1. Introduction

The paper considers graph completion inclusion isotone for least squares problem with uncertain data. The closed graph theorem in the sense of [1], and [2] for instance gives a basic result that characterizes continuous functions in terms of their graphs. For any function we define the graph , as the map to be the Cartessian product

.

There is a metric topology for which is defined. When is Cauchy, for it follows that in as. In other words, for a closed graph F, it holds that forcing implying the graph of is enclosed by the convex hull of control points. In a nutshell, a graph is continuous at the point if it pulls open sets back to open sets and carries open sets over to open sets.

Fundamental to this discussion are the basic principles of advanced topology. Good reference texts may be [1], [3] and [4]. Following [1], a linear map F from a linear topological space X to a linear topological space Y will be called bounded if it maps bounded sets to bounded sets. A map which is a linear topological space X to a linear topological space Y will be called sequentially continuous if for every sequence converging to some point x of E has a bounded envelope and such a sequentially continuous map is ultrabarrelled [1]. A function F is called approxable in the sense of [5] if for a multi valued mapping for every there is continuous single valued mapping with graph . A function is said to be upper semi-continuous at if for any neighbourhood of there exists a neighbourhood of such that . A similar definition goes for a lower semi continuous function.

By reasons due to recurrence and category theorem, the map has a measure preserving homeomorphism and hence its set of recurrent point is residual and of full measure. In other words it can be said that such a map has a generic measure preserving homeomorphism whose square has a dense orbit. In some cases one often comes across some functions with distortions at some points which necessitate the following definition.

Definition 1.1, [6]. Let be a map between two locally Euclidean metric spaces. The quantity

, is called the radius distortion of at x.

As a consequence following, we introduce the nonlinear system of equation

(1.1)

where , m>n, is an interval vector. It is supposed that the function F has at least where . Therefore, in a Frechet space E, every continuous linear map from a Frechet space E into F has a closed range and such a map is finite dimensional.

Applications of nonlinear systems for example are well documented in the work of [7] which includes the following: Aircraft stability problems, Inverse Elastic rod problems, Equations of radiative transfer, Elliptic boundary value problems, Power flow problems, Distribution of water on a pipeline, Discretization of evolution problems using implicit schemes, Chemical plant equilibrium problems and, Nonlinear programming problems. We often adopt the concept of divided difference from a univariate function which is extended over to multivariate vector valued function by defining slope as

, provided that F is differentiable on the line .

Motivated by the above details, we state the following:

Lemma 1.1, [8]. Let be convex and let be continuously differentiable in D.

(i) If then

(This is a strong form of Mean-value theorem);

(ii) if then and

(This is a weak form of Mean-value theorem);

(iii) if is Lipschitz continuous in D, that is the relation holds for some , then

(This is truncated Taylor expansion with remainder term).

It is a result to follow up to the discussion that we have the following theorems.

Theorem 1.1, [9]. Suppose has an F-derivative at each point of an open neighbourhood of. Then, is strong at x if and only if is continuous at .

Because system 1.1 is over determined, we often transform to a linear system by the process of 2-norm assuming that the Jacobian matrix which is rectangular exists in the form

(1.2)

This operator is required to be everywhere defined that also holds verbatim in Banach space, provided its graph is closed in with respect to its product topology.

Of paramount interest to us is a mapping that is balanced and absorbent whose inductive limit is ultra barrelled for which contraction mapping of Newton-Mysovskii theorem follows:

Theorem 1.2, [9]. Supposing that is F-differentiable on a convex set and that for each is non-singular and satisfies

.

Assuming further that such that ; and for which

.

Then the iterated contraction of Newton operator is in the form:

(1.3)

which remains in and converges to a solution of .

Moreover,

where

.

We expect the weak pre-image and the strong pre-image coincide simultaneously such that the orbit of F is a sequence for which

(1.4)

holds good that be coercive and a local homeomorphism of to itself, every zero of is in the intersection of convex hull with the hyper plane.

As a result of equation 1.4 the norm reducing property of Newton operator for system 1.1 is given by for k=0,1,.., .

The quantity will be called the order of convergence for assuming the limit exists which is again equal to the R-Order of convergence of [9].

