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    Preconditioning of Taylor models, implementation and test cases
    (2021)
    Bünger, Florian
    Makino and Berz introduced the Taylor model approach for validated integration of initial value problems (IVPs) for ordinary differential equations (ODEs). Especially, they invented preconditioning of Taylor models for stabilizing the integration and proposed the following different types: parallelepiped preconditioning (with and without blunting), QR preconditioning, and curvilinear preconditioning. We review these types of preconditioning and show how they are implemented in INTLAB's verified ODE solver verifyode by stating explicit MATLAB code. Finally, we test our implementation with several examples.
    Publicationtype: Journal Article
      51
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    Interval arithmetic with fixed rounding mode
    (The Institute of Electronics, Information and Communication Engineers ; J-Stage, 2016-07-01)
    Rump, Siegfried M.
    ;
    Ogita, Takeshi
    ;
    Morikura, Yusuke
    ;
    Oishi, Shin’ichi
    We discuss several methods to simulate interval arithmetic operations using floating-point operations with fixed rounding mode. In particular we present formulas using only rounding to nearest and using only chop rounding (towards zero). The latter was the default and only rounding on GPU (Graphics Processing Unit) and cell processors, which in turn are very fast and therefore attractive in scientific computations.
    Publicationtype: Journal Article
      128
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    Implementation and improvements of affine arithmetic
    (2015-07-01)
    Rump, Siegfried M.
    ;
    Kashiwagi, Masahide
    Affine arithmetic is a well-known tool to reduce the wrapping effect of ordinary interval arithmetic. We discuss several improvements both in theory and in terms of practical implementation. In particular details of INTLAB's affine arithmetic toolbox are presented. Computational examples demonstrate advantages and weaknesses of the approach.
    Publicationtype: Journal Article
      24
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    Accelerating interval matrix multiplication by mixed precision arithmetic
    (2015-07-01)
    Ozaki, Katsuhisa
    ;
    Ogita, Takeshi
    ;
    Bünger, Florian
    ;
    Oishi, Shin’ichi
    This paper is concerned with real interval arithmetic. We focus on interval matrix multiplication. Well-known algorithms for this purpose require the evaluation of several point matrix products to compute one interval matrix product. In order to save computing time we propose a method that modifies such known algorithm by partially using low-precision floating-point arithmetic. The modified algorithms work without significant loss of tightness of the computed interval matrix product but are about 30% faster than their corresponding original versions. The negligible loss of accuracy is rigorously estimated.
    Publicationtype: Journal Article
      41
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    Computable backward error bounds for basic algorithms in linear algebra
    (2015-07-01)
    Rump, Siegfried M.
    Standard error estimates in numerical linear algebra are often of the form γk|R||S| where R,S are known matrices and γk:=ku/(1-u) with u denoting the relative rounding error unit. Recently we showed that for a number of standard problems γk can be replaced by ku for any order of computation and without restriction on the dimension. Such problems include LU- and Cholesky decomposition, triangular system solving by substitution, matrix multiplication and more. The theoretical bound implies a practically computable bound by estimating the error in the floating-point computation of ku|R||S|. Standard techniques, however, imply again a restriction on the dimension. In this note we derive simple computable bounds being valid without restriction on the dimension. As the bounds are mathematically rigorous, they may serve in computer assisted proofs.
    Publicationtype: Journal Article
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