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Runge-Kutta methods for orthogonal and isospectral flows
Abstract
Orthogonal and isospectral flows occur in many applications and they possess important invariants. However, a naive application of Runge-Kutta methods is bound to render these invariants incorrectly. In this paper we describe how to retain relevant invariants with Runge-Kutta methods or, alternatively, incur an error in the rendition of the invariants which is significantly smaller than the overall numerical error.
... Recently a certain effort has been devoted to the numerical solution of isospectral flows (see [2,3,7,12,14]). The usual form of an isospectral flow is the differential system: L' ...
... Furthermore, they have proposed a new approach, based on the Flaschka formulation, and a class of numerical procedures called modified Gauss--Legendre Runge-Kutta methods (MGLRKs, the integer s denotes the stages of GLRK). In [3] the same authors proposed a cheaper but not symmetry preserving method and discussed the effect of symmetry-breaking on the solution. ...
... is skew-symmetric for symmetric arguments, the matrix solution of V' [2,3]). Then, a single iteration of GLRKs will be applied to the linearized ...
This paper deals with the numerical solution of the Lax system L′ = [B(L),L], L(0) = L0 (∗), where L0 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.
... In the last few years a certain effort has been devoted to the numerical solution of isospectral flows (see [2,3,5,9,11,12]). The usual form of these dynamical systems is given by ...
... The theoretical solution L(t) of (1) and (2) is a symmetric real matrix preserving the eigenvalues, that is a(L(t))= a(L0) for t~>0, where a(.) is the spectrum (see [3,7,10,14]). The flow (1), (2) is said to be isospectral. Special cases are the Toda flow and the double-bracket flow. ...
... where, since B(L) is skew-symmetric, U(t) is an unitary matrix for t~>0 (see [3,4,7,14]). ...
In 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, [if[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.
... In recent years, numerical analysts have increasingly appreciated the great importance of qualitative behavior of a system of di erential equations. In fact, several numerical methods have been developed to integrate systems of ODEs whose solutions preserve, during the evolution, certain qualitative features of the initial condition, such as orthogonality, dissipativity, isospectrality, etc (see [2,7,9,8,11,14]). In this paper, we shall look at explicit Runge-Kutta (R-K) methods and how they can preserve any quadratic conservation law when the coe cients of the method are made solution dependent. ...
... with y 0 ∈ R q and f : R q → R q , su ciently smooth function and a partition of the time interval given by t n+1 = t n + h, for n ¿ 0, where t 0 = 0 and h ¿ 0 is the step size. Then applying a s-stage R-K scheme, deÿned by the Butcher tableau c A b T (2) with b = (b 1 ; : : : ; b s ) T , c = (c 1 ; : : : ; c s ) T ∈ R s and A = {a ij } ∈ R s×s , we get ...
... It is well known that no explicit R-K method veriÿes automatically that M = 0 (see for instance [2,5,10]). However, we may impose on any R-K method to be S-conservative, by requiring that the quantity ...
This paper concerns the study of explicit 4-stage fourth order Runge–Kutta methods which preserve quadratic conservation laws when their weights are made solution dependent. Application to orthogonal and Hamiltonian linear systems and numerical tests are also reported.
... Assume that (2) is satisÿed. Since Y 0 is Hermitian and B is skew-Hermitian, then because of (2) we obtain B k Y 0 = (−1) k Y 0 B k , for any k ¿ 0, hence Y (t) = Y H (t) for all t ¿ 0. The converse is trivially true because if Y (t) is Hermitian then Y (t 0 ) = [Y (t 0 )] H and hence (2) follows. ...
... with B(t; Y ) = 1 2 A(t; Y ) (see [2]). Since V (t) is unitary, from (8) it follows that Y (t) is similar to Y 0 for all t, hence it preserves the eigenvalues and the determinant of Y 0 . ...
