comp 391-079 Numerical Analysis

Trapezoid Rule & Multiple-applicaiton [top]

The integral of a function can estimated by dividing the range of integration from a to b into sections and summing the area associated with every section. The area if one section can be estimated as a trapezoid, in which one non-parallel side of the trapezoid connects the end points of the section that lie on the function to be integrated. Another interpretation of the trapezoid is the average of the maximum and minimum area that is in the section. The integral is estimated more accurately as the sections become small and the trapezoid better estimates the integrated function.

The error of the multiple application trapezoid rule shown below. F’’ bar is the sum of second derivates at the points used in the trapezoid rule. Notice the error reduces quadratically with the increase in the number of sections.

1/3 Simpson Rule & Multiple-applicaiton [top]

Simpson’s rule estimates the perimeter of a section by a polynomial rather than the line approximation of the trapezoid rule. The 1/3 rule uses a quadratic polynomial to estimate the function to be integrated. The name is from the 1/3 ratio in the final equation of integrated area. The LaGrange equation for interpolating points can be simplified for three points.

The integration of several sections is as follows. Notice that there need to be an even number of points.

The error for 1/3 Simpson Rule is as follows. Notice that there is less error than for the trapezoid rule.

The function f is evaluated at zhi which maximizes f that is somewhere in interval integrated over. Finding the fourth derivative in the interval is a bit involved.

3/8 Simpson Rule [top]

Simpson’s 3/8 rule is similar to 1/3 Simpson rule except a cubic polynomial is used to estimate the integrated function. Again it is named after the 3/8 factor in the equation of integrated area. In this case four points are needed instead of three.

Again with the LaGrange representation of a cubic polynomial and four points estimates one section of the integral as follows:

The error for one section improves over the other methods and is:

Observations [top]

Unequal segments.

The trapezoid rules apply to sections of different widths. The points used in one Simpson Rule section need equal separation; specifically, 1/3 needs three equally spaced points and 3/8 needs four equally spaced points.
Multiple integrals
The segmentation integral techniques can be applied to multiple integrals. The second integral will sum over the results from the first integral.

More techniques with known functions [top]

There is another set of methods for evaluating functions that cannot easily be integrated. The previous methods worked for data sets where only a certain set of points is known. In this case the methods take advantage of evaluating the functions at any point.
Romberg integration
The Richardson’s extrapolation technique computes a more accurate integral approximation from two less accurate measurements. The Romberg integration algorithm uses Richardson’s technique. The algorithm is very eficent when implemented as a program. The error can be evaluated from one iteration to the next. This will be useful to determine when the iteration should stop.
Gauss Quadrature
Gauss Quadrature formula is an extension of the Newton-Cotes equation. To achieve higher accuracy, however, the segments integrated are not equal width.
Improper integrals
Improper integrals include bounds that go to infinity. The technique is to divide the integration limits into sections to which existing techniques can be applied.