1,721,013 research outputs found
A note on estimates of diagonal elements of the inverse of diagonally dominant tridiagonal matrices
No full text
Numerical procedures based on Runge-Kutta methods for solving isospectral flows
No full textThis paper deals with the numerical solution of the Lax system L' = [B(L), L], L(0) = L-0 (*), where L-0 is a constant symmetric matrix, B(-) maps symmetric matrices into skew-symmetric matrices, and [B(L), L] is the commutator of B(L) and L. Here two different procedures, based on the approach recently proposed by Calvo, Iserles and Zanna (the MGLRK methods), are suggested. Such an approach is a computational form for the Flaschka formulation of (*). Our numerical procedures consist in solving (*) by a Runge-Kutta method, then, a single step of a Gauss-Legendre Runge-Kutta (GLRK) method may be applied to the Flaschka formulation of (*). In the first procedure we compute the approximation of the Lax system by a continuous explicit RK method, instead, the second procedure computes the approximation of the Lax system by a GLRK method (the same method used for the Flaschka system). The computational costs have been derived and compared with the ones of the MGLRK methods. Finally, several numerical tests and computational comparisons will be shown
PARALLEL METHODS IN THE NUMERICAL TREATMENT OF POPULATION-DYNAMIC MODELS
No full textIn the numerical treatment of population dynamic models a great number of large linear systems must be solved at each time step. Thus the application of parallel algorithms may result to be convenient. In this paper we analyze how it can be done and the problems that arise in doing it. We propose two different ways of parallel implementation for a numerical method suitable to solve the model considered. As numerical tests we have considered the speedup obtained for a distributed memory parallel computer and the speedup extimated for shared memory parallel computers. These numerical results show high speedup for the second kind of computer
Applications of the Cayley approach in the numerical solution of matrix differential systems on quadratic groups
No full textn recent years several numerical methods have been developed to integrate matrix differential systems of ODEs whose solutions remain on a certain Lie group throughout the evolution. In this paper some results, derived for the orthogonal group in by Diele et al. (1998), will be extended to the class of quadratic groups including the symplectic and Lorentz group. We will show how this approach also applies to ODEs on the Stiefel manifold and the orthogonal factorization of the Lorentz group will be derived. Furthermore, we will consider the numerical solution of important problems such as the Penrose regression problem, the calculation of Lyapunov exponents of Hamiltonian systems, the solution of Hamiltonian isospectral problems. Numerical tests will show the performance of our numerical methods
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
Variable Step-Size Techniques in Continuous Runge-Kutta Methods for Isospectral Dynamical Systems
No full textIn this paper we consider numerical methods for the dynamical system L′ = [B(L), L], L(0) = L0, (*) where L0 is a n × n symmetric matrix, [B(L), L] is the commutator of B(L) and L, and B(L) is a skew-symmetric matrix for each symmetric matrix L. The differential system is isospectral, i.e., L(t) preserves the eigenvalues of L0, for t ≥ 0. The matrix B(L) characterizes the flow, and for special B(·), the solution matrix L(t) tends, as t increases, to a diagonal matrix with the same eigenvalues of L0. In [11] a modification of the MGLRK methods, introduced in [2], has been proposed. These procedures are based on a numerical approximation of the Flaschka formulation of (*) by Runge-Kutta (RK) methods. Our numerical schemes (denoted by EdGLRKs) consist in solving the system (*) by a continuous explicit Runge-Kutta method (CERK) and then performing a single step of a Gauss-Legendre RK method, for the Flaschka formulation of (*), in order to convert the approximation of L(t) to an isospectral solution. The problems of choosing a constant time step or a variable time step strategy are both of great importance in the application of these methods. In this paper, we introduce a definition of stability for the isospectral numerical methods. This definition involves a potential function associated to the isospectral flow. For the class EdGLRKs we propose a variable step-size strategy, based on this potential function, and an optimal constant time step h in the stability interval. The variable time step strategy will be compared with a known variable step-size strategy for RK methods applied to these dynamical systems. Numerical tests will be given and a comparison with the QR algorithm will be show
A Newton Type Method for Solving Nonlinear Equations on Quadratic Matrix Groups
No full textIn this paper we consider numerical methods for solving nonlinear equations on matrix Lie groups. Recently Owren and Welfert (Technical Report Numerics, No 3/1996, Norwegian University of Science and Technology, Trondheim, Norway, 1996) have proposed a method where the original nonlinear equation F(Y)=0 is transformed into a nonlinear equation on the Lie algebra of the group, thus Newton-type methods may be applied which require the evaluation of exponentials of matrices. Here the previous transformation will be performed by the Cayley approximant of the exponential map. This approach has the advantage that no exponentials of matrices are needed. The numerical tests reported in the last section seem to show that our approach is less expensive and provides a larger convergence region than the method of Owren and Welfert
Variable step-size techniques in continuous Runge-Kutta methods for isospectral dynamical systems
No full textIn this paper we consider numerical methods for the dynamical system L′ = [B(L), L], L(0) = L0, (*) where L0 is a n × n symmetric matrix, [B(L), L] is the commutator of B(L) and L, and B(L) is a skew-symmetric matrix for each symmetric matrix L. The differential system is isospectral, i.e., L(t) preserves the eigenvalues of L0, for t ≥ 0. The matrix B(L) characterizes the flow, and for special B(·), the solution matrix L(t) tends, as t increases, to a diagonal matrix with the same eigenvalues of L0. In [11] a modification of the MGLRK methods, introduced in [2], has been proposed. These procedures are based on a numerical approximation of the Flaschka formulation of (*) by Runge-Kutta (RK) methods. Our numerical schemes (denoted by EdGLRKs) consist in solving the system (*) by a continuous explicit Runge-Kutta method (CERK) and then performing a single step of a Gauss-Legendre RK method, for the Flaschka formulation of (*), in order to convert the approximation of L(t) to an isospectral solution. The problems of choosing a constant time step or a variable time step strategy are both of great importance in the application of these methods. In this paper, we introduce a definition of stability for the isospectral numerical methods. This definition involves a potential function associated to the isospectral flow. For the class EdGLRKs we propose a variable step-size strategy, based on this potential function, and an optimal constant time step h in the stability interval. The variable time step strategy will be compared with a known variable step-size strategy for RK methods applied to these dynamical systems. Numerical tests will be given and a comparison with the QR algorithm will be shown
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