Generalized Eigenvalue Problems for Rectangular Matrices

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Abstract
Two classes of algorithms are considered for finding a zero of the smallest singular value of rectangular matrix M(λ) as a function of the parameter λ. These zeros are called generalized eigenvalues. The first class of methods works directly with M(λ) and is an adaptation of certain inverse iteration procedures available when M is square. In these methods it is important that the smallest singular value has a well determined zero. The second approach considers the minimization of a penalty objective function, and two exact penalty functions are given. This class of methods will find a minimum of the smallest singular value which need not be restricted to zero. Numerical results are given for a family of problems in which it is possible to control both the degree of illconditioning of the generalized eigenvalue problem and the size of the smallest singular value.

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