Inverse Eigenvalue Problem Research Articles

Parameter Reconstruction of a Circuit System from Partial Eigeninformation

In this paper, the inverse eigenvalue problem of reconstructing a LC circuit system from its some parameters and incomplete eigenpairs is considered. The necessary and sufficient condition for the solvability of the problem is derived by solving a kind of general inverse eigenvalue problems. The computational expressions of the solution, the related numerical algorithm and a numercical example are given.

On two generalized inverse eigenvalue problems for Hessenberg-upper triangular pencils and their application to the study of GMRES convergence

We discuss two generalized inverse eigenvalue problems. The first one: For a given unreduced upper Hessenberg matrix H, find a nonsingular upper triangular matrix T such that all the pencils Hk−λTk have prescribed eigenvalues, where Hk and Tk are the leading k×k principal submatrices of H and T, respectively. The second one: For a given unitary unreduced upper Hessenberg matrix Q, find a nonsingular upper triangular matrix T such that all the pencils Tk−θQk⁎ have prescribed eigenvalues, where Tk is the leading k×k principal submatrix of T, and Qk⁎ is the conjugate transpose of the leading k×k principal submatrix of Q. We present the necessary and sufficient conditions for the solvability of the two problems. Our results lead to an alternative proof for the statement that any admissible Ritz value set or admissible harmonic Ritz value set is possible for the prescribed GMRES residual norms. Here, the term “admissible” means there are some restrictions on the sets if GMRES stagnates at some iterations.

The solvability conditions for the inverse eigenvalue problem of normal skew J-Hamiltonian matrices

Let J in{mathbb{R}}^{ntimes n} be a normal matrix such that J^{2}=-I_{n}, where I_{n} is an n-by-n identity matrix. In (S. Gigola, L. Lebtahi, N. Thome in Appl. Math. Lett. 48:36–40, 2015) it was introduced that a matrix A in{mathbb {C}}^{ntimes n} is referred to as normal J-Hamiltonian if and only if {(AJ)}^{*}=AJ and AA^{*}=A^{*}A. Furthermore, the necessary and sufficient conditions for the inverse eigenvalue problem of such matrices to be solvable were given. We present some alternative conditions to those given in the aforementioned paper for normal skew J-Hamiltonian matrices. By using Moore–Penrose generalized inverse and generalized singular value decomposition, the necessary and sufficient conditions of its solvability are obtained and a solvable general representation is presented.

The inverse eigenvalue problem for entanglement witnesses

We consider the inverse eigenvalue problem for entanglement witnesses, which asks for a characterization of their possible spectra (or equivalently, of the possible spectra resulting from positive linear maps of matrices). We completely solve this problem in the two-qubit case and we derive a large family of new necessary conditions on the spectra in arbitrary dimensions. We also establish a natural duality relationship with the set of absolutely separable states, and we completely characterize witnesses (i.e., separating hyperplanes) of that set when one of the local dimensions is 2.

A new method for multi-bit and qudit transfer based on commensurate waveguide arrays

The faithful state transfer is an important requirement in the construction of classical and quantum computers. While the high-speed transfer is realized by optical-fibre interconnects, its implementation in integrated optical circuits is affected by cross-talk. The cross-talk between densely packed optical waveguides limits the transfer fidelity and distorts the signal in each channel, thus severely impeding the parallel transfer of states such as classical registers, multiple qubits and qudits. Here, we leverage on the suitably engineered cross-talk between waveguides to achieve the parallel transfer on optical chip. Waveguide coupling coefficients are designed to yield commensurate eigenvalues of the array and hence, periodic revivals of the input state. While, in general, polynomially complex, the inverse eigenvalue problem permits analytic solutions for small number of waveguides. We present exact solutions for arrays of up to nine waveguides and use them to design realistic buses for multi-(qu)bit and qudit transfer. Advantages and limitations of the proposed solution are discussed in the context of available fabrication techniques.

A quadratically convergent algorithm for inverse eigenvalue problems with multiple eigenvalues

In 2017, for inverse symmetric eigenvalue problems, a new quadratically convergent algorithm has been derived from simple matrix equations. Although this algorithm has some nice features compared with the other quadratically convergent methods, it is not applied to multiple eigenvalues. In this paper, we improve this algorithm with the aid of an optimization problem for the eigenvectors associated with multiple eigenvalues. The proposed algorithm is adapted to an arbitrary set of given eigenvalues. The main contribution is our convergence theorem formulated in a different manner from previous work for the existing quadratically convergent methods. Our theorem ensures the quadratic convergence in a neighborhood of the solutions that satisfy a mild condition.

