NUMERICAL METHODS FOR APPROXIMATING EIGENVALUES OF BOUNDARY VALUE PROBLEMS

This paper describes some new finite difference methods for the approxi- mation of eigenvalues of a two point boundary value problem associated with a fourth order linear differential equation of the type (py') (q y') + (r s)y O. The smallest positive eigenvalue of some typical eigensystems is computed to demon- strate the practical usefulness of the numerical techniques developed.

3) together with some of their modifications occur frequently in applied mathematics, modern physics and electrical engineering, see [5,7,8,12].In (I.I), we assne that the real-valued functions p(x), r(x) and s(x) are continuous on [a,b] and satisfy the conditions p(x) C C 2 [a,b], q(x) CC'[a,b] and p(x), q(x), s(x) > O, r(x) > 0, x [a,b Recently, numerical techniques of order 2 and 4 have been developed for computing approximate values of % for the boundary value problem (1.1-(1.3)with p(x) l, q(x) E O, see [1,2].In the fourth order method, the problem is dlscretlzed to yield a generalized seven-band symmetric matrix elgenvalue problem of the form AY Ah BY (1. 5)   where A is an approximate value of % with elgenvector Y and B is a diagonal matrix depending on the function s(x).Consequently, the eigenvalue problem (1. 5)   can be converted to the usual standard problem of the type MY AY without any excessive amount of computational effort.There are at present several areas of research activity surrounding the development and analysis of numerical methods for approximating A satisfying the generalized matrix elgenproblems of the type (1.5), see [4, 9, I0, 11, 13].Usmanl has developed [14] some new finite difference methods of order 2 and 4 for computing elgenvalues of the differential system (I.I) with p(x) E I, q(x) 0 (for p(x) 0, q(x) E I, see [15]) associated with the boundary conditions y(a) y'(a) y"(b) y'"( O. (1.6) The purpose of this work is to present some new finite difference methods for computing approximate values of % for the boundary value problems (1.1)-(1.2) and (1.1)-(1.3).These methods lead to generalized elgenvalue problems of the form (1.5 where A is a flve-band or seven-band matrix and B is a diagonal positive definite matrix. We preface the numerical methods by some analytical properties of the elgenvalues and elgenfunctions of the boundary value problems under discussion.If %1 and %2 are two distinct eigenvalues of the problem I[ aand Yl(X), Y2(X) are the corresponding eigenfunctions, then b s(x) yl(x) y2(x) dx 0. a PROOF.
The proof is a direct consequence of Green's identity (see [3], p. 86) and the boundary conditions (1.2) and (1.3).LEMMA 2.2.If y(x) is an elgenfunction belonging to the elgenvalue % of the boundary value problem , then y(x) is an eigenfunction belonging to the eigenvalue PROOF.The proof is trivial and follows from LY 0.
(2.2) s y2dx a PROOF.Let + i, , C R, be an eigenvalue of the problem I[ with eigenfunction y(x) u(x) + iv(x) where u(x) and v(x) are real-valued functions.
From Lemma 2.2, it follows that is also an eigenvalue of the problem I with respect to the eigenfunction y(x) u(x) iv(x).Now, from Theorem 2.1, it follows that b b 0 f s(x)y(x)(x) dx f s(x) ly(x) 12dx > 0 a a (2.3) because s(x) > 0 and y(x) # 0. The contradiction in (2.3) suggests that % can- not be complex, hence it is real as required.
In order to prove (2.2), we multiply (I.I) by y(x) and integrate the resulting equation twice from x a to x b.
1 Thus, our method for computing approximations A for l satisfying (I.I)-(1.2) can be expressed as a generalized five-band symmetric matrix eigenvalue problem (A + h4R)y--Ah4Sy. (3.5) It can be proved that A is a positive definite matrix and hence for any stepslze h > 0, the approximations A for by (3.5) are real and positive for all p, q, s > 0 and r > 0. That our method provides 0(h 2) convergent approximations A for k can be established following Grigorieff [6].We omit the long and tedious details of conver- gence proof for brevity.
Normally, only one or a few of the extreme elgenvalues of (1.1)-(1.2) are needed in applications.In what follows, we will compute only the smallest eigenvalue of the system to illustrate our method based on (3.1).We consider the eigenvalue problems with boundary conditions (1.2) at a 0, b I.We computed approximations to the smallest eigenbalue A of these eigensystems by our method (3.1) or equivalently (3.5).
The corresponding relative errors are shown in Table I.
It is evident from the entries of the accompanying table that our numberical method provides 0(h2) con- vergent approximations.In computing the relative errors, we assumed that kl A1 with h 2-8, because exact value of k cannot be obtained by analytical methods for these eigensystems.We omit the lengthy details of the development of our numerical method but we re- mark that a second order method for computing approximations A to satisfying (I.I)-(1.3) is based on (A' + h4R)y Ah4SY (4. where A' differs from A introduced in (3.5) in the first and the last rows only.
Another second order method for computing approximations A to satisfying (I.I)-(1.3) is based on (A" + h4R)y--Ah4Sy where A", as before, differs from A in the first and the last rows only.
The numerical results for computing A based on (4.2) are slightly better than those based on (4.1), but the matrix A" is no longer a symmetric matrix.
We illustrate our methods based on (4.1) and (4.In this section we consider the linear differential equation (4) y q(x)y" + (r(x) % s(x))y 0 associated with the boundary conditions (1.2) or (1.3).For n > 7, let the step- size h and the sequence {x i} be defined as in section 3.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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by computing A satisfying (3.7) and the boundary conditions (1.3) with a 0 and b I.The smallest elgenvalue A is computed employing inverse power iteration method and the numerical results are summarized in TableII

Table II
5. HIGHER ORDER METHODS FOR SPECIAL CASE OF (I.I).
The boundary value problem (5.1)-(1.2) is discretized by the following

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation