];
}
 
 
Index last(const ArrayOfIndex& x) {
  ARTS_ASSERT(x.nelem() > 0);
  return x[x.nelem() - 1];
}
 
 
void linspace(Vector& x,
              const Numeric start,
              const Numeric stop,
              const Numeric step) {
  Index n = (Index)floor((stop - start) / step) + 1;
  if (n < 1) n = 1;
  x.resize(n);
  for (Index i = 0; i < n; i++) x[i] = start + (double)i * step;
}
 
 
void nlinspace(Vector& x,
               const Numeric start,
               const Numeric stop,
               const Index n) {
  ARTS_ASSERT(1 < n);  // Number of points must be greater 1.
  x.resize(n);
  Numeric step = (stop - start) / ((double)n - 1);
  for (Index i = 0; i < n - 1; i++) x[i] = start + (double)i * step;
  x[n - 1] = stop;
}
void nlinspace(VectorView x,
               const Numeric start,
               const Numeric stop,
               const Index n) {
  Numeric step = (stop - start) / ((double)n - 1);
  for (Index i = 0; i < n - 1; i++) x[i] = start + (double)i * step;
  x[n - 1] = stop;
}
 
 
void nlogspace(Vector& x,
               const Numeric start,
               const Numeric stop,
               const Index n) {
  // Number of points must be greater than 1:
  ARTS_ASSERT(1 < n);
  // Only positive numbers are allowed for start and stop:
  ARTS_ASSERT(0 < start);
  ARTS_ASSERT(0 < stop);
 
  x.resize(n);
  Numeric a = log(start);
  Numeric step = (log(stop) - a) / ((double)n - 1);
  x[0] = start;
  for (Index i = 1; i < n - 1; i++) x[i] = exp(a + (double)i * step);
  x[n - 1] = stop;
}
 
 
Numeric trapz(ConstVectorView x,
              ConstVectorView y)
{
  const Index n = x.nelem();
  ARTS_ASSERT(y.nelem() == n);
  Numeric sum = 0.0;
  for (Index i=1; i<n; ++i)
    sum += (x[i]-x[i-1]) * (y[i]+y[i-1]);
  return sum/2.0;
}
 
 
void cumsum(VectorView csum,
            ConstVectorView x)
{
  const Index n = x.nelem();
  ARTS_ASSERT(csum.nelem() == n);
  csum[0] = x[0];
  for (Index i=1; i<n; ++i)
    csum[i] = csum[i-1] + x[i];
}
 
 
Numeric AngIntegrate_trapezoid(ConstMatrixView Integrand,
                               ConstVectorView za_grid,
                               ConstVectorView aa_grid) {
  Index n = za_grid.nelem();
  Index m = aa_grid.nelem();
  Vector res1(n);
  ARTS_ASSERT(is_size(Integrand, n, m));
 
  for (Index i = 0; i < n; ++i) {
    res1[i] = 0.0;
 
    for (Index j = 0; j < m - 1; ++j) {
      res1[i] += 0.5 * DEG2RAD * (Integrand(i, j) + Integrand(i, j + 1)) *
                 (aa_grid[j + 1] - aa_grid[j]) * sin(za_grid[i] * DEG2RAD);
    }
  }
  Numeric res = 0.0;
  for (Index i = 0; i < n - 1; ++i) {
    res +=
        0.5 * DEG2RAD * (res1[i] + res1[i + 1]) * (za_grid[i + 1] - za_grid[i]);
  }
 
  return res;
}
 
 
Numeric AngIntegrate_trapezoid_opti(ConstMatrixView Integrand,
                                    ConstVectorView za_grid,
                                    ConstVectorView aa_grid,
                                    ConstVectorView grid_stepsize) {
  Numeric res = 0;
  if ((grid_stepsize[0] > 0) && (grid_stepsize[1] > 0)) {
    Index n = za_grid.nelem();
    Index m = aa_grid.nelem();
    Numeric stepsize_za = grid_stepsize[0];
    Numeric stepsize_aa = grid_stepsize[1];
    Vector res1(n);
    ARTS_ASSERT(is_size(Integrand, n, m));
 
    Numeric temp = 0.0;
 
    for (Index i = 0; i < n; ++i) {
      temp = Integrand(i, 0);
      for (Index j = 1; j < m - 1; j++) {
        temp += Integrand(i, j) * 2;
      }
      temp += Integrand(i, m - 1);
      temp *= 0.5 * DEG2RAD * stepsize_aa * sin(za_grid[i] * DEG2RAD);
      res1[i] = temp;
    }
 
    res = res1[0];
    for (Index i = 1; i < n - 1; i++) {
      res += res1[i] * 2;
    }
    res += res1[n - 1];
    res *= 0.5 * DEG2RAD * stepsize_za;
  } else {
    res = AngIntegrate_trapezoid(Integrand, za_grid, aa_grid);
  }
 
  return res;
}
 
 
Numeric AngIntegrate_trapezoid(ConstVectorView Integrand,
                               ConstVectorView za_grid) {
  Index n = za_grid.nelem();
  ARTS_ASSERT(is_size(Integrand, n));
 
  Numeric res = 0.0;
  for (Index i = 0; i < n - 1; ++i) {
    // in this place 0.5 * 2 * PI is calculated:
    res += PI * DEG2RAD *
           (Integrand[i] * sin(za_grid[i] * DEG2RAD) +
            Integrand[i + 1] * sin(za_grid[i + 1] * DEG2RAD)) *
           (za_grid[i + 1] - za_grid[i]);
  }
 
  return res;
}
 
 
Numeric sign(const Numeric& x) {
  if (x < 0)
    return -1.0;
  else if (x == 0)
    return 0.0;
  else
    return 1.0;
}
 
void mgd(VectorView psd,
         const Vector& x,
         const Numeric& n0,
         const Numeric& mu,
         const Numeric& la,
         const Numeric& ga) {
  const Index nx = x.nelem();
 
  ARTS_ASSERT(psd.nelem() == nx);
 
  if (ga == 1) {
    if (mu == 0) {
      // Exponential distribution
      for (Index ix = 0; ix < nx; ix++) {
        const Numeric eterm = exp(-la * x[ix]);
        psd[ix] = n0 * eterm;
      }
    } else {
      ARTS_USER_ERROR_IF (mu > 10,
          "Given mu is ", mu, '\n',
          "Seems unreasonable. Have you mixed up the inputs?")
      // Gamma distribution
      for (Index ix = 0; ix < nx; ix++) {
        const Numeric eterm = exp(-la * x[ix]);
        const Numeric xterm = pow(x[ix], mu);
        psd[ix] = n0 * xterm * eterm;
        psd[ix] = n0 * pow(x[ix], mu) * exp(-la * x[ix]);
      }
    }
  } else {
    // Complete MGD
    ARTS_USER_ERROR_IF (mu > 10,
        "Given mu is ", mu, '\n',
        "Seems unreasonable. Have you mixed up the inputs?")
    ARTS_USER_ERROR_IF (ga > 10,
        "Given gamma is ", ga, '\n',
        "Seems unreasonable. Have you mixed up the inputs?")
    for (Index ix = 0; ix < nx; ix++) {
      const Numeric pterm = pow(x[ix], ga);
      const Numeric eterm = exp(-la * pterm);
      const Numeric xterm = pow(x[ix], mu);
      psd[ix] = n0 * xterm * eterm;
    }
  }
}
 
void mgd_with_derivatives(VectorView psd,
                          MatrixView jac_data,
                          const Vector& x,
                          const Numeric& n0,
                          const Numeric& mu,
                          const Numeric& la,
                          const Numeric& ga,
                          const bool& do_n0_jac,
                          const bool& do_mu_jac,
                          const bool& do_la_jac,
                          const bool& do_ga_jac) {
  const Index nx = x.nelem();
 
  ARTS_ASSERT(psd.nelem() == nx);
  ARTS_ASSERT(jac_data.nrows() == 4);
  ARTS_ASSERT(jac_data.ncols() == nx);
 
  if (ga == 1 && !do_ga_jac) {
    if (mu == 0 && !do_mu_jac) {
      // Exponential distribution
      for (Index ix = 0; ix < nx; ix++) {
        const Numeric eterm = exp(-la * x[ix]);
        psd[ix] = n0 * eterm;
        if (do_n0_jac) {
          jac_data(0, ix) = eterm;
        }
        if (do_la_jac) {
          jac_data(2, ix) = -x[ix] * psd[ix];
        }
      }
    } else {
      ARTS_USER_ERROR_IF (mu > 10,
          "Given mu is ", mu, '\n',
          "Seems unreasonable. Have you mixed up the inputs?")
      // Gamma distribution
      for (Index ix = 0; ix < nx; ix++) {
        const Numeric eterm = exp(-la * x[ix]);
        const Numeric xterm = pow(x[ix], mu);
        psd[ix] = n0 * xterm * eterm;
        if (do_n0_jac) {
          jac_data(0, ix) = xterm * eterm;
        }
        if (do_mu_jac) {
          jac_data(1, ix) = log(x[ix]) * psd[ix];
        }
        if (do_la_jac) {
          jac_data(2, ix) = -x[ix] * psd[ix];
        }
        psd[ix] = n0 * pow(x[ix], mu) * exp(-la * x[ix]);
      }
    }
  } else {
    // Complete MGD
    ARTS_USER_ERROR_IF (mu > 10,
        "Given mu is ", mu, '\n',
        "Seems unreasonable. Have you mixed up the inputs?")
    ARTS_USER_ERROR_IF (ga > 10,
        "Given gamma is ", ga, '\n',
        "Seems unreasonable. Have you mixed up the inputs?")
    for (Index ix = 0; ix < nx; ix++) {
      const Numeric pterm = pow(x[ix], ga);
      const Numeric eterm = exp(-la * pterm);
      const Numeric xterm = pow(x[ix], mu);
      psd[ix] = n0 * xterm * eterm;
      if (do_n0_jac) {
        jac_data(0, ix) = xterm * eterm;
      }
      if (do_mu_jac) {
        jac_data(1, ix) = log(x[ix]) * psd[ix];
      }
      if (do_la_jac) {
        jac_data(2, ix) = -pterm * psd[ix];
      }
      if (do_ga_jac) {
        jac_data(3, ix) = -la * pterm * log(x[ix]) * psd[ix];
      }
    }
  }
}
 
