CodeCogs

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

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Interface

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

using namespace Maths::Calculus::Differential;

inline double taylor1(double (*f)(double), double x, double h, double gamma = 1.0)
inline double taylor2(double (*f)(double), double x, double h, double gamma = 1.0)

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

FUNCTION

taylor1

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

the function to differentiate

the abscissa at which you want to compute the first derivative

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.

FUNCTION

taylor2

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)