Method, Gauss

With the aid of effective Gauss method for solving linear equations with such matrices a direct method known as the elimination method has been designed and unveils its potential in solving difference equations,... 

There are many different methods for the task of fitting any number of parameters to a given measurement [14-16], We can put them into two groups (a) the direct methods, where the sum of squares is optimized directly, e.g., finding the minimum, similar to the example in Figure 7.4, and (b) the Newton-Gauss methods, where the residuals in r or R themselves are used to guide the iterative process toward the minimum. 

Figure 3.85 and Table 3.25 show the identified values of the effective diffusion coefficient for all adsorption experiments. The activation technique applied can be shown to allow the enhancement of Dg, so the speed of the transport process will be higher. Table 3.25 also contains the values of Dg identified by the Newton-Gauss method. 

Table 3.25 Identified values for all activated carbons (first line — from Fig. 3.85, second line — by Newton-Gauss method.

In ref. 167 the preservation of some structure properties of the flow of differential systems by numerical exponentially fitted Runge-Kutta (EFRK) methods is investigated. The sufficient conditions on symplecticity of EFRK methods are presented. A family of symplectic EFRK two-stage methods with order four has been produced. This new method includes the symplectic EFRK method proposed by Van de Vyver and a collocation method at variable nodes that can be considered as the natural collocation extension of the classical RK Gauss method. 

S. Gonzalez-Pinto, S. Perez-Rodriguez and R. Rojas-Bello, Efficient iterations for Gauss methods on second-order problems. Journal of Computational and Applied Mathematics,... 

For computation of a particular solution vector x, this method requires i(n -n)-l-n operations of multiplication or division, versus n —n) +n for the Gauss-Jordan method. Thus, the Gauss method takes about two-thirds as many operations. The computation of an inverse matrix takes 0(n ) operations of multiplication or division for either method. 

Depending on the space discretisation techniques used, the set of equations to be solved may be different, but for FD- and FV- based methods, the discretisation results in a set of linear or non-linear algebraic equations. These depend on the nature of these partial differential equations and how they are derived. For linear equations, it is well known that a Gauss elimination method can be used as a basic method to solve them. Further details of the Gauss method can be found in [60],... 

Several other methods have been proposed. Gauss method, for example, will be taken up later on. See also Hopkinson,. ... 

Of the most common implicit algorithms, the most useful ones are those adopting quadrature points, which are points used by the open Gauss method, semiopen Radau method, and the close Lobatto method (see Chapter 1). 

Runge-Kutta s method derived from the Gauss method with one intermediate point only is... 

The Newton-Gauss method consists of linearizing the model equation using a Taylor series expansion around a set of initial parameter values bo, also called preliminary estimates, whereby only the first-order partial derivatives are consider... 

In Section 9.2, the method of Newton-Gauss has been presented as a method with fast convergence, but low robustness convergence is ensured only if /behaves quasilinearly as a function of the parameter vector P over the full length of each parameter step. If one is sufficiently close to the solution, then the parameter steps still to be taken will be sufficiently short, so that the previously mentioned condition is fulfilled. In this case, the Newton-Gauss method can be employed advantageously to rapidly reach the solution. In practice, the initial parameter vector bo will usually be too far removed from the solution. 

As with the Newton-Gauss method,/is linearized around a tentative parameter vector bo ... 

As mentioned previously, the Newton-Gauss method converges very fast if the initial values bo can be chosen close to the optimum. Now the problem is that of obtaining good initial estimates. Naturally, no general techniques exist for this, for the location of the optimum is precisely what one is going to search for. Inspection of some types of current models can nevertheless indicate in which direction one can look for solving the problem of the initial estimates. 

The well-known Gauss method of elimination is older than the theory of vector spaces. It will be outlined briefly, only to brush up the reader s memory. Let us again forget, for a moment, what we know about vectors and matrices. Let us have a set of M linear equations in unknowns x, —, ... 

This set of equations can be treated by e.g. the Gauss method. The second approximation of values is obtained from the relation... 

The Gauss elimination method is also very useful in the calculation of determinants of matrices. The elementary operations used in the Gauss method are consistent with the... 

Therefore, a matrix whose determinant is to be evaluated should first be converted to the triangular form using the Gauss method, and then its determinant should be calculated from the product of the diagonal elements of the triangular matrix. 

We will apply the Gauss-Jordan procedure, without pivoting, to the setof Eqs. (2.I02) shown in Sec. 2..6.1 in order to observe the difference between the Gauss-Jordan and the Gauss method. Starting with the augmented matrix... 

Eq. (6.34) and the appropriate boundary conditions constitute a set of linear algebraic equations, so the Gauss methods for the solution of such equations may be used. Eq. (6.34) is actually a predominantly diagonal system therefore, the Gauss-Seidel method (see Sec. 2.7) is especially suitable for the solution of this problem. Rearranging Eq. (6.34) to solve for m,- ... 

What is the percent error produced by the Gauss method for the value of p generated ... 

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