Schur method for robust pole assignment of descriptor systems via proportional plus derivative state feedback

ABSTRACTThe pole assignment problem for descriptor systems is a classical inverse algebraic eigenvalue problem, which has attracted attention for decades. In this paper, we propose a direct method to solve the problem with the application of the proportional plus derivative state feedback, intending to obtain a robust closed-loop system. Theorems on the feasibility of our method will be presented. Numerical examples show that our method yields poles of high relative accuracy.

Universal realizability of spectra with two positive eigenvalues

A list of eigenvalues is said to be realizable if it is the spectrum of a nonnegative matrix, diagonalizably realizable (DR) if it is the spectrum of a diagonalizable nonnegative matrix, and universally realizable (UR) if there is a nonnegative matrix with this spectrum and any possible Jordan canonical form allowed by the spectrum. The nonnegative inverse eigenvalue problem (NIEP) asks which lists are realizable. It is known that there exist spectra that are realizable, but not DR. We raise the question of whether DR implies UR. This is known in a few cases, including n≤4, nonnegative spectra, and Suleimanova spectra. We add some new classes of spectra that are UR.

Inverse spectral problems for discontinuous Sturm–Liouville problems of Atkinson type

We investigate inverse spectral problems for discontinuous Sturm–Liouville problems of Atkinson type whose spectrum consists of a finite set of eigenvalues. For given two finite sets of interlacing real numbers, there exists a class of Sturm–Liouville equations such that the two sets of numbers are exactly the eigenvalues of their associated Sturm–Liouville problems with two different separated boundary conditions. The main approach is to give an equivalent relation between Sturm–Liouville problems of Atkinson type and matrix eigenvalue problems, and the theory of inverse matrix eigenvalue problems.

Inverse eigenvalue problems for acyclic matrices whose graph is a dense centipede

Abstract The reconstruction of a matrix having a pre-defined structure from given spectral data is known as an inverse eigenvalue problem (IEP). In this paper, we consider two IEPs involving the reconstruction of matrices whose graph is a special type of tree called a centipede. We introduce a special type of centipede called dense centipede.We study two IEPs concerning the reconstruction of matrices whose graph is a dense centipede from given partial eigen data. In order to solve these IEPs, a new system of nomenclature of dense centipedes is developed and a new scheme is adopted for labelling the vertices of a dense centipede as per this nomenclature . Using this scheme of labelling, any matrix of a dense centipede can be represented in a special form which we define as a connected arrow matrix. For such a matrix, we derive the recurrence relations among the characteristic polynomials of the leading principal submatrices and use them to solve the above problems. Some numerical results are also provided to illustrate the applicability of the solutions obtained in the paper.

Two equivalent inverse eigenvalue problems about graded-index optical fibers

Two equivalent inverse eigenvalue problems about graded-index optical fibers

Mathematical and numerical studies of an inverse eigenvalue problem on computing guided modes in optical fibers

Mathematical and numerical studies of an inverse eigenvalue problem on computing guided modes in optical fibers

On the bisymmetric nonnegative inverse eigenvalue problem

The problem of finding conditions for a list of n complex numbers to be the eigenvalues of an n×n nonnegative matrix is called the nonnegative inverse eigenvalue problem (NIEP). The SNIEP (BNIEP) is the NIEP when the desired matrices are symmetric (bisymmetric). Recently, some sufficient conditions for the BNIEP are given by Julio and Soto in [6]. In this article, we study the BNIEP for n=4 and prove that the BNIEP and SNIEP are different for n=6. Then we give some sufficient conditions for the BNIEP and for the NIEP for normal centrosymmetric matrices related to the sufficient conditions of Julio and Soto and we also provide certain sufficient conditions for the BNIEP analogous to the sufficient conditions for the NIEP given by Borobia in [1]. Finally, we give sufficient conditions for the BNIEP when the prescribed diagonal entries are required.

A class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth

In this paper, we propose and discuss a class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth. For an odd 2p+1 with a positive integer p, the problem is to construct an n×n real symmetric banded matrix with bandwidth 2p+1 whose m×m leading principal submatrix is a given m×m real symmetric banded matrix with bandwidth 2p+1 and spectrum is a given set of real numbers {λi}i=1n, where the number of distinct real numbers of {λi}i=1n is 2k when m=pk, or 2k+1 when pk<m<p(k+1), where m,n and k are positive integers and m<n. We point out that the well-known double dimensional (DD) problem is a special case of our proposed inverse eigenvalue problems. The necessary and sufficient condition for the solvability of the above inverse eigenvalue problem is derived, and the target real symmetric banded matrix can be constructed by the block Lanczos algorithms when the inverse eigenvalue problem is solvable. Several numerical examples show that our algorithms are feasible. Some concluding remarks are introduced.

