In this paper we are interested in the polynomial Krylov approximations for the computation of phi(A)upsilon, where A is a square matrix, v represents a given vector, and. is a suitable function which can be employed in modern integrators for differential problems. Our aim consists of proposing and analyzing innovative a posteriori error estimates which allow a good control of the approximation procedure. The effectiveness of the results we provide is tested on some numerical examples of interest.

We are concerned with the discretization of optimal control problems when a Runge-Kutta scheme is selected for the related Hamiltonian system. It is known that Lagrangian's first order conditions on the discrete model, require a symplectic partitioned Runge-Kutta scheme for state-costate equations. In the present paper this result is extended to growth models, widely used in Economics studies, where the system is described by a current Hamiltonian.

The Leontief model, originally developed for describing an economic system in terms of mutually interrelated and structurally conditioned simultaneous flows of commodities and services, has important applications to wide ranging disciplines. A basic model assumes the linear form x=Tx+d, where x represents the total output vector and d represents the final demand vector. The consumption matrix T plays the critical role of characterizing the entire input-output dynamics. Normally, T is determined by massive and arduous data gathering means which inadvertently bring in measurement noises. This paper considers the inverse problem of reconstructing the consumption matrix in the open Leontief model based on a sequence of inexact output vectors and demand vectors. Such a formulation might have two advantages: one is that no internal consumption measurements are required and the other is that inherent errors could be reduced by total least squares techniques. Several numerical methods are suggested. A comparison of performance and an application to real-world data are demonstrated.

This paper deals with the numerical solution of optimal control problems for ODEs. 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 in order 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.

The inverse problem of constructing a symmetric Toeplitz matrix with prescribed eigenvalues has been a challenge both theoretically and computationally in the literature. It is now known in theory that symmetric Toeplitz matrices can have arbitrary real spectra. This paper addresses a similar problem--can the three largest eigenvalues of symmetric pentadiagonal Toeplitz matrices be arbitrary? Given three real numbers ? ? ?, this paper finds that the ratio ? = ?-? ?-? , including infinity if ? = ?, determines whether there is a symmetric pentadiagonal Toeplitz matrix with ?, ? and ? as its three largest eigenvalues. It is shown that such a matrix of size n × n does not exist if n is even and ? is too large or if n is odd and ? is too close to 1. When such a matrix does exist, a numerical method is proposed for the construction.

Matrix completion with prescribed eigenvalues is a special type of inverse eigenvalue problem. The goal is to construct a matrix subject to both the structural constraint of prescribed entries and the spectral constraint of prescribed spectrum. The challenge of such a completion problem lies in the intertwining of the cardinality and the location of the prescribed entries so that the inverse problem is solvable. An intriguing question is whether matrices can have arbitrary entries at arbitrary locations with arbitrary eigenvalues and how to complete such a matrix. Constructive proofs exist to a certain point (and those proofs, such as the classical Schur-Horn theorem, are amazingly elegant enough in their own right) beyond which very few theories or numerical algorithms are available. In this paper the completion problem is recast as one of minimizing the distance between the isospectral matrices with the prescribed eigenvalues and the affined matrices with the prescribed entries. The gradient flow is proposed as a numerical means to tackle the construction. This approach is general enough that it can be used to explore the existence question when the prescribed entries are at arbitrary locations with arbitrary cardinalities.

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

In the present paper the discretization of a particular model arising in the economic field of innovation diffusion is developed. It consists of an optimal control problem governed by an ordinary differential equation. We propose a direct optimization approach characterized by an explicit, fixed step-size continuous Runge-Kutta integration for the state variable approximation. Moreover, high-order Gaussian quadrature rules are used to discretize the objective function. In this way, the optimal control problem is converted into a nonlinear programming one which is solved by means of classical algorithms.

A matrix Z ? R2n×2n is said to be in the standard symplectic form if Z enjoys a block LU-decomposition in the sense of A 0 -H I Z = I G 0 AT , where A is nonsingular and both G and H are symmetric and positive definite in Rn×n. Such a structure arises naturally in the discrete algebraic Riccati equations. This note contains two results. First, by means of a parameter representation it is shown that the set of all 2n × 2n standard symplectic matrices is closed undermultiplication and, thus, forms a semigroup. Secondly, block LU-decompositions of powers of Z can be derived in closed form which, in turn, can be employed recursively to induce an effective structure-preserving algorithm for solving the Riccati equations. The computational cost of doubling and tripling of the powers is investigated. It is concluded that doubling is the better strategy.

We introduce a degenerate nonlinear parabolic system that describes the chemical aggression of calcium carbonate stones under the attack of sulphur dioxide. For this system, we present some finite element and finite difference schemes to approximate its solutions. Numerical stability is given under suitable CFL conditions. Finally, by means of a formal scaling, the qualitative behavior of the solutions for large times is investigated, and a numerical verification of this asymptotics is given. Our results are in qualitative agreement with the experimental behavior observed in the chemical literature.

Some inverse eigenvalue problems for matrices with Toeplitz-related structure are considered in this paper. In particular, the solutions of the inverse eigenvalue problems for Toeplitz-plus-Hankel matrices and for Toeplitz matrices having all double eigenvalues are characterized, respectively, in close form. Being centrosymmetric itself, the Toeplitz-plus-Hankel solution can be used as an initial value in a continuation method to solve the more difficult inverse eigenvalue problem for symmetric Toeplitz matrices. Numerical testing results show a clear advantage of such an application.