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Validated Solutions Of Initial Value Problems For Ordinary Differential Equations
, 1996
"... . Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated methods for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an e ..."
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Cited by 69 (11 self)
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. Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated methods for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced. The authors survey Taylor series methods for validated solutions of IVPs for ODEs, describe several such methods in a common framework, and identify areas for future research. Key words. initial value problems, ordinary differential equations, interval arithmetic, Taylor series methods. AMS subject classifications. 65L05, 65G10, 65L60. 1. Introduction. We consider validated numerical methods for the solution of the autonomous initialvalue problems (IVPs) y 0 (t) = f(y); y(t 0 ) = y 0 ; (1.1) where t 2 [t 0 ; T ] for some T ? t 0 . Here t 0 ; T 2 R,f 2 C k\Gamma1 (D), D ` R n is an open set, f : D ! R n , and y 0 2 D. For expositional c...
Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation
, 1999
"... Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated (also called interval) methods for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique s ..."
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Cited by 31 (8 self)
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Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated (also called interval) methods for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced. To date, the only effective approach for computing guaranteed enclosures of the solution of an IVP for an ODE has been interval methods based on Taylor series. This thesis derives a new approach, an interval HermiteObreschkoff (IHO) method, for computing such enclosures. Compared to interval Taylor series (ITS) methods, for the same order and stepsize, our IHO scheme has a smaller truncation error and better...
An Interval HermiteObreschkoff Method for Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation
 Developments in Reliable Computing
, 1998
"... To date, the only effective approach for computing guaranteed bounds on the solution of an initial value problem (IVP) for an ordinary differential equation (ODE) has been interval methods based on Taylor series. This paper derives a new approach, an interval HermiteObreschkoff (IHO) method, for co ..."
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Cited by 14 (3 self)
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To date, the only effective approach for computing guaranteed bounds on the solution of an initial value problem (IVP) for an ordinary differential equation (ODE) has been interval methods based on Taylor series. This paper derives a new approach, an interval HermiteObreschkoff (IHO) method, for computing such enclosures. Compared to interval Taylor series (ITS) methods, for the same stepsize and order, our IHO scheme has a smaller truncation error, better stability, and requires fewer Taylor coefficients and highorder Jacobians. The stability properties of the ITS and IHO methods are investigated. We show as an important byproduct of this analysis that the stability of an interval method is determined not only by the stability function of the underlying formula, as in a standard method for an IVP for an ODE, but also by the associated formula for the truncation error.
Precise: Efficient multiprecision evaluation of algebraic roots and predicates for reliable geometric computation
, 2000
"... Many geometric problems like generalized Voronoi diagrams, medial axis computations and boundary evaluation involve computation and manipulation of nonlinear algebraic primitives like curves and surfaces. The algorithms designed for these problems make decisions based on signs of geometric predicat ..."
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Cited by 12 (3 self)
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Many geometric problems like generalized Voronoi diagrams, medial axis computations and boundary evaluation involve computation and manipulation of nonlinear algebraic primitives like curves and surfaces. The algorithms designed for these problems make decisions based on signs of geometric predicates or on the roots of polynomials characterizing the problem. The reliability of the algorithm depends on the accurate evaluation of these signs and roots. In this paper, we present a naive precisiondriven computational model to perform these computations reliably and demonstrate its effectiveness on a certain class of problems like sign of determinants with rational entries, boundary evaluation and curve arrangements. We also present a novel algorithm to compute all the roots of a univariate polynomial to any desired accuracy. The computational model along with the underlying number representation, precisiondriven arithmetic and all the algorithms are implemented as part of a standalone software library, PRECISE. 1.
On Taylor model based integration of ODEs
 SIAM J. Numer. Anal
"... Abstract. Interval methods for verified integration of initial value problems (IVPs) for ODEs have been used for more than 40 years. For many classes of IVPs, these methods are able to compute guaranteed error bounds for the flow of an ODE, where traditional methods provide only approximations to a ..."