... By using the fact that Hermitian unitary systems are equivalent to isospectral di erential systems (see (7) or (10)), Hermitian-preserving methods may be derived by known isospectral methods recently proposed for Toda and double bracket ows (see [1,2,4,7]). An isospectral method consists of a computable expression for the isospectral form (8) of the solution. ...
In recent years some numerical methods have been developed to integrate matrix differential systems whose solutions are unitary matrices. In this paper we study numerical methods for the special class of unitary systems the solutions of which are Hermitian matrices throughout the evolution. Several numerical comparisons with known unitary integrators are reported.
... Based on the idea of IEQ method, we propose to transform the finite dimensional Hamiltonian system (1) into the reformulated system (2) by the nonlinear transformation, which has the quadratic invariant energy function. Suppose, the quadratic invariant energy functioñ H(X(t)) is the positive definite quadratic function, we find that the reformulated system (2) can be written as the following Lie group equation by the linear transformation ...
... And as a kind of exponential integrator, it also has good stability. The RKMK method has been widely to solve the differential equation on manifolds [2,4,12,26]. In this paper, we present a new explicit energy preserving (EEP)-RKMK method for the nonlinear pendulum problem and the Hénon-Heiles system based on the nonlinear transformation and the RKMK method. ...
By the nonlinear transformation, some classical Hamiltonian systems can be written as the reformulation systems, which have the quadratic positive definite energy conserving function. These new reformulated systems are transformed into the ordinary differential equations on manifolds by the linear transformation. The explicit Runge-Kutta Munthe-Kaas (RKMK) method, which is a kind of Lie group method, is applied to solve the differential equations on manifolds. Therefore, a explicit energy preserving(EEP) RKMK method is proposed. The EEP-RKMK method is applied to solve the two classical dynamical systems: the pendulum problem and the Hénon-Heiles system. Thus, the explicit energy conserving schemes of the two classical dynamical systems are obtained. Numerical simulations investigate the effective of these new schemes in preserving energy conserving property of these equations and well simulating the dynamical behaviors.
AMS subject classification: 37K05 65M20 65M70
... As an example, one could now consider the Lie-Euler method (7) in this setting, which coincides with the exponential Euler method ...
... If the eigenvalues are distinct, then the isotropy group is discrete and consists of all matrices in SO(d) which are diagonal. Lie group integrators for isospectral flows have been extensively studied, see for example [7,8]. See also [10] for an application to the KdV equation. ...
We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed. Finally, a family of symplectic integrators on cotangent bundles of Lie groups is presented and the notion of discrete gradient methods is generalised to Lie groups.
... In the last few years numerical analysts have shown a growing interest in the numerical solution of matrix systems of ODEs whose solutions preserve, during the evolution, certain qualitative features of the initial matrices, such as orthogonality, symplecticness, isospectrality (see [1,3,7,9,11,15,18,21,25,27]). In general, if the theoretical solution evolves on a matrix Lie group G, then ad-hoc numerical procedures could be applied in order to obtain a numerical solution on the same matrix Lie group (see, for instance, [17,23,24,30,31]). ...
... The numerical solution of this Hamiltonian isospectral problem has been computed solving the associated ortho-symplectic system (38) either by the CayRK( 1 2 ) or by the method of the adjoint equation of order 2 proposed in [2]. Fig. 3 plots the Hamiltonian error of the numerical solution, that is, L T k J + J L k F , at t = 10 against the CPU time, obtained by the Cayley method (solid line) and by the method described in [1] (dashed line), for step-sizes h = 0.4, 0.2, 0.1, 0.05, 0.025, 0.0125. We observe that CayRK( 1 2 ) method provides better results. ...
In 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.
... where Y 0 is a unitary matrix and F (t, Y ) is skew-Hermitian. Such systems arise, for instance, in the computation of Lyapunov exponents, in the smooth singular value decomposition of parametrized matrices (see [7,8]), as well as in the numerical solution of isospectral flow (see [1,2,[13][14][15]). Some unitary integrators for such problems have recently been proposed (see [2,6,12]). ...
