1,721,011 research outputs found
Centrosymmetric isospectral flows and some inverse eigenvalue problems
No full textAbstractThis paper is concerned with the solution of some structured inverse eigenvalue problems in the class of centrosymmetric matrices. For this aim, isospectral flows evolving in the space of centrosymmetric matrices are considered to numerically construct a symmetric Toeplitz matrix or a persymmetric Hankel matrix from prescribed eigenvalues. We establish a link between the two problems and we investigate the use of simultaneously diagonalizable algebra based on sine transform [Linear Algebra Appl. 52/53 (1983) 992] to choose the starting centrosymmetric matrices for the isospectral flows. Some numerical tests show that our approach can tackle both problems when the solvability is guaranteed and it can give good insights when the existence of the solution is not guaranteed
The use of the factorization of five diagonal matrices by tridiagonal Toeplitz matrices
No full textThe aim of this paper is the use of the factorization of five-diagonal matrices as the product of two Toeplitz tridiagonal matrices. Either bounds for the inverse or numerical methods for solving linear systems may be derived. Some results will be extended to block five-diagonal matrices. Applications to the numerical solution of ODE and PDE together with numerical tests will be given
“The Cayley Method and the Inverse Eigenvalue Problem for Toeplitz Matrices”
No full textDespite the fact that symmetric Toeplitz matrices can have arbitrary eigenvalues, the numerical construction of such a matrix having prescribed eigenvalues remains to be a challenge. A two-step method using the continuation idea is proposed in this paper. The first step constructs a centro-symmetric Jacobi matrix with the prescribed eigenvalues in finitely many steps. The second step uses the Cayley transform to integrate flows in the linear subspace of skew-symmetric and centro-symmetric matrices. No special geometric integrators are needed. The convergence analysis is illustrated for the case of n = 3. Numerical examples are presented
The global error of Magnus methods based on the Cayley map for some oscillatory problems
No full textThis paper deals with numerical methods for the discretization of highly oscillatory systems. We approach the problem
by writing the solution in terms of the Magnus expansion based on the Cayley map. The global error, obtained when the
method is applied to the linear oscillator, is investigated. Moreover, we provide numerical experiments in order to validate
our theoretical results
Innovative numerical methods for solving double-bracket systems
No full textThis paper deals with numerical methods for double-bracket flows. This kind of problems has many applications, for instance in the field of linear programming, sorting of data, neural networking. We perform the Taylor expansion of the related skew-symmetric system solution obtained by means of the unitary Cayley transform. The resulting schemes are compared with the ones obtained on the exponential map (Iserles (2001)) and with some classical semi-explicit methods
Coupling quadrature and continuous Runge–Kutta methods for optimal control problems
No full textThis article deals with the numerical solution of optimal control problems for ordinary differential
equations. The approach is based on the coupling between quadrature rules and continuous Runge–
Kutta solvers, and it lies in the framework of direct optimization methods and recursive discretization
techniques. The analysis of discrete solution accuracy has been carried out and coupling criteria
are established to have global methods featured by a given accuracy order. Consequently, numerical
schemes are built up to high orders. The effectiveness of the proposed schemes has been validated on
several test problems arising in the field of economic applications. The search for optimal solutions has been performed by standard
algorithms in Matlab environment.This article deals with the numerical solution of optimal control problems for ordinary differential equations. The approach is based on the coupling between quadrature rules and continuous Runge-Kutta solvers, and it lies in the framework of direct optimization methods and recursive discretization techniques. The analysis of discrete solution accuracy has been carried out and coupling criteria are established to have global methods featured by a given accuracy order. Consequently, numerical schemes are built up to high orders. The effectiveness of the proposed schemes has been validated on several test problems arising in the field of economic applications. Results have been compared with the ones by classical Runge-Kutta methods, in terms of single function evaluations and average CPU time of the optimization process. The search for optimal solutions has been performed by standard algorithms in Matlab environment
One step semi-explicit methods based on the Cayley transform for solving isospectral flows
No full textThis note deals with the numerical solution of the matrix differential system Y′ = [B(t,Y), Y], Y(0) = Y0, t ⩾ 0, where Y0 is a real constant symmetric matrix, B maps symmetric into skew-symmetric matrices, and [B(t,Y),Y] is the Lie bracket commutator of B(t,Y) and Y, i.e. [B(t,Y),Y] = B(t,Y)Y − YB(t,Y). The unique solution of (1) is isospectral, that is the matrix Y(t) preserves the eigenvalues of Y0 and is symmetric for all t (see [1, 5]). Isospectral methods exploit the Flaschka formulation of (1) in which Y(t) is written as Y(t) = U(t)Y0UT(t), for t ⩾ 0, where U(t) is the orthogonal solution of the differential system U′ = B(t, UY0UT)U, U(0) = I, t ⩾ 0, (see [5]). Here a numerical procedure based on the Cayley transform is proposed and compared with known isospectral methods
“On Robust Matrix Completion with Prescribed Eigenvalues”
No full textMatrix completion with prescribed eigenvalues is a special kind of inverse eigenvalue problems. Thus far, only a handful of specific cases concerning its existence and construction have been studied in the literature. The general problem where the prescribed entries are at arbitrary locations with arbitrary cardinalities proves to be challenging both theoretically and computationally. This paper investigates some continuation techniques by recasting the completion problem as an optimization of the distance between the isospectral matrices with the prescribed eigenvalues and the affine matrices with the prescribed entries. The approach not only offers an avenue to solving the completion problem in its most general setting but also makes it possible to seek a robust solution that is least sensitive to perturbation
Numerical methods based on Gaussian quadrature and continuous Runge-Kutta integration for optimal control problems
No full textThis paper provides a numerical approach for solving optimal
control problems governed by ordinary differential equations. Continuous
extension of an explicit, fixed step-size Runge-Kutta scheme is
used in order to approximate state variables; moreover, the objective
function is discretized by means of Gaussian quadrature rules. The resulting
scheme represents a nonlinear programming problem, which can
be solved by optimization algorithms. With the aim to test the proposed
method, it is applied to different problems.This paper provides a numerical approach for solving optimal control problems governed by ordinary differential equations. Continuous extension of an explicit, fixed step-size Runge-Kutta scheme is used in order to approximate state variables; moreover, the objective function is discretized by means of Gaussian quadrature rules. The resulting scheme represents a nonlinear programming problem, which can be solved by optimization algorithms. With the aim to test the proposed method, it is applied to different problems. © Springer-Verlag 2004
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