taylor_table

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Contents
  1. Interface
  2. Overview
  3. Members

Computes the first and second derivatives of a function at multiple points.

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Calculus Maths Differential

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Interface

#include <codecogs/maths/calculus/differential/taylor_table.h>

using namespace Maths::Calculus::Differential;

Overview

This module computes either the first or the second numerical derivatives of a function at multiple points and returns a vector with each resulting value. In order to achieve this it uses the component taylor. The advantage is that the interval used in computing the numerical derivative need not be symmetrical around either one of the given points. Basically three abscissas x_1, x_*, x_2 are chosen such that

\displaystyle x_1 = x_* - h, \qquad x_2 = x_* + \gamma h
(1)

where h, \gamma are real positive constants corresponding to the precision and the symmetry of the interval of differentiation. The derivative is thus approximated at point x_*.

Notice however that the functions in this module consider the same precision and symmetry constants when computing the numerical derivative at each of the given points.

GPL Licence — free for non commercial use. See Licence details.

FUNCTION

taylor1_table

Example 1

#include <codecogs/maths/calculus/diff/taylor_table.h>
#include <math.h>
#include <stdio.h>

// precision constant
#define H 0.0001
                    
// function to differentiate
double f(double x)
{
  return cos(x);
}

// the first derivative of the function, to estimate errors
double df(double x)
{
  return -sin(x);
}

int main()
{
  // display precision
  printf("\n    h = %.4lf\n\n", H);

  printf(" f(x) = cos(x)\n");
  printf("f`(x) = -sin(x)\n\n");

  // initialise points table
  double P[10] = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10};

  // compute the first derivative at each point in the table
  std::vector<double> points(P, P+10),
  derivatives = Maths::Calculus::Diff::taylor1_table(f, points, H);

  // display the results, including error estimation
  printf("Point\tApproximation\t\tActual value\t\tError\n\n");
  for (int i = 0; i < 10; i++)
    printf("x = %d\t%.15lf\t%.15lf\t%.15lf\n", i, 
    derivatives[i], df(points[i]), fabs(df(points[i]) - derivatives[i]));
  printf("\n");

  return 0;
}

Output

h = 0.0001

 f(x) = cos(x)
f`(x) = -sin(x)

Point   Approximation           Actual value            Error

x = 0   -0.841470983405614      -0.841470984807897      0.000000001402283
x = 1   -0.909297425311170      -0.909297426825682      0.000000001514512
x = 2   -0.141120007824946      -0.141120008059867      0.000000000234921
x = 3   0.756802494046756       0.756802495307928       0.000000001261172
x = 4   0.958924273062761       0.958924274663138       0.000000001600378
x = 5   0.279415497732546       0.279415498198926       0.000000000466379
x = 6   -0.656986597622516      -0.656986598718789      0.000000001096273
x = 7   -0.989358244972195      -0.989358246623382      0.000000001651187
x = 8   -0.412118484553760      -0.412118485241757      0.000000000687997
x = 9   0.544021109981466       0.544021110889370       0.000000000907904

Parameters

f
the function to differentiate
points
the vector of abscissas at which to compute the derivative
h
the value of the precision constant h
gamma
Default value = 1.0

Returns

a vector containing the values of the first derivative of f evaluated at each of the given points

FUNCTION

taylor2_table

Example 1

#include <codecogs/maths/calculus/diff/taylor_table.h>
#include <math.h>
#include <stdio.h>
 
// precision constant
#define H 0.0001
                    
// function to differentiate
double f(double x)
{
  return cos(x);
}
 
// the second derivative of the function, to estimate errors
double d2f(double x)
{
  return -cos(x);
}
 
int main()
{
  // display precision
  printf("\n    h = %.4lf\n\n", H);
 
  printf("  f(x) = cos(x)\n");
  printf("f``(x) = -cos(x)\n\n");
 
  // initialise points table
  double P[10] = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
 
  // compute the second derivative at each point in the table
  std::vector<double> points(P, P+10),
  derivatives = Maths::Calculus::Diff::taylor2_table(f, points, H);
 
  // display the results, including error estimation
  printf("Point\tApproximation\t\tActual value\t\tError\n\n");
  for (int i = 0; i < 10; i++)
    printf("x = %d\t%.15lf\t%.15lf\t%.15lf\n", i, 
    derivatives[i], d2f(points[i]), fabs(d2f(points[i]) - derivatives[i]));
  printf("\n");
 
  return 0;
}

Output

h = 0.0001

  f(x) = cos(x)
f``(x) = -cos(x)

Point   Approximation           Actual value            Error

x = 0   -0.540302319866019      -0.540302305868140      0.000000013997879
x = 1   0.416146818496858       0.416146836547142       0.000000018050284
x = 2   0.989992487602587       0.989992496600445       0.000000008997858
x = 3   0.653643589818356       0.653643620863612       0.000000031045256
x = 4   -0.283662183922072      -0.283662185463226      0.000000001541154
x = 5   -0.960170275904858      -0.960170286650366      0.000000010745508
x = 6   -0.753902247021491      -0.753902254343305      0.000000007321814
x = 7   0.145500035223032       0.145500033808614       0.000000001414419
x = 8   0.911130256680583       0.911130261884677       0.000000005204094
x = 9   0.839071522055730       0.839071529076452       0.000000007020722

Parameters

f
the function to differentiate
points
the vector of abscissas at which to compute the derivative
h
the value of the precision constant h
gamma
Default value = 1.0

Returns

a vector containing the values of the second derivative of f evaluated at each of the given points