# simpson

Computes the definite integral of a function using Simpson's first rule.

Controller: **CodeCogs**

## Interface

C++

HTML

## 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 shaped as a parabola between each two consecutive points.{The background derivation of Simpon's first rule is given here:}
Simpon's first rule represents the function between as a second order polynomial. The function is described by three ordinates, , and . In order to identify the truncation error, a fourth order polynomial is used,
Once integrated this becomes,
From (1),
and
Therefore
or
Substituting into (2),
If we ignore the last term, this would create a truncation error proportional to the fourth derivative of the function f(x), which depends on .

Consider a four times differentiable function , two distinct abscissas and a positive integer . Then the following approximation holds:
where , with an error bound of:
This is a quadrature formula known as Simpson's rule. From the formula of the error bound you may notice that e.g. if one doubles the number of points *n*, the approximation error is decreased sixteen times. This rule approximates the area under

*f*between

*a*and

*b*by the area under a quadratic polynomial

*P*as seen in the following image, where

*m*is the midpoint between

*a*and

*b*. Below you will find examples that show how to use each version of the function and the error estimates in each case. Generally this module provides better estimates than using the trapezoidal rule as implemented in Quadrature/trapezoidal. To convince yourself, compare the approximations generated by this module with the estimates given by the trapezoidal rule module.

## References:

- Mihai Postolache - "Metode Numerice", Editura Sirius
- Wikipedia, http://en.wikipedia.org/wiki/Simpson's_rule

### Example 1

### References

### Authors

*Lucian Bentea (September 2006)*

## Simpson

doublesimpson( | int | n | |

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

double | a | ||

double | b | ) |

### Example 2

- In what follows an approximation is found for the definite integral
and the absolute error from its actual value is estimated.
#include <codecogs/maths/calculus/quadrature/simpson.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::simpson(N, f, 1, 3), // use the Leibniz-Newton formula to find a more precise estimate realfi = pf(3) - pf(1); // uses the Leibniz-Newton formula // 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) = 1.530294803828911 real value = 1.530294802468585 error = 0.000000001360326

### Parameters

n the number of sample points of the function f, from which to approximate (must be even) 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

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## Simpson

doublesimpson( | 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 3

- The code below gives an approximation for the definite integral
and estimates the absolute error from its actual value.
#include <codecogs/maths/calculus/quadrature/simpson.h> #include <stdio.h> #include <math.h> // the primitive of g, 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::simpson(7, ordinates, 1, 1.3), // use the Leibniz-Newton formula to find a more precise estimate realfi = pf(1.3) - pf(1); // uses the Leibniz-Newton formula // 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; }

### Output

f(x) = sqrt(x) points = 7 I(1, 1.3) = 0.321485166666667 real value = 0.321485368419253 error = 0.000000201752586

### Parameters

n the number of sample points of the function f, from which to approximate (must be odd) values an array with the value of the function at equally spaced abscissas 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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