... Such systems arise, for instance, in the computation of Lyapunov exponents, in the smooth singular value decomposition of parametrized matrices (see [7,8]), as well as in the numerical solution of isospectral flow (see [1,2,[13][14][15]). Some unitary integrators for such problems have recently been proposed (see [2,6,12]). Dieci et al. presented two different approaches: structural unitary integrators based on s-stage Gauss-Legendre-Runge-Kutta (GLRKs) methods, which automatically preserve unitariness, and projected unitary integrators (see [6]). The projected integrators proposed in [6] consist of a two-step procedure in which an approximation of the solution, provided by any explicit formulas, is computed and then projected onto the set of unitary matrices. ...
In recent years some numerical methods have been developed to integrate matrix differential systems whose solutions are unitary matrices. In this paper we propose a new approach that transforms the original problem into a skew-Hermitian differential system by means of the Cayley transform. The new methods are semi-explicit, that is, no iteration is required but the solution of a certain number of linear matrix systems at each step is needed. Several numerical comparisons with known unitary integrators are reported.
... Firstly, invariants (also known as conservation laws or integrals) often represent important physical quantities of the underlying system and, intuitively, we should distrust a numerical solution that renders them wrongly. Secondly, retention of invariants is often instrumental in decreasing the accumulation of error in longterm time-stepping discretization (Calvo, Iserles & Zanna 1996, Sanz-Serna & Calvo 1994 and in the recovery of correct asymptotic behaviour (Wisdom & Holman 1991). Thirdly, and perhaps most importantly, it makes sense to be guided by a`principle of avarice': having deduced, often after prolonged and di cult mathematical tour de force, qualitative features of an underlying system, it makes little sense to give them up during discretization : : : . ...
... orthogonal ows (Dieci, Russell & van Vleck 1994), Hamiltonian problems (Sanz-Serna & Calvo 1994), equations on the orthogonal groups O(d) and SO(d), on the unitary group U(d) and on Stiefel and Grassmann manifolds (Olver 1995). 4 Moreover, certain invariants can be represented as a group action of SO(d), and this can be exploited in their discretization by Runge{Kutta methods, isospectral ows being the most prominent example (Calvo et al. 1996, Calvo et al. 1997a, Calvo, Iserles & Zanna 1997b. ...
We explore the retention of invariants by a class of time-stepping discretization methods, inclusive of multistep methods and truncated Taylor expansions.
... The pioneering work of this topic was given by Crouch and Grossman [7], who were the first to introduce numerical methods that evolve on manifolds. They applied a classical Runge-Kutta (RK) method to Lie-group equations by repeatedly freezing and thawing coefficients and keeping the flow in the correct configuration space [3,15,16]. RK methods on Lie-groups have been further studied in [14,17,[25][26][27]. Among them, Munthe-Kaas considered applying the classical RK methods to the Lie algebra equation of Lie-group equations on manifolds and for related work see [8,[25][26][27]31]. ...
... Methods that do not preserve quadratic conservation laws would require a projection step or a very small stepsize to reduce the local error hence the unitarity error. Alternatively, one could use methods for orthogonal flows [15,2,11] or with Lie-group methods [3, 10, 1] but this is beside the scope of the present paper. As a proof of concept, we will use Gauss-Legendre Runge-Kutta methods, which are implicit. ...
... In recent years there has been an increasing attention to numerical methods for ordinary di erential equations that preserve some qualitative features of the underlying ow such as symplecticness, isospectrality or orthogonality [2,4]. This paper is concerned with modiÿed Gauss-Legendre Runge-Kutta (GLRK) type methods that preserve the orthogonality of the ow. ...
This paper deals with the numerical integration of matrix differential equations of type Y′(t)=F(t,Y(t))Y(t) where F maps, for all t, orthogonal to skew-symmetric matrices. It has been shown (Dieci et al., SIAM J. Numer. Anal. 31 (1994) 261–281; Iserles and Zanna, Technical Report NA5, Univ. of Cambridge, 1995) that Gauss–Legendre Runge–Kutta (GLRK) methods preserve the orthogonality of the flow generated by Y′=F(t,Y)Y whenever F(t,Y) is a skew-symmetric matrix, but the implicit nature of the methods is a serious drawback in practical applications. Recently, Higham (Appl. Numer. Math. 22 (1996) 217–223) has shown that there exist linearly implicit methods based on the GLRK methods with orders ⩽2 which preserve the orthogonality of the flow. The aim of this paper is to study the order and stability properties of a class of linearly implicit orthogonal methods of GLRK type obtained by extending Higham's approach. Also two particular linearly implicit schemes with orders 3 and 4 based on the two-stage GLRK method that minimize the local truncation error are proposed. In addition, the results of several numerical experiments are presented to test the behaviour of the new methods.
... If the eigenvalues are distinct, then the isotropy group is discrete and consists of all matrices in SO(d) which are diago- nal. Lie group integrators for isospectral flows have been extensively studied, see for example [11, 12]. See also [14] for an application to the KdV equation. ...
In this survey we discuss a wide variety of aspects related to Lie group
integrators. These numerical integration schemes for differential equations on
manifolds have been studied in a general and systematic manner since the 1990s
and the activity has since then branched out in several different subareas,
focussing both on theoretical and practical issues. From two alternative
setups, using either frames or Lie group actions on a manifold, we here
introduce the most important classes of schemes used to integrate nonlinear
ordinary differential equations on Lie groups and manifolds. We describe a
number of different applications where there is a natural action by a Lie group
on a manifold such that our integrators can be implemented. An issue which is
not well understood is the role of isotropy and how it affects the behaviour of
the numerical methods. The order theory of numerical Lie group integrators has
become an advanced subtopic in its own right, and here we give a brief
introduction on a somewhat elementary level. Finally, we shall discuss Lie
group integrators having the property that they preserve a symplectic structure
or a first integral.
... In recent years, however, there has been great interest in special-purpose methods, designed to exactly preserve the mathematical properties of various special classes of ODE's. This has led to symplectic integrators for Hamiltonian ODE's [1], volume-preserving integrators for divergence-free ODE's [2,3], symmetry-preserving integrators for ODE's possessing some symmetry group [4,5], and isospectral integrators for isospectral ODE's [6,7]. 1 In this paper, we construct integral-preserving integrators (IPI's) for ODE's possessing an arbitrary number of first integrals (i.e. conserved quantities). ...
Using Discrete Gradient Methods (Quispel & Turner, J. Phys. A29 (1996) L341-L349) we construct integration methods that solve ordinary difierential equa-tions numerically while preserving all their ˉrst integrals (i. e. constants of the motion) exactly.
... For simplicity, we neglect the computational costs of order lower than n 3, while we consider the cost for multiplying matrices, the cost for solving linear systems, the cost ~(n) for computing the Lie-bracket operator, the cost 6(n) for computing B(t, Y). We denote by ADJRKv the method of the adjoint equation (proposed in [2]) in which the Flaschka and adjoint equation are both solved by a v-stage explicit RK method. We denote by PRKv the semi-explicit method in which U1 is computed by a projection method based on the modified Gram-Schmidt algorithm (see [4]). ...
This 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.
... Isospectral ows arose in the study of ODEs with conservative properties. For example see [1,7,3]. We recall the following Deÿnition 3. ...
The inverse eigenvalue problem for Toeplitz matrices (ITEP), concerning the reconstruction of a symmetric Toeplitz matrix from prescribed spectral data, is considered. To numerically construct such a matrix the approach introduced by Chu in (SIAM Rev. 40(1) (1998) 1–39) is followed. He proposed to solve the ITEP by using an isospectral flow whose equilibria are symmetric Toeplitz matrices. In this paper we study the previous isospectral flow for reversed times and we obtain some formal properties of the solution. The case n=3 for ITEP is analytically investigated by following an approach different from the one in (Chu, SIAM Rev. 40(1) (1998) 1–39). We prove that the flow globally converges to a regular Toeplitz matrix starting from a tridiagonal symmetric and centro-symmetric matrix. Numerical experiments confirm the above results and suggest their extension in higher dimension.
... During the last few years there has been a growing interest in the use of geometric integrators for the numerical integration of ordinary differential equations (ODEs) (see [15], [6], [3] and references therein). These types of integrators are those that preserve some relevant geometric structure of the original solution (symplecticness, orthogonality, isospectrality, energy-conservation, etc.), which makes that, in most cases, the approximate solutions have smaller long-term qualitative error than other standard integrators. ...
The purpose of this paper is to construct a class of orthogonal integrators for stochastic differential equations (SDEs). The family of SDEs with orthogonal solutions is univocally characterized. For this, a class of orthogonal integrators is introduced by imposing constrains to Runge-Kutta (RK) matrices and weights of the standard stochastic RK schemes.The performance of the method is illustrated by means of numerical simulations.
... In discretization of a flow on O(n) it is difficult to ensure that the updates keep the answer always orthogonal. Different methods have been proposed to address this [4], [10], [12]. We mention that in the context of ICA an Euler discretization with small enough fixed step-size, which is equivalent to steepest descent algorithm, is promising. ...
We present a set of gradient based orthogonal and non- orthogonal matrix joint diagonalization algorithms. Our approach is to use the geometry of matrix Lie groups to develop continuous-time ∞ows for joint diagonalization and derive their discretized versions. We employ the developed methods to construct a class of Independent Component Analysis (ICA) algorithms based on non-orthogonal joint diagonaliza- tion. These algorithms pre-whiten or sphere the data but do not restrict the subsequent search for the (reduced) un-mixing matrix to orthogonal matrices, hence they make efiective use of both second and higher order statistics.
... Theorem 4.2 concerns methods thatàlmostthatàlmost' preserve an integral of the system; that is, they preserve the integral to a higher order than the classic order suggests. Examples of such methods have been proposed by Calvo, Iserles & Zanna (1996). Theorem 4.4 concerns the preservation of "xed points. ...
An arbitrary consistent one-step approximation of an ordinary differential equation is studied. If the scheme is assumed to be accurate of O(△t
r
) then it may be shown to be O(△t
r
+m
) accurate as an approximation to a modified equation for any integer m ≥ 1. A technique is introduced for proving that the modified equations inherit qualitative properties from the numerical method. For Hamiltonian problems the modified equation is shown to be Hamiltonian if the numerical method is symplectic, and for general problems the modified equation is shown to possess integrals shared between the numerical method and the underlying system. The technique for proving these results does not use the well-known theory for power series expansions of particular methods such as Runge-Kutta schemes, but instead uses a simple contradiction argument based on the approximation properties of the numerical method. Although the results presented are known, the method and generality of the proofs are new and may be of independent interest.
In this survey we discuss a wide variety of aspects related to Lie group integrators. These numerical integration schemes for differential equations on manifolds have been studied in a general and systematic manner since the 1990s and the activity has since then branched out in several different subareas, focussing both on theoretical and practical issues. From two alternative setups, using either frames or Lie group actions on a manifold, we here introduce the most important classes of schemes used to integrate nonlinear ordinary differential equations on Lie groups and manifolds. We describe a number of different applications where there is a natural action by a Lie group on a manifold such that our integrators can be implemented. An issue which is not well understood is the role of isotropy and how it affects the behaviour of the numerical methods. The order theory of numerical Lie group integrators has become an advanced subtopic in its own right, and here we give a brief introduction on a somewhat elementary level. Finally, we shall discuss Lie group integrators having the property that they preserve a symplectic structure or a first integral.