The rest part in the paper is arranged as follows: Section 2 discusses what is meant by the statistical meaning of the matrix . Section 3 describes completeness of graph in Banach space topology. Procedure for estimating eigenvalues of interval Jacobian matrix formed the basis of discussion in chapter 4. Section 5 in the paper gives numerical illustration of what has been discussed in previous sections.

2. The Statistical Meaning of the Matrix and Its Applications

Henceforth, we adopt that the matrix A denotes the matrix A(x). In line with ideas expressed in [10] we give the statistical meaning to the matrix. We note that the components are independent, normally distributed random variable with mean and all having the same variance which we describe in the form:

,

Therefore setting there holds that

The covariance matrix of the random vector y is given by. The first moments are and. For a rectangular matrix A for which is non singular Nuemaier [11] using decomposition proved that for all x.

Following [12], it was proved that over any field, and that holds over any field of characteristic 0. This means computing least squares solution or bounding the errors of computations are not defined over an arbitrary field. It thus holds that and that.

We shall be interested in the least squares problem where the coefficient matrix and right hand vector assume some kind of noise often known as white noise. This situation leads to what is called Total Least Squares Problem (TLS) whereby, tempered distribution to the coefficient matrix A is, and to the vector y by respectively for which holds

(2.1)

Subject to

(2.2)

The expression is the usual Frobenius norm which coincides with in the classical Banach space. In its simplest form, the linear inverse problem [13], [14] to which the least squares problem belongs was described in the form:

(2.3)

where, represents, , and is the realization of random noise. Thus when B is invertible the linear system 2.3 is said to be well posed whereas it is ill-posed when B is not invertible.

As a result the minimum -norm of the residual is given by

(2.4)

Where, it is understood that is the norm on with unit ball defined by ..

By substituting (2.4) into (1.1) we have that

(2.5)

In equation 2.5, the second term dominates in the ill-posed problem when the uncertainty is high thereby making practically useless in most cases of applications. This means that the induced map is open, nearly continuous, and nearly open if and only if has the same ill-posed property, [1]. Therefore via interval arithmetic this problem has been addressed [15].

We expect that both and be closed in the topology [1] so that, convergence implies convergence in the tempered distribution for which any neighbourhood in the metric space holds for satisfying system 1.1.

As pointed by some schools of thought, one drawback of Tikhonov regularization is that it tends to produce a solution that is often excessively smooth in image processing for which this method results in loss of sharpness. Nevertheless, the classical Tikhonov regularization method for minimizing has the solution as in the sense of [16]. The equation that determines in the restructured least squares sense was given by

(2.6)

where is well defined.

Thus when the optimal solution is given by

Where are the unique optimal point to the problem minimize

Note that and .

Since and is rank one, it holds that . This forces .

Equation 2.6 is used when there is no perturbation in y. In case of total least squares the solution to the perturbed least squares equation 2.1 is given by the

(2.7)

where is the smallest singular value of , ([ 17], e.g.,).

In most applications of interest we often perturb the matrix B such that perturbation satisfies the condition that and

(2.8)

Because of equation 2.8, it follows that

(2.9)

Where we used the fact that, and. The matrix is the Pseudo-Penrose inverse of .

As a result of equations 2.8 and 2.9 and in view of equation 2.7, it can be deduced that

(2.10)

The actual value of as well as is obtained in the form

(2.11)

(2.12)

Therefore when the matrix B is perturbed by , from well known result [18] there holds the estimate

(2.13)

3. The Question of Completeness of graph in Banach Space Topology

We are interested in the regularity property of graph that is equated to openness which relates regularity to inversion problems. By this we mean regularity of set valued maps, [19] for which openness and inversion properties of equation 1.1 form the basis of investigation. Inverse mapping theorem asserts that the inverse of an invertible bounded linear operator between Banach spaces is a continuous map.

As is well known, a complete metric space cannot be written as a countable union of nowhere dense sets. The Baire Category theorem provides that, an indication that F is open around and is the open -neighbourhood of x in . That is,

(3.1)

Fundamentally, what is required is the graph completion operator that is inclusion isotone with respect to functional argument. That is, the Hausdorff continuous operator satisfying the inclusion which is Dedekind order complete with respect to point wise defined partial ordering. As it were, we also assume that finite algebra of measurable sets holds verbatim.

As is standard, the completeness of the graph suffices to show. That implies that it is absolutely summable, which is that, . It follows that , a consequence of Banach space. We note in passing that if is not Hausdorff, an extreme point is not a supporting set.

