test_integration.cc

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/* Copyright (C) 2003-2012 Claas Teichmann <claas@sat.physik.uni-bremen.de>
                      
   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. 
   
*/
 
#include <sys/time.h>
#include <cmath>
#include <iostream>
#include <stdexcept>
 
#include "array.h"
#include "arts.h"
#include "logic.h"
#include "math_funcs.h"
#include "matpackI.h"
 
extern const Numeric DEG2RAD;
extern const Numeric PI;
 
void init_xy(float stepsize,
             int frequency,
             Matrix& Integrand,
             Vector& za_grid,
             Vector& aa_grid);
 
void init_x(
    int vsize, float stepsize, int frequency, Vector& Integrand, Vector& Theta);
 
Numeric AngIntegrate_trapezoid_original(MatrixView Integrand,
                                        ConstVectorView za_grid,
                                        ConstVectorView aa_grid);
 
Numeric AngIntegrate_trapezoid_opt(MatrixView Integrand,
                                   ConstVectorView za_grid,
                                   ConstVectorView aa_grid);
 
Numeric AngIntegrate_trapezoid_fixedstep(MatrixView Integrand,
                                         ConstVectorView za_grid,
                                         ConstVectorView aa_grid,
                                         Numeric stepsize);
 
Numeric AngIntegrate_trapezoid_fixedstep_opt(MatrixView Integrand,
                                             ConstVectorView za_grid,
                                             ConstVectorView aa_grid,
                                             Numeric stepsize);
 
Numeric AngIntegrate_trapezoid_fixedstep_opt2(MatrixView Integrand,
                                              ConstVectorView za_grid,
                                              ConstVectorView aa_grid,
                                              Numeric stepsize);
 
Numeric AngIntegrate_trapezoid_original(ConstVectorView Integrand,
                                        ConstVectorView za_grid);
 
Numeric AngIntegrate_trapezoid_fixedstep(ConstVectorView Integrand,
                                         ConstVectorView za_grid,
                                         Numeric stepsize);
 
Numeric test_xy(int z_size, int a_size, float stepsize, int frequency);
 
Numeric test_xy_opt(int z_size, int a_size, float stepsize, int frequency);
 
Numeric test_xy_fixedstep(int z_size,
                          int a_size,
                          float stepsize,
                          int frequency);
 
Numeric test_xy_fixedstep_opt(int z_size,
                              int a_size,
                              float stepsize,
                              int frequency);
 
Numeric test_xy_fixedstep_opt2(int z_size,
                               int a_size,
                               float stepsize,
                               int frequency);
 
Numeric test_AngIntegrate_trapezoid_opti(int z_size,
                                         int a_size,
                                         float stepsize,
                                         int frequency);
 
Numeric test_x(int vsize, float stepsize, int frequency);
 
Numeric test_x_fixedstep(int vsize, int frequency);
 
int main(int argc, char* argv[]) {
  if (argc == 1) {
    cerr << argv[0] << " requires one parameter" << endl;
    exit(1);
  }
 
  cout << "Uebergabewert von argv : " << argv[1] << endl;
 
  int frequency = (int)strtol(argv[1], NULL, 10);  // Zahl zur Basis 10
  cout << "Wert von frequency    : " << frequency << endl;
 
  //  test_x(1801, 0.1, frequency);
  //  test_x_fixedstep(1801, frequency);
  //  test_x(181, 1, frequency);
  //  test_x(19, 10, frequency);
 
  test_xy(181, 361, 1.0, frequency);
  //  test_xy_opt(1801, 3601, 0.1, frequency);
  //  test_xy_fixedstep(1801, 3601, 0.1, frequency);
  //  test_xy_fixedstep_opt(1801, 3601, 0.1, frequency);
  //  test_xy_fixedstep_opt(181, 361, 1.0, frequency);
  //  test_xy_fixedstep_opt(19, 37, 10, frequency);
  //  test_xy_fixedstep_opt2(1801, 3601, 0.1, frequency);
  //  test_xy_fixedstep_opt2(181, 361, 1.0, frequency);
  test_AngIntegrate_trapezoid_opti(181, 361, 1.0, frequency);
  //  test_xy_fixedstep_opt2(19, 37, 10, frequency);
}
 
 
void init_x(int vsize,
            float stepsize,
            int frequency,
            Vector& Integrand,
            Vector& Theta) {
  cout << "----------------init_x---------------\n";
 