A collection of inverse eigenvalue problems are identified and classified according to their characteristics. Current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper. This exposition also reveals many open questions that deserve further study. An extensive bibliography of pertinent literature is attached.
We present a new scale-invariant cost function for non-orthogonal joint-diagonalization of a set of symmetric matrices with application to Independent Component Analysis (ICA). We derive two gradient minimization schemes to minimize this cost function. We also consider their performance in the context of an ICA algorithm based on non-orthogonal joint diagonalization.
We present a general procedure for recursively improving the invariance of a
numerical integrator under a symmetry group. If h is a symmetry, we construct the
adjoint
method
h−1 h.
In each time step we apply either the original method or the adjoint method, according to a
prescription based on the Thue–Morse
sequence. The outcome is a solution sequence which
displays progressively smaller symmetry errors, to any desired order in the time-step. The
method can also be used to force the solution to stay close to a desired submanifold of phase
space, while retaining structural properties of the original method.
In the last years several numerical methods have been developed to integrate matrix differential equations which preserve
certain features of the theoretical solution such as orthogonality, eigenvalues, first integrals, etc. In this paper we approach
the numerical solution of a second order matrix differential system whose solution evolves on the Lie-group of the orthogonal
matrices
On
\mathcal{O}_n
. We study the orthogonality properties of classical Runge Kutta Nyström methods and non standard numeeical procedures for
second order ordinary differential equations.
In this paper we describe a Fortran90 routine for the numerical integration of orthogonal differential systems based on the
Cayley transform methods. Three different implementations of the methods are given: with restart, with restart at each step
and in composed form. Numerical tests will show the performances of the solver for the solution of orthogonal test problems,
of orthogonal rectangular problems and for the calculation of Lyapunov exponents in the linear and nonlinear cases, and finally
for solving inverse eigenvalue problem for Toeplitz matrices. The results obtained using Cayley methods are compared with
those given by Fortran90 version of Munthe-Kaas methods, which have been coded in a similar way.
In the paper, we introduce a kind of Lie group method to solve non-damping Landau–Lifshitz (LL) equations and the ferromagnet chain equations. The basic idea is to discretize the partial differential equation into the form dY/dt=F(t,Y), then apply the Lie group method to it. The iterative numerical algorithm can preserve the property of square conservation of the discrete equation. The algorithm has the same accuracy as the classical Runge–Kutta method. It shows that we have proposed a valid algorithm to solve ferromagnet chain equations. The method can be applied to other similar partial differential equations.
We show that the Lax equation L̇=[N,L] applied for calculating the eigenvalues of matrices is only one in a whole family of matrix differential equations related to eigenvalue problem. All these equations are built by means of various representations ρ of a Lie algebra g on Kn×n. A general connection between matrix differential equation L̇=ρN(L) and FG eigenvalue algorithm is shown. Remarks concerning practical applications of this connection to matrix eigenvalues calculations are also presented.
A model for analysis and visualization of asymmetric data—GIPSCAL—is reconsidered by means of the projected gradient approach. GIPSCAL problem is formulated as initial value problem for certain first order matrix ordinary differential equations. This results in a globally convergent algorithm for solving GIPSCAL. Additionally, first and second order optimality conditions for the solutions are established. A generalization of the GIPSCAL model for analyzing three-way arrays is also considered. Finally, results from simulation experiments are reported.
Thesis research directed by: Electrical Engineering. Title from t.p. of PDF. Thesis (M.S.) -- University of Maryland, College Park, 2004. Includes bibliographical references. Text.
This paper is concerned with the quasi-linear Burgers equation by using the exponential integral methods. Spatial discretization of this kind of partial differential equations leads to a system of ordinary differential equations, which can be solved by the numerical methods. Furthermore, numerical results show that the exponential integral method has explicit stability, and has the same accuracy as the corresponding Runge-Kutta method.
