# NUMERICAL METHODS FOR SOLVING INVERSE EIGENVALUE PROBLEMS

Original Publication Date: 1983-Dec-31

Included in the Prior Art Database: 2005-Sep-15

## Publishing Venue

Software Patent Institute

## Related People

Jorge Nocedal: AUTHOR [+4]

## Abstract

Consider the following inverse eigenvalue problem: Given n real symmetric matrices [Equation ommitted] find the coefficients cl .... c n such that the matrix [Equation ommitted] T has the given eigenvalues. Here [Equation ommitted] A slight reformulation of (1.1) describes the additive inverse eigenvalue problem, which arises in the solution of inverse Sturm-Liouville problems. In practice it happens frequently that only sone eigenvalues are given. However, for the purpose of analysis it is convenient to consider the case where the number of parameters, eigen-values and the order of the matrices is the same. Another area where problem (1.1) arises is in.shell model com-putations in nuclear spectroscopy (see Brussard and Glaudemans (1977)). There A is the Hamiltonian, the variables {c i I are the interactions of one and two bodies, and the matrices (A.1) represent the result of adding and symmetrizing (or antisymmetrizing) the effect of many particles. We will now briefly describe an inverse Sturm-Liouville problem.

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__Page 1 of 14__THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

**NUMERICAL METHODS FOR SOLVING INVERSE EIGENVALUE
PROBLEMS* **

Jorge Nocedal and Michael L. Overton

April 1983

Report No. 74

*This paper will appear in the Proceedings of the Third Inter-American Workshop on Numerical Methods (Caracas, Venezuela, June 1982), V. Pereyra, ed., Lecture Notes in Mathematics, Springer-Verlag. NUMERICAL METHODS FOR SOLVING INVERSE EIGENVALUE PROBLEMS t

Jorge Nocedal and Michael L. Overton Courant Institute of Mathematical Sciences New York University New York, N.Y. 10012

1. Introduction

Consider the following inverse eigenvalue problem: Given n real symmetric matrices

(Equation Omitted)

find the coefficients cl .... c n such that the matrix

(Equation Omitted)

T has the given eigenvalues. Here

(Equation Omitted)

A slight reformulation of (1.1) describes the additive inverse eigenvalue problem, which arises in the solution of inverse Sturm-Liouville problems. In practice it happens frequently that only sone eigenvalues are given. However, for the purpose of analysis it is convenient to consider the case where the number of parameters, eigen-values and the order of the matrices is the same. Another area where problem (1.1) arises is in.shell model com-putations in nuclear spectroscopy (see Brussard and Glaudemans (1977)). There A is the Hamiltonian, the variables {c i I are the interactions of one and two bodies, and the matrices (A.1) represent the result of adding and symmetrizing (or antisymmetrizing) the effect of many particles. We will now briefly describe an inverse Sturm-Liouville problem. Consider the boundary value problem

(Equation Omitted)

with some boundary conditions, for example

(Equation Omitted)

Suppose that p(x) is unknown. Given a spectrum for the problem (1.2), can we determine p(x)? Let us discretize (1. 2) let

Northwestern University Page 1 Dec 31, 1983

__Page 2 of 14__NUMERICAL METHODS FOR SOLVING INVERSE EIGENVALUE PROBLEMS

(Equation Omitted)

J. 'This work was supported in part by the Department of Energy, grant DEA C0276 ERO 3077- V, and in part by the National Science Foundation, grant MICS-81-01924. Then, using finite differences to approximate u",

(Equation Omitted)

(Equation Omitted)

In matrix form (1.3) can be written as

(Equation Omitted)

where

(Equation Omitted)

n The problem is therefore: given-A 0 and a spectrum {Ai)l find a diagonal matrix D such that

(Equation Omitted)

has the given spectrum. This is the additive inverse eigenvalue problem. It can be written in the form

(Equation Omitted)

(1.5)

T with

(Equation Omitted)

where e i is the i-th column of the identity matrix.

The numerical methods and the convergence results that we give for the form (1.1) also apply to

(1.5). The question that arises immediately is: when does there exist a solution to problem (1.1)
or (1.2), and when is it unique? The answer has been given fo= various special cases. We refer
the reader to Borg (1946), Hadeler (1968), Hald (1972) and...