void delanoe_shape_with_derivative(VectorView psd,
                                   MatrixView jac_data,
                                   const Vector& x,
                                   const Numeric& alpha,
                                   const Numeric& beta) {
  Numeric f_c = tgamma(4.0) / 256.0;
  f_c *= pow(tgamma((alpha + 5.0) / beta), 4 + alpha);
  f_c /= pow(tgamma((alpha + 4.0) / beta), 5 + alpha);
 
  Numeric f_d = tgamma((alpha + 5.0) / beta);
  f_d /= tgamma((alpha + 4.0) / beta);
 
  for (Index i = 0; i < x.nelem(); ++i) {
    Numeric xi = x[i];
    psd[i] = beta * f_c * pow(xi, alpha) * exp(-pow(f_d * xi, beta));
    jac_data(0, i) =
        psd[i] * (alpha / xi - beta * f_d * pow(f_d * xi, beta - 1.0));
  }
}
 
 
Numeric mod_gamma_dist(
    Numeric x, Numeric N0, Numeric Lambda, Numeric mu, Numeric gamma) {
  Numeric dN;
 
  if (x > 0. && N0 > 0. && Lambda > 0. && (mu + 1) / gamma > 0.) {
    //Distribution function
    dN = N0 * pow(x, mu) * exp(-Lambda * pow(x, gamma));
 
    return dN;
  } else {
    ARTS_USER_ERROR (
      "At least one argument is zero or negative.\n"
      "Modified gamma distribution can not be calculated.\n"
      "x      = ", x, "\n"
      "N0     = ", N0, "\n"
      "lambda = ", Lambda, "\n"
      "mu     = ", mu, "\n"
      "gamma  = ", gamma, "\n")
  }
}
 
 
void unitl(Vector& x) {
  ARTS_ASSERT(x.nelem() > 0);
 
  const Numeric l = sqrt(x * x);
  for (Index i = 0; i < x.nelem(); i++) x[i] /= l;
}
 
 
void flat(VectorView x, ConstMatrixView X) {
  ARTS_ASSERT(x.nelem() == X.nrows() * X.ncols());
 
  Index i = 0;
 
  for (Index c = 0; c < X.ncols(); c++) {
    for (Index r = 0; r < X.nrows(); r++) {
      x[i] = X(r, c);
      i += 1;
    }
  }
}
 
 
void flat(VectorView x, ConstTensor3View X) {
  ARTS_ASSERT(x.nelem() == X.nrows() * X.ncols() * X.npages());
 
  Index i = 0;
 
  for (Index c = 0; c < X.ncols(); c++) {
    for (Index r = 0; r < X.nrows(); r++) {
      for (Index p = 0; p < X.npages(); p++) {
        x[i] = X(p, r, c);
        i += 1;
      }
    }
  }
}
 
 
void reshape(Tensor3View X, ConstVectorView x) {
  ARTS_ASSERT(x.nelem() == X.nrows() * X.ncols() * X.npages());
 
  Index i = 0;
 
  for (Index c = 0; c < X.ncols(); c++) {
    for (Index r = 0; r < X.nrows(); r++) {
      for (Index p = 0; p < X.npages(); p++) {
        X(p, r, c) = x[i];
        i += 1;
      }
    }
  }
}
 
 
void reshape(MatrixView X, ConstVectorView x) {
  ARTS_ASSERT(x.nelem() == X.nrows() * X.ncols());
 
  Index i = 0;
 
  for (Index c = 0; c < X.ncols(); c++) {
    for (Index r = 0; r < X.nrows(); r++) {
      X(r, c) = x[i];
      i += 1;
    }
  }
}
 
 
void calculate_weights_linear(Vector& x, Vector& w, const Index nph) {
  Index N = nph * 2;
 
  //directions in total
  nlinspace(x, -1, 1, N);
 
  //allocate
  w.resize(x.nelem());
 
  // calculate weights
  w[0] = (x[1] - x[0]) / 2.;
 
  for (Index i = 1; i < nph * 2 - 1; i++) {
    w[i] = (x[i + 1] - x[i - 1]) / 2.;
  }
  w[x.nelem() - 1] = (x[x.nelem() - 1] - x[x.nelem() - 2]) / 2.;
}
 
void calculate_int_weights_arbitrary_grid(Vector& w, const Vector& x) {
 
  ARTS_USER_ERROR_IF(x.nelem() <1, "Grid needs at least 2 points." );
 
  Index N = x.nelem();
 
  w.resize(N);
 
  w[0] = (x[1] - x[0])/2;
  if (N>2){
 
    for (Index i = 1; i < N-1; i++ ){
      w[i] = (x[i+1] - x[i-1])/2;
    }
 
  }
  w[N-1] = (x[N-1] - x[N-2])/2;
}