- Newton-Cotes Integration of Equations

- Intergrations of Equations

- Numerical Differentiation
- High-accuracy differentiation formulas
- Richardson extrapolation
- Derivatives of unequally spaced data
- Numerical integration/differentiation formulas with libraties and packages

- Engineering Applications: Numerical Integration and Differentiation

(6.1) |

The integration means the total value, or summation, of over the range to .

(6.2) |

(6.3) |

where , with , the uniform spacing in -values.

The Newton-Cotes formulas are based on the strategy of replacing a complicated function or tabulated data with an approximating function that is easy to integrate:

(6.4) |

where is the Newton-Gregory interpolating polynomial. For

(6.5) |

For

(6.6) |

See the figure 21.1 in the textbook.

(6.7) |

where

(6.8) |

The result of integration is

(6.9) |

which is called as trapezoidal rule.

One way to improve the accuracy of the trapezoidal rule is to divide the integration interval from to into a number of segments and apply the method to each segment. The width of segments

(6.10) |

The integration is

(6.11) |

Simpson's 1/3 rule : use a second-order polynomial

(6.12) |

Simpson's 1/3 rule is

(6.13) |

The label ``1/3'' stems from the fact that is divided by 3.

Simpson's 3/8 rule : use a third-order Lagrange polynomial

(6.14) |

Simpson's 3/8 rule is

(6.15) |

See the figure 21.11 in the textbook.

Two separate estimate using step sizes of and

The error of the multiple-application trapezoidal rule is

(6.17) |

Assume that is constant regardless of step size

(6.18) |

Rearranage the above equation

(6.19) |

which can be substituted into eq. (6.16)

(6.20) |

which can be solved for

(6.21) |

Thus, we have developed an estimate of the truncation error in terms of the integral estimates and their step sizes. This estimate can then be substituted into

(6.22) |

to yield an improved estimate of the integral:

(6.23) |

For the special case where the interval is halved

(6.24) |

or

(6.25) |

The Romberg integration algorithm

(6.26) |

where and are the more and less accurate integral and is the improved integral.

- Trapezoidal rule : two parameters model
(6.27)

where the 's are the unknown parameters. - Gauss Quadrature : four parameters model
(6.28)

where the 's, , are the unknown parameters.

The trapezoidal rule's formula can be derived from another point of view, the method of undetermined coefficients. Because the trapezoidal rule is a two parameters model, we need two relationships that connect two parameters.

(6.29) |

and

(6.30) |

(6.31) |

The trapezoidal rule must pass through the end point and results in a large error. But suppose that the constraints of fixed base points was removed and we were freely evaluate the area under a straight line joining any two points on the curve. See the figure 22.5 to figure out the differences.

The object of Gauss quadrature is to determine the coefficients of an equation of the form

with assuming that eq. (6.32) fit the integral of a constant, a linear, a parabolic, and a cubic function

(6.33) | ||

(6.34) | ||

(6.35) | ||

(6.36) |

These relationships yield the two-point Gauss-Legendre formula

(6.37) |

Because Gauss quadrature requires function evalutions at nonuniformly spaced points within the integration interval, it is not appropriate for cases where the function is unknown.

- Decrease the step size
- Use a higher-order formula
- Combine two derivative estimates to compute more accurate approximation

(6.38) |

which can be solved for

If we truncate the second- and higher-derivative terms

(6.40) |

The accuracy of the above equation depend on the step size .

In contrast to this approach, substitue the second-derivative term

(6.41) |

into eq. (6.39) to yield

(6.42) |

or, by collecting terms,

(6.43) |

Notice that inclusion of the second-derivative term has improved the accuracy to .

(6.44) |

For centered difference approximations with , the application of this formula will yield a new derivative estimate of .

2001-11-29