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Maths › Calculus › Differential ›

taylor

Computes the first and second derivatives of a function using the Taylor formula.

Controller: CodeCogs

Contents

Interface
Taylor1
Taylor2
Page Comments

Interface

C++

Overview

This module computes the first or second numerical derivatives of a function at a particular point using the Taylor formula. The advantage is that the interval used in computing the numerical derivative need not be symmetrical around the given point.

References:

Mihai Postolache - "Metode Numerice", Editura Sirius

Authors

Lucian Bentea (September 2006)

Taylor1

doubletaylor1(	double	(`f`)(double)[function pointer]*
	double	`x`
	double	`h`
	double	`gamma` `= 1.0`	)[inline]

Consider a function $f:[a,b] \rightarrow \mathbb{R}$ that is three times differentiable on its domain, and three abscissas $x_1, x_*, x_2 \in [a,b]$ such that

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

(1)

where $h, \gamma$ are real positive constants. Then the approximate value of the first derivative $\displaystyle \frac{df}{dx}$ at the abscissa $x_*$ is given by:

$\frac{df}{dx}(x_*) \approx \frac{f(x_2) - f(x_1)}{2h} \qquad \mbox{if } \gamma = 1$

(2)

$\frac{df}{dx}(x_*) \approx \frac{f(x_2) - f(x_1)}{(\gamma + 1)h} \qquad \mbox{if } \gamma \neq 1.$

(3)

In the case $\gamma = 1$ we have an error bound of:

$\epsilon \leq \frac{h^2}{6} \sup_{a<x<b} \left|\frac{d^3f}{dx^3}\right|$

(4)

while in the case when $\gamma \neq 1$ the error is given by:

$\epsilon = (1 - \gamma) \frac{h}{2}\frac{d^2f}{dx^2}(x_*) - \frac{h^2}{3!(\gamma + 1)} \left[\gamma^3 \frac{d^3f}{dx^3}(\xi_2) + \frac{d^3f}{dx^3}(\xi_1)\right]$

(5)

where $\xi_1 \in (x_1, x_*)$ and $\xi_2 \in (x_*, x_2)$ . The error estimate for $\gamma \neq 1$ comes from the Taylor formula applied to the given function $f$ at point $x_*$ .

On account of the above relations you may notice that the error approaches zero as $h$ approaches zero for $\gamma \neq 1$ , and as $h^2$ approaches zero for $\gamma = 1$ .

Example 1

Below we give an example of how to compute the first numerical derivative of $\cos x$ at point $x = 1$ , considering $\gamma = 1$ (equally spaced abscissas). The absolute error from the actual value of the derivative at that point is also displayed.

#include <codecogs/maths/calculus/diff/taylor.h>
#include <stdio.h>
#include <math.h>
 
// precision constant
#define H 0.001
 
// function to differentiate
double f(double x)
{
  return cos(x);
}
 
// the derivative of the function, to estimate errors
double df(double x)
{
  return -sin(x);
}
 
int main()
{
  // display precision
  printf("         h = %.3lf\n\n", H);
 
  // compute the numerical derivative
  double fh = Maths::Calculus::Diff::taylor1(f, 1, H);
 
  // display the result and error estimate
  printf("      f(x) = cos(x)\n");
  printf("     f`(1) = %.15lf\n", fh);
  printf("real value = %.15lf\n", df(1));
  printf("     error = %.15lf\n\n", fabs(fh - df(1)));
 
  return 0;
}

Output

h = 0.001
 
      f(x) = cos(x)
     f`(1) = -0.841470844562695
real value = -0.841470984807897
     error = 0.000000140245202

Parameters

f	the function to differentiate
x	the abscissa at which you want to compute the first derivative
h	the value of the precision constant h
gamma	Default value = 1.0

Returns

The first numerical derivative of the given function at the x abscissa.

Source Code

Source code is available when you buy a Commercial licence.

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Taylor2

doubletaylor2(	double	(`f`)(double)[function pointer]*
	double	`x`
	double	`h`
	double	`gamma` `= 1.0`	)[inline]

Consider a function $f:[a,b] \rightarrow \mathbb{R}$ that is four times differentiable on its domain, and three abscissas $x_1, x_*, x_2 \in [a,b]$ such that

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

(6)

where $h, \gamma$ are real positive constants. Then the approximate value of the second derivative $\displaystyle \frac{d^2f}{dx^2}$ at the abscissa $x_*$ is given by:

$\frac{d^2f}{dx^2}(x_*) \approx \frac{f(x_1) - 2f(x_*) + f(x_2)}{h^2} \qquad \mbox{if } \gamma = 1$

(7)

$\frac{d^2f}{dx^2}(x_*) \approx \frac{2}{h^2\gamma(\gamma + 1)} [\gamma f(x_1) - (1 + \gamma) f(x_*) + f(x_2)] \qquad \mbox{if } \gamma \neq 1.$

(8)

In the case $\gamma = 1$ we have an error bound of:

$\epsilon \leq \frac{h^2}{12} \sup_{a<x<b} \left|\frac{d^4f}{dx^4}(x)\right|$

(9)

while in the case when $\gamma \neq 1$ the error is given by:

$\epsilon = \frac{h}{3(\gamma + 1)} \left[\gamma^2 \frac{d^3f}{dx^3}(\xi_2) - \frac{d^3f}{dx^3}(\xi_1)\right]$

(10)

where $\xi_1 \in (x_1, x_*)$ and $\xi_2 \in (x_*, x_2)$ . The error estimate for $\gamma \neq 1$ comes from the Taylor formula applied to the given function $f$ at point $x_*$ .

On account of the above relations you may notice that the error approaches zero as $h$ approaches zero for $\gamma \neq 1$ , and as $h^2$ approaches zero for $\gamma = 1$ .

Example 2

Below we give an example of how to compute the second numerical derivative of $\cos x$ at $x = 1$ , considering $\gamma = 1$ (equally spaced abscissas). The absolute error from the actual value of the derivative at that point is also displayed.

#include <codecogs/maths/calculus/diff/taylor.h>
#include <stdio.h>
#include <math.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("         h = %.4lf\n\n", H);
 
  // compute the numerical derivative
  double fh = Maths::Calculus::Diff::taylor2(f, 1, H);
 
  // display the result and error estimate
  printf("      f(x) = cos(x)\n");
  printf("    f``(1) = %.15lf\n", fh);
  printf("real value = %.15lf\n", d2f(1));
  printf("     error = %.15lf\n\n", fabs(fh - d2f(1)));
 
  return 0;
}

Output

h = 0.0001
 
      f(x) = cos(x)
    f``(1) = -0.540302319866019
real value = -0.540302305868140
     error = 0.000000013997879

Parameters

f	the function to differentiate
x	the abscissa at which you want to compute the second derivative
h	the value of the precision constant h
gamma	Default value = 1.0

Returns

The second numerical derivative of the given function at the x abscissa.

Source Code

Source code is available when you buy a Commercial licence.

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Page Comments

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double	`taylor1` (double (f)(double), double x, double h, double gamma = 1.0)[inline]* Computes the value of the first derivative of a function using the Taylor formula.
double	`taylor2` (double (f)(double), double x, double h, double gamma = 1.0)[inline]* Computes the value of the second derivative of a function using the Taylor formula.