Gauss Formulae

This F1 scries with unit argument can be summed by Gauss formula 7. 2) and the expression for S found. It the follows that... 

In view of the Gauss formula, expression (8.105) can be cast in the form... 

Now let us transform the surface integral in (13.133) into a volume integral, using the Gauss formula. According to equation (13.15), the normal stresses, P , can be expressed by the stress tensor as follows ... 

Substituting expression (13.134) into the surface integral and applying the Gauss formula, we find ... 

One important question regarding the distribution of measurements about their mean is the expected frequency of occurrence of an error as a function of the error magnitude. The most commonly utilized function, which describes well the relative frequency of occurrence of random errors in large sets of measurements, is given by Gauss formula (see also rel. (9) Section 5.2) ... 

Sk and Si are vectors defining the position of tesserae k and I, respectively, while Rk is the radius of the sphere which tessera k belongs to. The diagonal term of D, which collects the contribution of the reaction field induced by the charge placed on tessera k on itself, is derived by the Gauss formula for an infinite charged plane with a correction term iccounting for the curvature of the convex tessera. 

The equation (5.69), with the use of the Gauss formula (5.1), transforms to the mass conservation law for each phase in differential form... 

Many algorithms have been proposed to perform the numerical integration of functions. We consider only two femilies of algorithms, which are the basis for the development and implementation of an even number of general programs for the numerical integration the Newton-Cotes and the Gauss formulae. 

The other open and semiopen Newton-Cotes formulae are of purely historical interest the open formulae are less effective than the Gauss formulae, and both the open and semiopen formulae are harder than the close formulae in their... 

Gauss formulae exploit the positions of Xi as degrees of freedom to increase precision by preserving the number of points where the function is evaluated. 

To use the Gauss formulae, it is necessary to know the zeroes of the polynomial Z and the weights Wi. They are tabled for many families of orthogonal polynomials and for many combinations of a and b and r x). 

The order of Gauss formulae that uses n internal points isp = 2 - n. 

Gauss formulae have certain advantages with respect to Newton-Cotes formulae. 

Gauss formulae are open. They can be used even when the function has numerical problems at the interval extremes. 

Whereas the Newton-Cotes formulae become less efficient as the order increases and thus are rarely used beyond the Boole formula, the Gauss formulae become increasingly efficient for almost every problem. 

This problem derives from the fact that orthogonal polynomials (which Gauss formulae are based on) have distinct zeroes for different polynomial orders. Since the points where the function must be calculated are the zeroes of an orthogonal polynomial, they are different in every formula from the same family (except for the central point). 

A direct consequence of this fact is the difficulty in estimating the error using a specific Gauss formula. 

If we want to control a Gauss formula with a formula with a higher order, it is necessary to calculate the function in all the points of the two formulae (sometimes with the exception of the central point). 

As already mentioned, it is useful to use the weight r(x) in an optimal way so as to exploit the Gauss formulae to the fullest the remaining portion of the function should be well representable by a polynomial. 

Table 1.5 Values ofx/ and w,- for the Gauss formula with seven points.

The couple of formulae commonly used consist of the Gauss formula with 7 points together with the Kronrod one with 15 points (see Tables 1.5 and 1.6). 

If one wishes to estimate the variance (or standard deviation) in a result calculated from a number of data f x,y,z...), each of which have certain standard deviations themselves, we can use Gauss formula for error propagation ... 

The two last expressions incorporate the mass flow of the overhead solids, the first does not. Using Gauss formula with the first expression (10.A.6a) for 77 gives ... 

We have to qualify this discussion, however. If the feed and collected fractions can be determined a lot more precisely than the overflow fraction, we might come to different conclusion when using Gauss formula with the new estimates of measurement errors. This could result, for instance, if we were able to accurately weigh the feed and the captured solids, while being forced to perform on-line sampling of the overflow fraction. 

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