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Maths › Calculus › Differential ›

taylor

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

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Interface
Taylor1
Taylor2
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Interface

C++

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

using namespace Maths::Calculus::Diff;

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.

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

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

(2)

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

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

(3)

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

(4)

In the case \inline \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|

(5)

while in the case when \inline \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]

(6)

where \inline \xi_1 \in (x_1, x_*) and \inline \xi_2 \in (x_*, x_2). The error estimate for \inline \gamma \neq 1 comes from the Taylor formula applied to the given function \inline f at point \inline x_*. On account of the above relations you may notice that the error approaches zero as \inline h approaches zero for \inline \gamma \neq 1, and as \inline h^2 approaches zero for \inline \gamma = 1.

Example 1

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

#include <>
#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

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Taylor2

doubletaylor2( double (*f)(double)[function pointer]

double x

double h

double gamma = 1.0 )[inline]

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

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

(7)

where \inline h, \gamma are real positive constants. Then the approximate value of the second derivative \inline \displaystyle \frac{d^2f}{dx^2} at the abscissa \inline 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

(8)

\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.

(9)

In the case \inline \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|

(10)

while in the case when \inline \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]

(11)

Example 2

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

#include <>
#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;
}