  //  Integrand.resize(vsize); // function to be integrated
  //  Theta.resize(vsize);     // Theta values
 
  for (int i = 0; i < Integrand.nelem(); i++)
    Integrand[i] = (float)i * stepsize;
 
  //Theta is between 0 and 180
  for (int i = 0; i < Theta.nelem(); i++) Theta[i] = (float)i * stepsize;
 
  cout << "function Y = X" << endl
       << "vsize = " << vsize << endl
       << "stepsize = " << stepsize << endl
       << "frequency = " << frequency << endl
       << "Integrand: von " << Integrand[0] << " bis "
       << Integrand[Integrand.nelem() - 1] << endl
       << "Theta: von " << Theta[0] << " bis " << Theta[Theta.nelem() - 1]
       << endl;
}
 
 
void init_xy(float stepsize,
             int frequency,
             Matrix& Integrand,
             Vector& za_grid,
             Vector& aa_grid) {
  cout << ">>>>>-----------init_xy---------------\n";
  Index n_za = za_grid.nelem();
  Index n_aa = aa_grid.nelem();
 
  // The r=1 so we get a circle
  for (Index i = 0; i < n_za; i++)
    for (Index j = 0; j < n_aa; j++) Integrand(i, j) = 1;
 
  //za_grid (Theta) is between 0 and 180
  for (Index i = 0; i < n_za; i++) za_grid[i] = (float)i * stepsize;
 
  //aa_grid (Phi) is between 0 and 360
  for (Index i = 0; i < n_aa; i++) aa_grid[i] = (float)i * stepsize;
 
  cout << "function x^2 + y^2 + z^2 = 1" << endl
       << "n_za = " << n_za << endl
       << "n_aa = " << n_aa << endl
       << "stepsize = " << stepsize << endl
       << "frequency = " << frequency << endl
       << "Integrand(*,0): von " << Integrand(0, 0) << " bis "
       << Integrand(n_za - 1, 0) << endl
       << "Integrand(0,*): von " << Integrand(0, 0) << " bis "
       << Integrand(0, n_aa - 1) << endl
       << "za_grid (Theta): von " << za_grid[0] << " bis "
       << za_grid[za_grid.nelem() - 1] << endl
       << "aa_grid (Phi)  : von " << aa_grid[0] << " bis "
       << aa_grid[aa_grid.nelem() - 1] << endl;
  cout << "---------------init_xy---------------<<<<<\n";
}
 
 
Numeric AngIntegrate_trapezoid_original(MatrixView 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]);
  }
 
  //cout<<res<<"\n";
  return res;
}
 
Numeric AngIntegrate_trapezoid_opt(MatrixView 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]);
  }
 
  //cout<<res<<"\n";
  return res;
}
 
 
Numeric AngIntegrate_trapezoid_fixedstep(MatrixView Integrand,
                                         ConstVectorView za_grid,
                                         ConstVectorView aa_grid,
                                         Numeric stepsize) {
  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)) *
                 stepsize * 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]) * stepsize;
  }
 
  //cout<<res<<"\n";
  return res;
}
 
 
Numeric AngIntegrate_trapezoid_fixedstep_opt(MatrixView Integrand,
                                             ConstVectorView za_grid,
                                             ConstVectorView aa_grid,
                                             Numeric stepsize) {
  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;
 
    res1[i] += Integrand(i, 0);
    for (Index j = 1; j < m - 1; j++) {
      res1[i] += Integrand(i, j) * 2;
    }
    res1[i] += Integrand(i, m - 1);
    res1[i] *= 0.5 * DEG2RAD * stepsize * sin(za_grid[i] * DEG2RAD);
  }
  Numeric res = 0.0;
  res += res1[0];
  for (Index i = 1; i < n - 1; i++) {
    res += res1[i] * 2;
  }
  res += res1[n - 1];
  res *= 0.5 * DEG2RAD * stepsize;
 
  //cout<<res<<"\n";
  return res;
}
 
 
Numeric AngIntegrate_trapezoid_fixedstep_opt2(MatrixView Integrand,
                                              ConstVectorView za_grid,
                                              ConstVectorView aa_grid,
                                              Numeric stepsize) {
  Index n = za_grid.nelem();
  Index m = aa_grid.nelem();
  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 * sin(za_grid[i] * DEG2RAD);
    res1[i] = temp;
  }
 
  Numeric res = res1[0];
  for (Index i = 1; i < n - 1; i++) {
    res += res1[i] * 2;
  }
  res += res1[n - 1];
  res *= 0.5 * DEG2RAD * stepsize;
 
  //cout<<res<<"\n";
  return res;
}
 
 
Numeric AngIntegrate_trapezoid_original(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]);
  }
 
  //cout<<res<<"\n";
  return res;
}
 
 
Numeric AngIntegrate_trapezoid_fixedstep(ConstVectorView Integrand,
                                         ConstVectorView za_grid,
                                         Numeric stepsize) {
  Index n = za_grid.nelem();
  ARTS_ASSERT(is_size(Integrand, n));
 
  Numeric res = 0.0;
  // cout << "Stepsize: " << stepsize << endl;
  res += (Integrand[0] * sin(za_grid[0] * DEG2RAD));
  for (Index i = 1; i < n - 1; ++i) {
    res += (Integrand[i] * sin(za_grid[i] * DEG2RAD) * 2);
    // cout << i << endl;
  }
  res += ((Integrand[n - 1] * sin(za_grid[n - 1] * DEG2RAD)));
  // cout << n-1 << endl;
  // normally ther would be a 2* here, but it's already in the equations above
  res *= PI * DEG2RAD * stepsize;
 
  //cout<<res<<"\n";
  return res;
}
 
 
Numeric test_xy(int z_size, int a_size, float stepsize, int frequency) {
  cout << ">>>>>-----------test_xy---------------\n";
  Matrix Integrand(z_size, a_size);  // function to be integrated
  Vector za_grid(z_size);            // zenith (Theta) values
  Vector aa_grid(a_size);            // azimuth (Phi) values
 
  init_xy(stepsize, frequency, Integrand, za_grid, aa_grid);
 
  Numeric result = 0;
 
  struct timeval start;
  struct timeval ende;
  gettimeofday(&start, NULL);
  //  cout << "Sekunden : " << start.tv_sec << endl
  //   << "Milisekunden: " << start.tv_usec << endl;
 
  for (int i = 0; i < frequency; i++)
    result = AngIntegrate_trapezoid_original(Integrand, za_grid, aa_grid);
 
  gettimeofday(&ende, NULL);
 
  double error = result / (4 * PI) - 1;
 
  double diffs = (double)(ende.tv_sec - start.tv_sec) +
                 (double)(ende.tv_usec - start.tv_usec) / 1000000.0;
  cout.precision(15);
  cout << "stepsize is    : " << stepsize << endl
       << "z_size         : " << z_size << endl
       << "a_size         : " << a_size << endl
       << "1 is           : " << result / (4 * PI) << endl
       << "The result is  : " << result << endl
       << "The error is   : " << error * 100 << " %\n"
       << "Number of loops: " << frequency << endl
       << "elapsed time   : " << diffs << "s" << endl
       << "----------------test_xy----------<<<<<\n";
 
  return result;
}
 
 
Numeric test_xy_opt(int z_size, int a_size, float stepsize, int frequency) {
  cout << ">>>>>-----------test_xy_opt---------------\n";
  Matrix Integrand(z_size, a_size);  // function to be integrated
  Vector za_grid(z_size);            // zenith (Theta) values
  Vector aa_grid(a_size);            // azimuth (Phi) values
 
  init_xy(stepsize, frequency, Integrand, za_grid, aa_grid);
 
