Overdetermined system

Equation (6.79) is valid for exactly determined systems (m = n). In case of overdetermined systems, m > n, the condition number is given by... 

In case of serious overlappings, multivariate techniques (see Sect. 6.4) are used and p ) > n sensors (measuring points zjt) are measured for n components. From this an overdetermined systems of equations results and, therefore, non-squared sensitivity matrixes. Then the total multicomponent sensitivity is given by... 

Thus, when the attention of the mathematicians of the time turned to the description of overdetermined systems, such as we are dealing with here, it was natural for them to seek the desired solution in terms of probabilistic descriptions. They then defined the best fitting equation for an overdetermined set of data as being the most probable equation, or, in more formal terminology, the maximum likelihood equation. 

We will begin our discussion by demonstrating that, for a non-overdetermined system of equations, the algebraic approach and the least-square approach provide the same solution. We will then extend the discussion to the case of an overdetermined system of equations. 

Arranging the whole-rock mineral concentrations for each element i (i=l,...,m) in a vector y, and putting the concentration of element i in the phase j at the ith row and jth column of the matrix Amxn (mineral matrix), the usual overdetermined system is obtained... 

No single spectrometer is the perfect input to Eq. (2). Each experimental plan has limitations and many of these are associated with accessible compositions, temperatures and pressures. In addition, there is the issue of having an overdetermined system as remarked previously. Taken together, we arrive at the conclusion ... 

The only way to determine all five constants w+, w., v+,Di, D2 in the above problem is to supply an additional jump condition. This jump condition cannot be universal since, if applied at both discontinuities, it leads to an overdetermined system. We must therefore differentiate between the waves moving with the speeds and A. Notice that only the first... 

Thus, we have i = 0 at the equilibrium. Equation (28) together with the linear balance Equation (27b) form an overdetermined system of As we... 

One more important property of the self-dual Yang-Mills equations is that they are equivalent to the compatibility conditions of some overdetermined system of linear partial differential equations [11,12]. In other words, the selfdual Yang-Mills equations admit the Lax representation and, in this sense, are integrable. For this very reason it is possible to reduce Eq. (2) to the widely studied solitonic equations, such as the Euler-Amold, Burgers, and Devy-Stuardson equations [13,14] and Liouville and sine-Gordon equations [15] by use of the symmetry reduction method. 

They are in line with the traditional Lie approach to the reduction of partial differential equations, since they exploit symmetry properties of the equation under study in order to construct its invariant solutions. And again, any deviation from the standard Lie approach requires solving overdetermined system of nonlinear determining equations. A more profound analysis of similarities and differences between these approaches can be found elsewhere [33,56,64]. 

Consequently, to describe all the ansatzes of the form (53),(54) reducing the Yang-Mills equations to a system of ordinary differential equations, one has to construct the general solution of the overdetermined system of partial differential equations (54),(86). Let us emphasize that system (54),(86) is compatible since the ansatzes for the Yang-Mills field ( ) invariant under the three-parameter subgroups of the Poincare group satisfy equations (54),(86) with some specific choice of the functions F, F2, , 7Mv, [35]. 

The least squares method is the best possible solution to an overdetermined system of equations, where we have more equations than unknowns. This is a very common situation in problems, in which, for example, we have a set of N measurements which we want to fit to an equation with M parameters (M < N). For example if we measure temperatures as a function of time... 

For two unknowns the overdetermined system contains at least three equations, in matrix notation... 

More generally, the reaction field factors may either be determined numerically, since they appear in an overdetermined system of linear equations,23 or they may be computed analytically for certain idealized cavities (e.g., sphere and ellipsoid).30 66,213,214 Efficient optimization of solvated geometries motivates the latter approach,2i3,235-237 but the formalism has also been ap-... 

The system (3.3) is called underdetermined if < L. The system (3.3) is called overdetermined if > L. Very often in geophysical applications, we work with an overdetermined system wherein the number of observations exceeds the number of model parameters. At the same time, in many situations it may be necessary to work with an underdetermined system. We will examine both types of linear equation systems below. 

For a system with N atoms, there are N(N — I)/2 interatomic distances and only 3N — 6 independent degrees of freedom (the x, y, z of atoms). Therefore, equation 25 is an overdetermined systems of equations. In the LST approach equation 25... 

Generalized inverse (pseudoinverse) To obtain an approximate solntion of an overdetermined system of linear eqnations, i.e., when the number of equations is greater than the number of unknowns (m > n), a vector x is sought to minimize the square of residuals, r r, where... 

This definition holds for exactly determined systems where the number of rows in X is equal to the number of columns, that is, n=p (cf. Eq. (6.41)). In the case of overdetermined systems, with n>p, the condition number is obtained from... 

The number of observations is always much greater than the minimum necessary to compute the unknowns. Because of this, relations between unknown values and observations lead to an overdetermined system of non-linear equations ... 

A common procedure for solving this overdetermined system is the method of variation of parameters (also referred to in the mathematical literature as Gauss-Newton non-linear least squares algorithm) (Vanicek and Krakiwsky 1982), and this procedure is described in the following. As approximate values of coordinates x° are known a priori, by Taylor s series expansion of the function / about point x°. 

The usual way to solve (1) is to multiply the overdetermined system of equations... 

In the rest of this chapter other approaches to the solution of this problem will be discussed. For immediate solution either the system of normal equations or the overdetermined system may be used, taking into account the possibilities for use of sparseness of the coefficient matrix from the point of view of effective use of modern computers. 

The procedure consists of decomposition of the coefficient matrix A of the overdetermined system with m equations and n unknowns (3) into the product... 

In order to eliminate these disadvantages recent attention has been paid in the literature to the immediate solution of the overdetermined system of equations (2). The reason is that, theoretically in terms of the solution s error, the methods for the solution of systems of linear equations based on orthogonal decomposition are considered more accurate than the methods based on triangular decomposition (Wilkinson 1965). 

Based on previous considerations the following may be concluded. Problem (1) can be solved using two approaches the method of normal equations, and immediated solution of the overdetermined system of equations (2) using the methods of orthogonal decomposition. 

It is not clear, from a practical point of view if there is an advantage in using the methods based on orthogonal decomposition for the immediate solutionof the overdetermined system of euqations (2) over the method of normal equations, so it is necessary to compare experimentally all these methods using a test stream formed from a number of large practical problems. 

Duff I.S., Reid J.K. (1976) A Comparison of some Methods for the Solution of Sparse Overdetermined Systems of Linear Equations. J. Inst. Math. Appl. Vol. 17 pp 267-280. 

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