Multiple Precision Interval Packages: Comparing Different Approaches

Grimmer, Markus; Petras, Knut; Revol, Nathalie

doi:10.1007/978-3-540-24738-8_4

Cited by 21 publications

(8 citation statements)

References 21 publications

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“…31 In addition, software described in ref. 32 can be used to obtain abscissa values and weights for GKP "nested chains," e.g., 3, 7, and 15 nodes. 33 The cubic spline integration was carried out using the SciPy software (www.scipy.org), which in turn relies on the FITPACK library.…”

Section: Some Practical Aspects Of the Numerical Quadrature Schemes Usedmentioning

confidence: 99%

Efficiency of alchemical free energy simulations. II. Improvements for thermodynamic integration

2010

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We attempt to optimize the efficiency of thermodynamic integration, as defined by the minimal number of unphysical intermediate states required for the computation of accurate and precise free energy differences. The suitability of various numerical quadrature methods is tested. In particular, we compare the trapezoidal rule, Simpson's rule, Gauss-Legendre, Gauss-Kronrod-Patterson, and Clenshaw-Curtis integration, as well as integration based on a cubic spline approximation of the integrand. We find that Simpson's rule and spline integration are already significantly more efficient that the trapezoidal rule, i.e., correct free energy differences can be obtained using fewer λ-states. We demonstrate that Simpson's rule can be used advantageously with nonequidistant values of the abscissa, which increases the flexibility of the method. Efficiency is enhanced even further if higher order methods, such as Gauss-Legendre, Gauss-Kronrod-Patterson, or Clenshaw-Curtis integration, are used; no more than seven λ-states, which in the case of Clenshaw-Curtis integration include the physical end states, were required for accurate results in all test problems studied. Thus, the performance of thermodynamic integration can equal that of Bennett's acceptance ratio method. We also show, however, that the high efficiency found here relies on the particular functional form of the soft-core potential used; overall, thermodynamic integration is more susceptible to the details of the hybrid Hamiltonian used than Bennett's acceptance ratio method. Therefore, we recommend Bennett's acceptance ratio method as the most robust method to compute alchemical free energy differences; nevertheless, scenarios when thermodynamic integration may be preferable are discussed.

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Section: Some Practical Aspects Of the Numerical Quadrature Schemes Usedmentioning

confidence: 99%

Efficiency of alchemical free energy simulations. II. Improvements for thermodynamic integration

2010

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“…GMP and exflib realize correct rounding for the four rules. For interval multiple-precision arithmetic, one can find a list in [11], for example. Among them, MPFI would be the most reliable package using the inf-sup form, as well as the interval version of exflib.…”

Section: The Center-radius Form 21 the Center-radius Form Of Intlabmentioning

confidence: 99%

On the basic operations of interval multiple-precision arithmetic with center-radius form

Matsuda

Yamamoto

2011

NOLTA

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Multiple-precision arithmetic with interval variables has been developed for computation with guaranteed high accuracy. There are several computer program packages which deal with interval variables of the inf-sup form, e.g. MPFI, etc. On the other hand, it is impressed by INTLAB that interval multiple-precision arithmetic using the center-radius form has advantages on memory size and computing time. However, arithmetic of the center-radius form sometimes makes the radius of an interval larger than the inf-sup form does, which would be one of the reasons why there is no practical program package for interval multiple-precision arithmetic with the center-radius form. The authors intend to establish a computer program package for multiple-precision arithmetic using intervals of the center-radius form which is still under construction. The present paper treats the problem of the center-radius form about the expansion of radii caused by the fundamental rules and the operation of square root. We propose several methods for calculation of multiplication, division, and square root, among which one can choose an appropriate method according to one's situation. Theoretical consideration and numerical examples are given for these methods.

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“…In addition, intervals can be used to implement the theoretical semantics Examples of interval arithmetic libraries are Boost [6] and MPFI [30], the latter being based on the multiple precision library MPFR [18]. Implementations of multiple precision interval libraries are compared in [17].…”

Section: Intervalsmentioning

confidence: 99%

An Overview of Semantics for the Validation of Numerical Programs

Martel

2005

Lecture Notes in Computer Science

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Abstract. In this article, we introduce a simple formal semantics for floating-point numbers with errors which is expressive enough to be formally compared to the other methods. Next, we define formal semantics for interval, stochastic, automatic differentiation and error series methods. This enables us to formally compare the properties calculated in each semantics to our reference, simple semantics. Most of these methods having been developed to verify numerical intensive codes, we also discuss their adequacy to the formal validation of softwares and to static analysis. Finally, this study is completed by experimental results.

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Multiple Precision Interval Packages: Comparing Different Approaches

Cited by 21 publications

References 21 publications

Efficiency of alchemical free energy simulations. II. Improvements for thermodynamic integration

Efficiency of alchemical free energy simulations. II. Improvements for thermodynamic integration

On the basic operations of interval multiple-precision arithmetic with center-radius form

An Overview of Semantics for the Validation of Numerical Programs

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