  Numeric result = 0;
 
  struct timeval start;
  struct timeval ende;
  gettimeofday(&start, NULL);
  //  cout << "Sekunden : " << start.tv_sec << endl
  //   << "Milisekunden: " << start.tv_usec << endl;
 
  for (int i = 0; i < frequency; i++)
    result = AngIntegrate_trapezoid_opt(Integrand, za_grid, aa_grid);
 
  gettimeofday(&ende, NULL);
 
  double error = result / (4 * PI) - 1;
 
  double diffs = (double)(ende.tv_sec - start.tv_sec) +
                 (double)(ende.tv_usec - start.tv_usec) / 1000000.0;
  cout.precision(15);
  cout << "stepsize is    : " << stepsize << endl
       << "z_size         : " << z_size << endl
       << "a_size         : " << a_size << endl
       << "1 is           : " << result / (4 * PI) << endl
       << "The result is  : " << result << endl
       << "The error is   : " << error * 100 << " %\n"
       << "Number of loops: " << frequency << endl
       << "elapsed time   : " << diffs << "s" << endl
       << "----------------test_xy_opt----------<<<<<\n";
 
  return result;
}
 
 
Numeric test_xy_fixedstep(int z_size,
                          int a_size,
                          float stepsize,
                          int frequency) {
  cout << ">>>>>-----------test_xy_fixedstep---------------\n";
  Matrix Integrand(z_size, a_size);  // function to be integrated
  Vector za_grid(z_size);            // zenith (Theta) values
  Vector aa_grid(a_size);            // azimuth (Phi) values
 
  init_xy(stepsize, frequency, Integrand, za_grid, aa_grid);
 
  Numeric result = 0;
 
  struct timeval start;
  struct timeval ende;
  gettimeofday(&start, NULL);
  //  cout << "Sekunden : " << start.tv_sec << endl
  //   << "Milisekunden: " << start.tv_usec << endl;
 
  for (int i = 0; i < frequency; i++)
    result =
        AngIntegrate_trapezoid_fixedstep(Integrand, za_grid, aa_grid, stepsize);
 
  gettimeofday(&ende, NULL);
 
  double error = result / (4 * PI) - 1;
 
  double diffs = (double)(ende.tv_sec - start.tv_sec) +
                 (double)(ende.tv_usec - start.tv_usec) / 1000000.0;
  cout.precision(15);
  cout << diffs << endl;
  cout << "stepsize is    : " << stepsize << endl
       << "z_size         : " << z_size << endl
       << "a_size         : " << a_size << endl
       << "1 is           : " << result / (4 * PI) << endl
       << "The result is  : " << result << endl
       << "The error is   : " << error * 100 << " %\n"
       << "Number of loops: " << frequency << endl
       << "elapsed time   : " << diffs << "s" << endl
       << "----------------test_xy_fixedstep----------<<<<<\n";
 
  return result;
}
 
 
Numeric test_xy_fixedstep_opt(int z_size,
                              int a_size,
                              float stepsize,
                              int frequency) {
  cout << ">>>>>-----------test_xy_fixedstep_opt---------------\n";
  Matrix Integrand(z_size, a_size);  // function to be integrated
  Vector za_grid(z_size);            // zenith (Theta) values
  Vector aa_grid(a_size);            // azimuth (Phi) values
 
  init_xy(stepsize, frequency, Integrand, za_grid, aa_grid);
 
  Numeric result = 0;
 
  struct timeval start;
  struct timeval ende;
  gettimeofday(&start, NULL);
  //  cout << "Sekunden : " << start.tv_sec << endl
  //   << "Milisekunden: " << start.tv_usec << endl;
 
  for (int i = 0; i < frequency; i++)
    result = AngIntegrate_trapezoid_fixedstep_opt(
        Integrand, za_grid, aa_grid, stepsize);
 
  gettimeofday(&ende, NULL);
 
  double error = result / (4 * PI) - 1;
 
