A Numerical Approach to the Inverse Toeplitz Eigenproblem

Laurie, Dirk

doi:10.1137/0909026

Cited by 28 publications

(21 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…T h us (71) is proved. We nally remark that one step of the above (Newton) method is a descent direction for the objective function (15). Thus it is possible to combine the objective function with some step-length control to improve the global convergence properties of our iterative method.…”

Section: Similar To (24) Any Tangent V Ector T(x)mentioning

confidence: 99%

Numerical Methods for Inverse Singular Value Problems

Chu¹

1992

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

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.

show abstract

Section: Similar To (24) Any Tangent V Ector T(x)mentioning

confidence: 99%

Numerical Methods for Inverse Singular Value Problems

Chu¹

1992

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

show abstract

“…We note from (12) that the condition v rG'(x)v >-0 for every v TxM is precisely the well-known second-order necessary optimality condition for problem (5 It is well known (and easy to prove) that under the Frobenius inner product the orthogonal complement of S(n) is given by (15) S(n)-{all skew-symmetric matrices}.…”

mentioning

confidence: 99%

The Projected Gradient Method for Least Squares Matrix Approximations with Spectral Constraints

Chu¹,

Driessel²

1990

SIAM J. Numer. Anal.

122

View full text Add to dashboard Cite

Abstract. The problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure. This paper describes a general procedure in using the projected gradient method. It is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly. This gives rise to the construction of a descent flow that can be followed numerically. The explicit form also facilitates the computation of the second-order optimality conditions. Examples of applications are discussed. With slight modifications, the procedure can be extended to solve least squares problems for general matrices subject to singular-value constraints.

show abstract

“…Find a symmetric non-negative matrix P that has the prescribed set as its spectrum. Discrete Method: A few locally convergent Newton-like algorithms are available for the rst problem 26,33]. Little is known for the non-negative matrix problem 3].…”

Section: Inverse Eigenvalue Flowsmentioning

confidence: 99%

A list of matrix flows with applications

Chu¹

1995

Hamiltonian and Gradient Flows, Algorithms and Control

View full text Add to dashboard Cite

Many mathematical problems, such as existence questions, are studied by using an appropriate realization process, either iteratively or continuously. This article is a collection of di erential equations that have been proposed as special continuous realization processes. In some cases, there are remarkable connections betwe e n s m o o t h o ws and discrete numerical algorithms. In other cases, the ow approach seems advantageous in tackling very di cult problems. The ow approach has potential applications ranging from new development o f n umerical algorithms to the theoretical solution of open problems. Various aspects of the recent d e v elopment and applications of the ow approach are reviewed in this article.

show abstract

A Numerical Approach to the Inverse Toeplitz Eigenproblem

Cited by 28 publications

References 3 publications

Numerical Methods for Inverse Singular Value Problems

Numerical Methods for Inverse Singular Value Problems

The Projected Gradient Method for Least Squares Matrix Approximations with Spectral Constraints

A list of matrix flows with applications

Contact Info

Product

Resources

About