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Item type:Publication, Access status: with files , On a quality measure for interval inclusionsVerification methods compute intervals which contain the solution of a given problem with mathematical rigour. In order to compare the quality of intervals some measure is desirable. We identify some anticipated properties and propose a method avoiding drawbacks of previous definitions.Publicationtype: Journal Article11 21 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Verified error bounds for matrix decompositions(2024); In this note we consider common matrix factorizations such as LU decomposition of a square and rectangular matrix, Cholesky and QR decomposition, singular value decomposition for square and rectangular matrices, and eigen-, Schur, and Takagi decomposition. We first note that well-conditioned factors tend to be sensitive to perturbations of the input matrix, while ill-conditioned factors tend to be insensitive. It seems that this behavior has not been recognized in numerical analysis. We develop a formula for the relation between the condition number of the factor and its sensitivity with respect to input perturbations, and give reasons for that. Our main focus is to describe verification methods for the factors of the mentioned decompositions. That means proving existence of the factorization together with rigorous entrywise error bounds for the factors. Our goal is to develop algorithms requiring \scrO(Pp2) operations for an m \times n matrix with P := max(m,n) and p := min(m,n). Moreover, bounds of high quality are aimed for, often not far from maximal accuracy. A main tool to achieve that is accurate dot products based on error-free transformations. Since preconditioning based on approximate inverses is used, our methods are restricted to full matrices.Publicationtype: Journal Article16 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , 35 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Modified error bounds for approximate solutions of dense linear systems(2020-05-01); ; ; We derive verified error bounds for approximate solutions of dense linear systems. There are verification methods using an approximate inverse of a coefficient matrix as a preconditioner, where the preconditioned coefficient matrix is likely to be anH-matrix (also known as a generalized diagonally dominant matrix). We focus on two inclusion methods of matrix multiplication for the preconditioning and propose verified error bounds adapted to the inclusion methods. These proposed error bounds are tighter than conventional ones, especially in critically ill-conditioned cases. Numerical results are presented showing the effectiveness of the proposed error bounds.Publicationtype: Journal Article175 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , 265 221 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Interval arithmetic with fixed rounding mode(The Institute of Electronics, Information and Communication Engineers ; J-Stage, 2016-07-01); ; ; 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 Article127 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Simple floating-point filters for the two-dimensional orientation problem(Springer, 2015-07-25); ; ; ; This paper is concerned with floating-point filters for a two dimensional orientation problem which is a basic problem in the field of computational geometry. If this problem is only approximately solved by floating-point arithmetic, then an incorrect result may be obtained due to accumulation of rounding errors. A floating-point filter can quickly guarantee the correctness of the computed result if the problem is well-conditioned. In this paper, a simple semi-static floating-point filter which handles floating-point exceptions such as overflow and underflow by only one branch is developed. In addition, an improved fully-static filter is developed.Publicationtype: Journal Article71 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Accelerating interval matrix multiplication by mixed precision arithmetic(2015-07-01); ; ; 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 Article41 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Generalization of error-free transformation for matrix multiplication and its applicationPublicationtype: Journal Article41 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Fast algorithms for floating-point interval matrix multiplicationWe discuss several methods for real interval matrix multiplication. First, earlier studies of fast algorithms for interval matrix multiplication are introduced: naive interval arithmetic, interval arithmetic by midpoint-radius form by Oishi-Rump and its fast variant by Ogita-Oishi. Next, three new and fast algorithms are developed. The proposed algorithms require one, two or three matrix products, respectively. The point is that our algorithms quickly predict which terms become dominant radii in interval computations. We propose a hybrid method to predict which algorithm is suitable for optimizing performance and width of the result. Numerical examples are presented to show the efficiency of the proposed algorithms.Publicationtype: Journal Article83