Orthogonal Rational Functions on the Unit Circle with Prescribed Poles not on the Unit Circle

Orthogonal rational functions (ORF) on the unit circle generalize orthogonal polynomials (poles at infinity) and Laurent polynomials (poles at zero and infinity). In this paper we investigate the properties of and the relation between these ORF when the poles are all outside or all inside the unit disk, or when they can be anywhere in the extended complex plane outside the unit circle. Some properties of matrices that are the product of elementary unitary transformations will be proved and some connections with related algorithms for direct and inverse eigenvalue problems will be explained.

On a local solvability and stability of the inverse transmission eigenvalue problem

We prove a local solvability and stability of the inverse transmission eigenvalue problem posed by McLaughlin and Polyakov (1994 J. Diff. Equ. 107 351–82). In particular, this result establishes the minimality of the data used therein. The proof is constructive.

A recursive method for constructing doubly stochastic matrices and inverse eigenvalue problem

This article presents a technique for combining two doubly stochastic matrices with known spectra to create a new matrix. As application of this, we obtain a recursive method for constructing doubly stochastic matrices for the inverse eigenvalue problem and find new sufficient conditions for this problem. In addition, we improve the Soules' condition.

Modification of natural frequencies of an automotive belt drive system based on eigen-sensitivity analysis of its configuration parameters

Abstract A methodology is presented to simultaneously modify natural frequencies of an automotive belt drive system via configuration modification to match natural frequencies of the system with its working conditions. In this methodology, the configuration modification is regarded as an inverse eigenvalue problem in which necessary changes of a set of configuration parameters are determined for achieving a desired natural frequency variation relative to the original design value. Natural frequencies and mode shapes of the system are obtained by solving an eigenvalue problem that is directly derived from established governing equations of motion of a typical automotive belt drive system. Explicit expressions of the first and second order derivatives of natural frequencies and eigenvectors with respect to the configuration parameters are derived by differentiating the eigenvalue equation. Several configuration parameters are selected as design variables and their corresponding modification regions and directions are given based on frequency sensitivity analysis. In the inverse eigenvalue problem, variations of natural frequencies caused by changes of multiple design variables are approximated using the first and second order Taylor series expansions, which are based on explicit expressions of the first and second order derivatives of natural frequencies and eigenvectors with respect to the configuration parameters. By solving the inverse eigenvalue problem, necessary changes of design variables are determined and the desired natural frequencies are achieved so that resonance conditions are successfully avoided.

Approximate equiangular tight frames for compressed sensing and CDMA applications

Performance guarantees for recovery algorithms employed in sparse representations, and compressed sensing highlights the importance of incoherence. Optimal bounds of incoherence are attained by equiangular unit norm tight frames (ETFs). Although ETFs are important in many applications, they do not exist for all dimensions, while their construction has been proven extremely difficult. In this paper, we construct frames that are close to ETFs. According to results from frame and graph theory, the existence of an ETF depends on the existence of its signature matrix, that is, a symmetric matrix with certain structure and spectrum consisting of two distinct eigenvalues. We view the construction of a signature matrix as an inverse eigenvalue problem and propose a method that produces frames of any dimensions that are close to ETFs. Due to the achieved equiangularity property, the so obtained frames can be employed as spreading sequences in synchronous code-division multiple access (s-CDMA) systems, besides compressed sensing.

An inverse eigenvalue problem. Application: Graded-index optical fibers

The main objective of this paper is to solve an inverse eigenvalue problem that consists (for a given frequency) in seeking for a suitable graded-index optical fiber able to carry electromagnetic fields having known propagation constants. This is an ill-posed problem. We present an original iterative method, based on Tikhonov regularization in order to demonstrate the existence of a unique solution (the true refractive index or at least a better approximation of it). Otherwise, we compute the optimal regularization parameter thanks to L-curve method. Numerical results from various test cases show that the regularized (unique reconstructed) solution, which corresponds simultaneously to the corner of the L-curve and minimum of relative error (and even together with the minimum of absolute error at this corner), is in good agreement with the true solution.