taylor table

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

Controller: CodeCogs

Contents

Interface
Taylor1 Table
Taylor2 Table
Page Comments

Dependents

Interface

C++

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 Maths/Calculus/Diff/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 $\inline x_1, x_*, x_2$ are chosen such that

$\displaystyle x_1 = x_* - h, \qquad x_2 = x_* + \gamma h$

(2)

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

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.

Authors

Lucian Bentea (November 2006)

Taylor1 Table

std::vector<double>taylor1_table(	double	(`f`)(double)[function pointer]*
	std::vector<double>&	`points`
	double	`h`
	double	`gamma` `= 1.0`	)

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

Source Code

Source code is available when you buy a Commercial licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.

Taylor2 Table

std::vector<double>taylor2_table(	double	(`f`)(double)[function pointer]*
	std::vector<double>&	`points`
	double	`h`
	double	`gamma` `= 1.0`	)

Example 2

#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;
}