. In this paper we are concerned with the problem of solving numerically isospectral flows. These flows are characterized by the differential equation L 0 = [B(L);L]; L(0) = L 0 ; where L 0 is a d Theta d symmetric matrix, B(L) is a skew-symmetric matrix function of L and [B; L] is the Lie bracket operator. We show that standard Runge--Kutta schemes fail in recovering the main qualitative feature of these flows, that is isospectrality, since they cannot recover arbitrary cubic conservation laws. This failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order. 1. Background and notation 1.1. Introduction. The interest in solving isospectral flows is motivated by their relevance in a wide range of applications, from molecular dynamics to micromagnetics to linear algebra. The general form of an isospectral flow is the differential equation L 0 = [B(L); L]; L(0) = L 0 ; (1) where L 0 is a give...
The traditional goal of numerical analysis is to devise and understand algorithms that approximate fast, robustly and within given error tolerance the solution of an underlying mathematical problem. In this paper we advance a philosophy that, while not at variance with this goal, calls for an important change of emphasis. In many problems of interest, mathematical analysis provides us with a partial knowledge of the solution, its properties and possibly its invariants. We suggest that such problems be computed by numerical methods that share their qualitative attributes. Thus, the recorvery of the correct qualitative behaviour of a mathematical problem should be seen as a major criterion in devising and analysing numerical algorithms. We specialise our discussion to three examples, all within the broader framework of ordinary differential equations and Runge-Kutta methods: orthogonal flows, symplectic flows and isospectral flows.
Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This new edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems.
1. Preliminaries.- 1.1 A Simple Example: Motion in a Potential Field.- 1.2 Poisson Structure and Hamiltonian Systems.- 1.3 Symplectic Manifolds.- 1.4 Homogeneous Symplectic Spaces.- 1.5 The Moment Map.- 1.6 Hamiltonian Systems with Symmetry.- 1.7 Reduction of Hamiltonian Systems with Symmetry.- 1.8 Integrable Hamiltonian Systems.- 1.9 The Projection Method.- 1.10 The Isospectral Deformation Method.- 1.11 Hamiltonian Systems on Coadjoint Orbits of Lie Groups.- 1.12 Constructions of Hamiltonian Systems with Large Families of Integrals of Motion.- 1.13 Completeness of Involutive Systems.- 1.14 Hamiltonian Systems and Algebraic Curves.- 2. Simplest Systems.- 2.1 Systems with One Degree of Freedom.- 2.2 Systems with Two Degrees of Freedom.- 2.3 Separation of Variables.- 2.4 Systems with Quadratic Integrals of Motion.- 2.5 Motion in a Central Field.- 2.6 Systems with Closed Trajectories.- 2.7 The Harmonic Oscillator.- 2.8 The Kepler Problem.- 2.9 Motion in Coupled Newtonian and Homogeneous Fields.- 2.10 Motion in the Field of Two Newtonian Centers.- 3. Many-Body Systems.- 3.1 Lax Representation for Many-Body Systems.- 3.2 Completely Integrable Many-Body Systems.- 3.3 Explicit Integration of the Equations of Motion for Systems of Type I and V via the Projection Method.- 3.4 Relationship Between the Solutions of the Equations of Motion for Systems of Type I and V.- 3.5 Explicit Integration of the Equations of Motion for Systems of Type II and III.- 3.6 Integration of the Equations of Motion for Systems with Two Types of Particles.- 3.7 Many-Body Systems as Reduced Systems.- 3.8 Generalizations of Many-Body Systems of Type I-III to the Case of the Root Systems of Simple Lie Algebras.- 3.9 Complete Integrability of the Systems of Section 3.8.- 3.10 Anisotropic Harmonic Oscillator in the Field of a Quartic Central Potential (the Garnier System).- 3.11 A Family of Integrable Quartic Potentials Related to Symmetric Spaces.- 4. The Toda Lattice.- 4.1 The Ordinary Toda Lattice. Lax Representation. Complete Integrability.- 4.2 The Toda Lattice as a Dynamical System on a Coadjoint Orbit of the Group of Triangular Matrices.