CodeCogs

Computes the definite integral of a function using the Gauss quadrature for 3 points.

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Interface

#include <codecogs/maths/calculus/quadrature/gauss.h>

using namespace Maths::Calculus::Quadrature;

inline double gauss(double (*f)(double), double a, double b)

This module computes the area beneath a user supplied function using an approximation given by a certain weighted sum of function values.

Consider a function $f:I \subset \mathbb{R} \rightarrow \mathbb{R}$ and two distinct abscissas $a < b \in I$ . In order to compute an approximation of the definite integral:

$I = \int_a^b f(x) \mathrm{d}x$

(1)

we will use the Gauss-Legendre quadrature formula for 3 points:

$I \approx \frac{b-a}{18} \left[5f\left(\frac{b+a}{2} + \frac{a-b}{2}\sqrt{\frac{3}{5}}\right) + 8f\left(\frac{b+a}{2}\right) + 5f\left(\frac{b+a}{2} + \frac{b-a}{2}\sqrt{\frac{3}{5}}\right)\right].$

(2)

For fixed $\displaystyle \xi \in (a, b)$ the error of this approximation is given by

$\epsilon = \frac{(b-a)^7}{31500} \frac{d^6}{dx^6} f(\xi).$

(3)

Example 1

In what follows an approximation is found for the definite integral

$\int_2^3 \sin x \,dx$

(4)

and the absolute error from its actual value is estimated.

#include <codecogs/maths/calculus/quadrature/gauss.h>
#include <stdio.h>
#include <math.h>

// function to integrate
double f(double x)
{
  return sin(x);
}

// the primitive of f, to estimate errors
double pf(double x)
{
  return -cos(x);
}

int main()
{
  // compute the approximate area
  double fi = Maths::Calculus::Quadrature::gauss(f, 2, 3),

  // use the Leibniz-Newton formula to find a more precise estimate
  realfi = pf(3) - pf(2);

  // display the result and error estimate
  printf("      f(x) = sin(x)\n");
  printf("   I(2, 3) = %.15lf\n", fi);
  printf("real value = %.15lf\n", realfi);
  printf("     error = %.15lf\n\n", fabs(fi - realfi));
 
  return 0;
}

Output

f(x) = sin(x)
   I(2, 3) = 0.573845954654464
real value = 0.573845660053303
     error = 0.000000294601161

Parameters

the function to integrate

the inferior limit of integration

the superior limit of integration

Returns

The definite integral of the given function from a to b.

References

Mihai Postolache - "Metode Numerice", Editura Sirius