Gauss integration

White, J.IT. (1969) Self-linking and the Gauss integral in higher dimensions. Am. J. Math. 91,... 

The Gauss integral is an integer number Ny that can be calculated by considering the field B, along the curve C, ... 

As explained earlier, the limits of integration of the above equation make it ideal for applying a Gauss integration scheme, discussed in Chapter 7. Equation (9.82) can therefore be approximated as... 

This can be found after transforming the partition function into a Gauss integral.23 Then the value of det (1) is required, not its roots. The classical limit might be important in physiology. 

White, J.H. (1969). Self-Linking and the Gauss Integral in Higher Dimension. Am.J.Math., 91, 693. 

The integral which extends over the area of the region has been converted to the volume integral of the divergence of q according to Gauss integral theorem. 

This integral is transformed by the substitution x/a + Trixa — y into the Gauss integral ... 

There are theoretical reasons for this consideration, which have been largely confirmed in practice also. The extended trapezoid method is often superior to the Gauss integration only for the periodical functions that have a period equal to the integration interval. 

This derivation was first applied by Van Orstrand and Dewey [97], who solved Equation 10.14 for diffusion from a satnrated solntion into pure solvent. Since in the Gauss integral is a number. 

The underlying method of numerical integration is often called the GAUSS integration or Gauss quadrature. 

For this calculation, one has to take into consideration the fact that P(n, r), the probability function, is normalized and satisfies the criterion of an integral equal to 1. Thus, the properties of the Gauss integrals apply to this case, which for P n, r) gives... 

There are many more sophisticated methods of quadrature (another name for numerical integration). For reasons already given, the trapezoidal rule is sufficient for most engineering applications. In some specialized cases. Gauss integration (see Section 4.2.3) is the most advantageous method. 

Error analysis is not so simple. The above formula is exact if fix) is a cubic polynomial (or a simpler one). A rule of thumb is that the order of accuracy of Gauss integration is twice that of equally spaced methods using the same number of data points. 

Consider the same problem as in Example 4.2 using four-point Gauss integration. 

This example illustrates the power of Gauss integration. A four-point formula gives essentially the exact result. [This is due to the fact that f x) is a quartic function, and the four-point formula is exact for polynomials up to degree 7.]... 

In summary, the trapezoidal rule is easy to implement and usually accurate enough. Furthermore, it can be used with nonequally spaced data (such as real experimental data). When there is a limit to the number of sample points, but they can be placed at will, then Gauss integration is often the best choice. For example, suppose a restriction is that only four samples can be obtained on a process during a test whose duration is 1 h. When during the hour should the samples be collected The answer to this is left as an exercise. 

Exercise 4.3 Evaluate the following integral using three- and four-point Gauss integrations ... 

Figure 13.7. Function evaluation points for selected Gauss integration formulas over triangle areas. Points are exact for (a) linear (b) quadratic or (c) cubic function variations over the areas.

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