  double diffs = (double)(ende.tv_sec - start.tv_sec) +
                 (double)(ende.tv_usec - start.tv_usec) / 1000000.0;
  cout.precision(15);
  cout << diffs << endl;
  cout << "stepsize is    : " << stepsize << endl
       << "z_size         : " << z_size << endl
       << "a_size         : " << a_size << endl
       << "1 is           : " << result / (4 * PI) << endl
       << "The result is  : " << result << endl
       << "The error is   : " << error * 100 << " %\n"
       << "Number of loops: " << frequency << endl
       << "elapsed time   : " << diffs << "s" << endl
       << "----------------test_xy_fixedstep_opt----------<<<<<\n";
 
  return result;
}
 
 
Numeric test_xy_fixedstep_opt2(int z_size,
                               int a_size,
                               float stepsize,
                               int frequency) {
  cout << ">>>>>-----------test_xy_fixedstep_opt2---------------\n";
  Matrix Integrand(z_size, a_size);  // function to be integrated
  Vector za_grid(z_size);            // zenith (Theta) values
  Vector aa_grid(a_size);            // azimuth (Phi) values
 
  init_xy(stepsize, frequency, Integrand, za_grid, aa_grid);
 
  Numeric result = 0;
 
  struct timeval start;
  struct timeval ende;
  gettimeofday(&start, NULL);
  //  cout << "Sekunden : " << start.tv_sec << endl
  //   << "Milisekunden: " << start.tv_usec << endl;
 
  for (int i = 0; i < frequency; i++)
    result = AngIntegrate_trapezoid_fixedstep_opt2(
        Integrand, za_grid, aa_grid, stepsize);
 
  gettimeofday(&ende, NULL);
 
  double error = result / (4 * PI) - 1;
 
  double diffs = (double)(ende.tv_sec - start.tv_sec) +
                 (double)(ende.tv_usec - start.tv_usec) / 1000000.0;
  cout.precision(15);
  cout << diffs << endl;
  cout << "stepsize is    : " << stepsize << endl
       << "z_size         : " << z_size << endl
       << "a_size         : " << a_size << endl
       << "1 is           : " << result / (4 * PI) << endl
       << "The result is  : " << result << endl
       << "The error is   : " << error * 100 << " %\n"
       << "Number of loops: " << frequency << endl
       << "elapsed time   : " << diffs << "s" << endl
       << "----------------test_xy_fixedstep_opt2----------<<<<<\n";
 
  return result;
}
 
 
Numeric test_AngIntegrate_trapezoid_opti(int z_size,
                                         int a_size,
                                         float stepsize,
                                         int frequency) {
  cout << ">>>>>-----------test_AngIntegrate_trapezoid_opti---------------\n";
  Matrix Integrand(z_size, a_size);  // function to be integrated
  Vector za_grid(z_size);            // zenith (Theta) values
  Vector aa_grid(a_size);            // azimuth (Phi) values
 
  Vector grid_stepsize(2);
 
  init_xy(stepsize, frequency, Integrand, za_grid, aa_grid);
 
  grid_stepsize[0] = za_grid[1] - za_grid[0];
  grid_stepsize[1] = aa_grid[1] - aa_grid[0];
  //grid_stepsize[0] = -1;
  //grid_stepsize[1] = -1;
 
  //  cout << za_grid << endl;
  //  cout << grid_stepsize[0] << endl;
  //  cout << grid_stepsize[1] << endl;
 
  Numeric result = 0;
 
  struct timeval start;
  struct timeval ende;
  gettimeofday(&start, NULL);
  //  cout << "Sekunden : " << start.tv_sec << endl
  //   << "Milisekunden: " << start.tv_usec << endl;
 
  for (int i = 0; i < frequency; i++)
    result =
        AngIntegrate_trapezoid_opti(Integrand, za_grid, aa_grid, grid_stepsize);
 
  gettimeofday(&ende, NULL);
 
  double error = result / (4 * PI) - 1;
 