4. Distribution of Eigenvalues of the Jacobian Matrix

A very important issue in engineering application has always been the occurrence of a saddle node bifurcation , Hopf bifurcation or solution near such bifurcation points,[20].

Denoting A(x) as the Jacobian matrix of partial derivative of F(x) assuming that system 1.1 is of order n, the eigenvalues of A(x) is represented by.

Let which does not contain eigenvalues of A other than. Then . The number of zeros inside is given by argument principle [10]

(4.1)

Thus , the integral is analytic function of and of .

In other words assuming is analytic in the sense of [21] inside and on a closed contour which encloses , then will be defined by the equation

(4.2)

Supposing the eigenvalues are able to discriminate their goals such that

(4.3)

And assuming further one can find , then is called a saddle-node bifurcation point of the nonlinear system of equation 1.1. Furthermore, if A(x) has a pair of conjugate eigenvalues passing the imaginary axis while the other eigenvalues have negative real parts, then is called a Hopf bifurcation point.

The solution to nonlinear system 1.1 is said to be stable if the eigenvalues of A(x) have negative real part.

Using [17], the 2-norm, for unsymmetrical matrix is the numerical abscissa where in its application, the behaviour of may be different in the initial, transient, and asymptotic phase. In other words, the asymptotic behaviour depends on as whenever. In any case, the bound given by is the best possible.

The cosine angle between two matrices using Frobenius inner product is given by

(4.4)

Where

, [20] and is often used in practice in which case

(4.5)

The relative condition numbers for the matrix sine and cosine in the sense of [22] satisfy

(4.6)

After all these, the estimation of eigenvalues of interval Jacobian matrices will be computed. Popular such methods for estimating bounds of eigenvalues have been the Gesrchgorin disks or Ovals of Cassini, [23]. Unfortunately the bounds these methods produce for the case of interval matrices had been known to be too wide for any meaningful uses. We proceed in the same spirit similar to [24] as well as [25] to provide realistic bounds for eigenvalues of interval matrices coming from the Jacobian of system 1.1.

For general treatment of eigenvalues, consider unsymmetric matrices of order n where for clarity, we adopt the following notation:

, , and after verification of Rohn showed that a necessary and sufficient condition for to be eigenvalue of interval matrix [A] is that is singular. That is to say a number is an eigenvalue of if the two conditions below can be verified

then is a real eigenvalue of A

then is not a real eigenvalue of A

[22] proved that eigenvalues of symmetric interval matrix A lie in the interval where .

The respectively denote minimal and maximal eigenvalue of . Let us take note that a rectangular matrix has full column rank if it possible to compute . For example the upper end point of the desired interval indicates how fast a population can grow or how fast a disease can spread in any experimental data analysis. As pointed out by [24] the estimated eigenvalue bound provided by Rohn’s method has the drawback of still not being empty even when the set of eigenvalues is empty.

5. Numerical Examples

Problem 1.

Consider a set of two-dimensional points:

Using quadratic polynomial fit for the data set and if we take notice of the resulting Vandermode matrix, and using MATLAB version 2007, the solution for equation 2.1 is obtained as

, with eigenvalues to the symmetric matrix .

Providing solution to the interval linear system a procedure earlier described in [15] applies. As a consequence, we omit repeating it here.

We are concerned with providing estimates of the eigenvalues computed for interval matrix which was derived from the above statistical data set

And this was computed to be

.

Using Rohn’s method [24] we also obtained eigenvalues bound to be

Using a 20% impurity as data noise we obtained bounds for eigenvalues of interval matrix as

With condition number and it can be seen that matrix eigenvalues may be affected by level of impurities in the statistical data set.

6. Conclusions

The paper studied graph completion operator for interval least squares problem. We discussed the statistical meaning of the matrix obtained from the statistical data entries of observation. It is shown that convergence of inverse operators for the resulting regularized Tikhonov parameter implies convergence in the tempered distribution of data noise wherein the earlier procedure described in [15] is applicable. Our emphasis was placed on estimating interval matrix which has great applications in the study of growth rate of a system which was applied on statistical least squares problem. As pointed out by [24] the estimated eigenvalue bound provided by Rohn’s method [24] has the drawback of still not being empty even when the set of eigenvalues may be empty as demonstrated by numerical example. This may be found useful in both Scientific and Engineering designs.