- 4.3 Explicit Integration of the Equations of Motion for the Ordinary Nonperiodic Toda Lattice.- 4.4 The Toda Lattice as a Reduced System.- 4.5 Generalized Nonperiodic Toda Lattices Related to Simple Lie Algebras.- 4.6 Toda-like Systems on Coadjoint Orbits of Borel Subgroups.- 4.7 Canonical Coordinates for Systems of Toda Type.- 4.8 Integrability of Toda-like Systems on Generic Orbits.- 5. Miscellanea.- 5.1 Equilibrium Configurations and Small Oscillations of Some Integrable Hamiltonian Systems.- 5.2 Motion of the Poles of Solutions of Nonlinear Evolution Equations and Related Many-Body Problems.- 5.3 Motion of the Zeros of Solutions of Linear Evolution Equations and Related Many-Body Problems.- 5.4 Concluding Remarks.- Appendix A.- Examples of Symplectic Non-Kahlerian Manifolds.- Appendix B.- Solution of the Functional Equation (3.1.9).- Appendix C.- Semisimple Lie Algebras and Root Systems.- Appendix D.- Symmetric Spaces.- References.
The Toda flow is a dynamical system whose dependent variables may be viewed as the entries of a symmetric tridiagonal matrix. The spectrum of the matrix is invariant in time and, as $t \to \infty $, the off-diagonal entries tend to zero, exposing the eigenvalues on the main diagonal. Recently a wave of interest in the Toda and related isospectral flows was sparked in the numerical analysis community by the suggestion that to integrate the Toda flow numerically might be a cost effective way to calculate the eigenvalues of large symmetric tridiagonal matrices. A second source of interest was the recently discovered connection between the Toda flow and the $QR$ algorithm. The present paper has two principal aims. The first is to point out that the Toda and related flows are intimately connected with the power method. This connection clarifies completely the convergence properties of the flows. The relationship of the Toda flow to both the $QR$ algorithm and the power method is based on a connection with the $QR$ matrix factorization. The second aim of this paper is to introduce and discuss families of isospectral flows associated with two other well-known matrix factorizations, the $LU$ factorization and the Cholesky factorization.
In this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed. Such systems have skew-Hermitian coefficient matrices in the linear case and a related structure in the nonlinear case. These skew systems arise in a number of applications, and interest originates from application to continuous orthogonal decoupling techniques. In this case, the matrix system has a cubic nonlinearity.
Numerical integration schemes that compute a unitary approximate solution for all stepsizes are studied. These schemes can be characterized as being of two classes: automatic and projected unitary schemes. In the former class, there belong those standard finite difference schemes which give a unitary solution; the only ones are in fact the Gauss–Legendre point Runge–Kutta (Gauss RK) schemes. The second class of schemes is created by projecting approximations computed by an arbitrary scheme into the set of unitary matrices. In the analysis of these unitary schemes, the stability considerations are guided by the skew-Hermitian character of the problem. Various error and implementation issues are considered, and the methods are tested on a number of examples.
A solution of a system of m autonomous differential equations defines a trajectory in m -dimensional space and, in particular, may give a closed orbital path. Typical trajectories are described by a model nonlinear problem introduced in this article. For this problem, a trajectory lies on a surface characterized by a real symmetric matrix. It is shown that some Runge-Kutta methods possess a property which ensures that, for this model problem, the numerical solution lies on the same surface as the trajectory. When m = 2, the numerical solution lies on the trajectory. This property is related to algebraic stability. A weaker property suffices for normalized differential systems.
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