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trapezoidal

Computes the definite integral of a function using the trapezoidal rule.

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Interface
Trapezoidal
Trapezoidal
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Interface

C++

Excel

Overview

This module computes the area beneath either a user supplied function or a set of discrete points, using an approximation which assumes the function is linear between each two consecutive points.

Consider a two times differentiable function $\inline f:I \subset \mathbb{R} \rightarrow \mathbb{R}$ , two distinct abscissas $\inline a < b \in I$ and a positive integer $\inline n$ . Then the following approximation holds:

$\int_a^b f(x) \mathrm{d}x \approx \frac{b-a}{n} \left[ \frac{f(a) + f(b)}{2} + \sum_{k=1}^{n-1} f \left(a + \frac{b-a}{n} k\right) \right]$

(2)

with an error bound of:

$\epsilon \leq \frac{(b-a)^3}{12n^2} \sup_{x \in [a,b]} \left|\frac{d^2}{dx^2}f(x)\right|.$

(3)

This is a quadrature formula known as the trapezoidal rule. From the formula of the error bound you may notice that e.g. if one doubles the number of points $\inline n$ , the approximation error is decreased four times. To better understand how this rule works, observe the following picture - the area between abscissas $\inline x_1$ and $\inline x_2$ is approximated by a trapezoid.

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Below you will find examples that show how to use each version of the function and the error estimates in each case.

References:

Mihai Postolache - "Metode Numerice", Editura Sirius
Mathworld, http://mathworld.wolfram.com/TrapezoidalRule.html

Authors

Lucian Bentea (September 2006)

Trapezoidal

doubletrapezoidal(	int	`n`
	double	(`f`)(double)[function pointer]*
	double	`a`
	double	`b`	)

This version is to be used when you wish to pass the function to integrate as an argument, instead of passing an array with its values at distinct points.

Example 1

In what follows an approximation is found for the definite integral

$\int_1^3 \sin x \,dx$

(4)

and the absolute error from its actual value is estimated.

#include <codecogs/maths/calculus/quadrature/trapezoidal.h>
#include <stdio.h>
#include <math.h>
 
// number of points
#define N 100
 
// 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::trapezoidal(N, f, 1, 3),
 
  // use the Leibniz-Newton formula to find a more precise estimate
  realfi = pf(3) - pf(1);
 
  // display problem data
  printf("      f(x) = sin(x)\n");
  printf("    points = %d\n\n", N);
 
  // display the result and error estimate
  printf("   I(1, 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)
    points = 100
 
   I(1, 3) = 1.530243792301766
real value = 1.530294802468585
     error = 0.000051010166819

Parameters

n	the number of sample points of the function f, from which to approximate
f	the function to integrate
a	the inferior limit of integration
b	the superior limit of integration

Returns

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

Source Code

Source code is available when you buy a Commercial licence.

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Trapezoidal

doubletrapezoidal(	int	`n`
	const double*	`values`
	double	`a`
	double	`b`	)

This version is to be used when the function is not known analytically, but a table with its values at equally spaced abscissas is available.

Example 2

The code below gives an approximation for the definite integral

$\int_1^{1.3} \sqrt{x} \,dx$

(5)

and estimates the absolute error from its actual value.

#include <codecogs/maths/calculus/quadrature/trapezoidal.h>
#include <stdio.h>
#include <math.h>
 
// the primitive of f, to estimate errors
double pf(double x)
{
  return 2*sqrt(x*x*x)/3;
}
 
int main()
{
  // values of the function at equally spaced abscissas
  double ordinates[7] = {1, 1.0247, 1.04881, 1.07238, 
  1.09544, 1.11803, 1.14017},
 
  // compute the approximate area
  fi = Maths::Calculus::Quadrature::trapezoidal(7, ordinates, 1, 1.3),
 
  // use the Leibniz-Newton formula to find a more precise estimate
  realfi = pf(1.3) - pf(1);
 
  // display problem data
  printf("      f(x) = sqrt(x)\n");
  printf("    points = 7\n\n");
 
  // display the result and error estimate
  printf(" I(1, 1.3) = %.15lf\n", fi);
  printf("real value = %.15lf\n", realfi);
  printf("     error = %.15lf\n\n", fabs(fi - realfi));
 
  return 0;
}