  double diffs = (double)(ende.tv_sec - start.tv_sec) +
                 (double)(ende.tv_usec - start.tv_usec) / 1000000.0;
  cout.precision(15);
  cout << diffs << endl;
  cout << "stepsize is    : " << stepsize << endl
       << "z_size         : " << z_size << endl
       << "a_size         : " << a_size << endl
       << "1 is           : " << result / (4 * PI) << endl
       << "The result is  : " << result << endl
       << "The error is   : " << error * 100 << " %\n"
       << "Number of loops: " << frequency << endl
       << "elapsed time   : " << diffs << "s" << endl
       << "----------------test_AngIntegrate_trapezoid_opti----------<<<<<\n";
 
  return result;
}
 
 
Numeric test_x(int vsize, float stepsize, int frequency) {
  cout << ">>>>>-----------test_x---------------\n";
  Vector Integrand(vsize);  // function to be integrated
  Vector Theta(vsize);      // Theta values
 
  init_x(vsize, stepsize, frequency, Integrand, Theta);
 
  Numeric result = 0;
 
  struct timeval start;
  struct timeval ende;
  gettimeofday(&start, NULL);
  cout << "Sekunden : " << start.tv_sec << endl
       << "Milisekunden: " << start.tv_usec << endl;
 
  for (int i = 0; i < frequency; i++)
    result = AngIntegrate_trapezoid_original(Integrand, Theta);
 
  gettimeofday(&ende, NULL);
 
  // norming the result to Pi
  result = result / (2 * 180);
 
  double error = result / PI - 1;
 
  double diffs = (double)(ende.tv_sec - start.tv_sec) +
                 (double)(ende.tv_usec - start.tv_usec) / 1000000.0;
  cout.precision(15);
  cout << diffs << endl;
  cout << "stepsize is    : " << stepsize << endl
       << "number of steps: " << vsize << endl
       << "1 is           : " << PI / PI << endl
       << "The result is  : " << result / PI << endl
       << "The error is   : " << error * 100 << "%\n"
       << "Number of loops: " << frequency << endl
       << "elapsed time   : " << diffs << "s" << endl
       << "---------------test_x-----------<<<<<\n";
 
  return result;
}
 
Numeric test_x_fixedstep(int vsize, int frequency) {
  cout << ">>>>>-----------test_x_fixedstep---------------\n";
  cout.precision(12);
  Vector Integrand(vsize);  // function to be integrated
  Vector Theta(vsize);      // Theta values
 
  double stepsize;
  stepsize =
      180.0 /
      (vsize - 1);  // attention this only works with eaqually spaced intervals
  cout << "Neue berechnete Stepsize: " << stepsize << endl;
 
  for (int i = 0; i < Integrand.nelem(); i++) Integrand[i] = i * stepsize;
 
  //Theta is between 0 and 180
  for (int i = 0; i < Theta.nelem(); i++) Theta[i] = i * stepsize;
 
  cout << "Integrand: von " << Integrand[0] << " bis "
       << Integrand[Integrand.nelem() - 1] << endl
       << "Theta: von " << Theta[0] << " bis " << Theta[Theta.nelem() - 1]
       << endl;
 
  Numeric result = 0;
 
  struct timeval start;
  struct timeval ende;
  gettimeofday(&start, NULL);
  cout << "Sekunden : " << start.tv_sec << endl
       << "Milisekunden: " << start.tv_usec << endl;
 
  for (int i = 0; i < frequency; i++)
    result = AngIntegrate_trapezoid_fixedstep(Integrand, Theta, stepsize);
 
  gettimeofday(&ende, NULL);
 
  // norming the result to Pi
  result = result / (2 * 180);
 
  double error = result / PI - 1;
 
  double diffs = (double)(ende.tv_sec - start.tv_sec) +
                 (double)(ende.tv_usec - start.tv_usec) / 1000000.0;
  cout.precision(15);
  cout << diffs << endl;
  cout << "stepsize is    : " << stepsize << endl
       << "number of steps: " << vsize << endl
       << "1 is          : " << PI / PI << endl
       << "The result is  : " << result / PI << endl
       << "The error is   : " << error * 100 << "%\n"
       << "Number of loops: " << frequency << endl
       << "elapsed time   : " << diffs << "s" << endl
       << "---------------test_x_fixedstep----------<<<<<\n";
 
  return result;
}