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Cited by 12 (0 self)
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Abstract. Interval methods for verified integration of initial value problems (IVPs) for ODEs have been used for more than 40 years. For many classes of IVPs, these methods are able to compute guaranteed error bounds for the flow of an ODE, where traditional methods provide only approximations to a solution. Overestimation, however, is a potential drawback of verified methods. For some problems, the computed error bounds become overly pessimistic, or the integration even breaks down. The dependency problem and the wrapping effect are particular sources of overestimations in interval computations. Berz and his coworkers have developed Taylor model methods, which extend interval arithmetic with symbolic computations. The latter is an effective tool for reducing both the dependency problem and the wrapping effect. By construction, Taylor model methods appear particularly suitable for integrating nonlinear ODEs. We analyze Taylor model based integration of ODEs and compare Taylor model methods with traditional enclosure methods for IVPs for ODEs. AMS subject classifications. 65G40, 65L05, 65L70. Key words. Taylor model methods, verified integration, ODEs, IVPs.
Implicit Methods for Enclosing Solutions of ODEs
 J. of Universal Computer Science
, 1998
"... Abstract: The paper presents a new enclosure method for initial value problems in systems of ordinary di erential equations. Like the common enclosure methods (eg Lohner's algorithm AWA), it is based on Taylor expansion. In contrast to them, however, it is an implicit method. The solution sets of no ..."
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Cited by 5 (0 self)
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Abstract: The paper presents a new enclosure method for initial value problems in systems of ordinary di erential equations. Like the common enclosure methods (eg Lohner's algorithm AWA), it is based on Taylor expansion. In contrast to them, however, it is an implicit method. The solution sets of nonlinear inequalities have to be enclosed by a Newtonlike algorithm. As the presented examples show, the new method sometimes yields much tighter bounds than any of the common explicit methods.
Interval Computations as an Important Part of Granular Computing: An Introduction
"... This chapter provides a general introduction to interval computations, especially to interval computations as an important part of granular computing. ..."
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Cited by 1 (0 self)
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This chapter provides a general introduction to interval computations, especially to interval computations as an important part of granular computing.
On the Blunting Method in the Verified Integration of ODEs
, 2009
"... Verified methods for the integration of initial value problems (IVPs) for ODEs aim at computing guaranteed error bounds for the flow of an ODE and maintaining a low level of overestimation at the same time. This paper is concerned with one of the sources of overestimation: a matrixvector product des ..."
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Cited by 1 (0 self)
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Verified methods for the integration of initial value problems (IVPs) for ODEs aim at computing guaranteed error bounds for the flow of an ODE and maintaining a low level of overestimation at the same time. This paper is concerned with one of the sources of overestimation: a matrixvector product describing a parallelepiped in the phase space. We analyze the socalled blunting method developed by Berz and Makino, which consists of a special choice of the matrix in this product. For the linear model problem u ′ = Au, u(0) = u0 ∈ u0, where u ∈ R m, A ∈ R m×m, m ≥ 2, and u0 is a given interval vector, we compare the convergence behavior of the blunting method with that of the wellknown QR interval method. For both methods, the amount of overestimation of the flow of the initial set depends on the spectral radius of some welldefined matrix. We show that under certain conditions, the spectral radii of the matrices that describe the excess propagation in the QR method and in the blunting method, respectively, have the same limits, so that the excess propagation in both methods is similar.
For UnknownButBounded Errors,
, 1993
"... For many measuring devices, the only information that we have about them is their biggest possible error " ? 0. In other words, we know that the error \Deltax = ~ x \Gamma x (i.e., the difference between the measured value ~ x and the actual values x) is random, that this error can sometimes become ..."
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For many measuring devices, the only information that we have about them is their biggest possible error " ? 0. In other words, we know that the error \Deltax = ~ x \Gamma x (i.e., the difference between the measured value ~ x and the actual values x) is random, that this error can sometimes become as big as " or \Gamma", but we do not have any information about the probabilities of different values of error.