Numerical Methods for Inverse Singular Value Problems

Abstract

Two n umerical methods | one continuous and the other discrete | are proposed for solving inverse singular value problems. The rst method consists of solving an ordinary di erential equation obtained from an explicit calculation of the projected gradient of a certain objective function. The second method generalizes an iterative process proposed originally by F riedland et al. for solving inverse eigenvalue problems. With the geometry understood from the rst method, it is shown that the second method (also, the method proposed by F riedland et al. for inverse eigenvalue problems) is a variation of the Newton method. While the continuous method is expected to converge globally at a slower rate (in nding a stationary point o f t h e objective function), the discrete method is proved to converge locally at a quadratic rate (if there is a solution). Some numerical examples are presented.

Projected Pseudotransient Continuation

“…Since F (u) is in the tangent space of Ω at u [7], the solution of (1.2) is in Ω if u 0 ∈ Ω. Since g is analytic in u, the results of [28] will apply, and so (1.1) holds for all initial vectors u 0 ∈ Ω.…”

A Newton-like method for solving rank constrained linear matrix inequalities

“…Though the dimension constraint (5) is not explicitly discussed in [6], it is satisfied by the problem studied in that paper and application of the tangent and lift method to that problem results in a (locally) quadratically convergent algorithm. Other applications of tangent and lift are given in [7].…”

“…This controller can be found by considering system and controller transfer functions and requiring that the denominator of the closed loop transfer function equal the polynomial (s + α) 6 . For comparison purposes, we also tried the cone complementarity linearization algorithm of [11] on the same problem.…”

“…Our method is based on the "tangent and lift" methodology [6], a generalization of Newton's method. While the classical Newton algorithm can be used to find zeroes of functions, the tangent and lift method is more general and can be used to find a point in the intersection of an affine subspace and a manifold.…”

Constructing a Hermitian Matrix from Its Diagonal Entries and Eigenvalues

“…In terms of the Frobenius inner product < A B > := X i j a ij b ij (10) the Fr echet derivative o f F at Q acting on an arbitrary matrix U 2 R n n can be calculated as follows:…”