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Start Over Descriptor "Underflow" Topic floating-point arithmetic
7 results on '"Underflow"'

1. Accurate Calculation of Euclidean Norms Using Double-word Arithmetic.

Author

LEFÈVRE, VINCENT, LOUVET, NICOLAS, MULLER, JEAN-MICHEL, PICOT, JORIS, and RIDEAU, LAURENCE

Subjects
*FLOATING-point arithmetic, *ARITHMETIC, *SQUARE root
Abstract

We consider the computation of the Euclidean (or L2) norm of an n-dimensional vector in floating-point arithmetic. We review the classical solutions used to avoid spurious overflow or underflow and/or to obtain very accurate results. We modify a recently published algorithm (that uses double-word arithmetic) to allow for a very accurate solution, free of spurious overflows and underflows. To that purpose, we use a double-word square-root algorithm of which we provide a tight error analysis. The returned L2 norm will be within very slightly more than 0.5 ulp from the exact result, which means that we will almost always provide correct rounding. [ABSTRACT FROM AUTHOR]

Published
2023
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2. SQUEEZING A MATRIX INTO HALF PRECISION, WITH AN APPLICATION TO SOLVING LINEAR SYSTEMS.

Author

HIGHAM, NICHOLAS J., PRANESH, SRIKARA, and ZOUNON, MAWUSSI

Subjects
*ARITHMETIC mean, *LINEAR systems, *FLOATING-point arithmetic, *MACHINE learning, *MATRICES (Mathematics)
Abstract

Motivated by the demand in machine learning, modern computer hardware is increasingly supporting reduced precision floating-point arithmetic, which provides advantages in speed, energy, and memory usage over single and double precision. Given the availability of such hardware, mixed precision algorithms that work in single or double precision but carry out part of a computationin half precision are now of great interest for general scientific computing tasks. Because ofthe limited range of half precision arithmetic, in which positive numbers lie between 6 × 10-8 and 7 ×104, a straightforward rounding of single or double precision data into half precision can leadto overflow, underflow, or subnormal numbers being generated, all of which are undesirable. Wedevelop an algorithm for converting a matrix from single or double precision to half precision. It firstapplies two-sided diagonal scaling in order to equilibrate the matrix (that is, to ensure that every rowand column has ∞-norm 1), then multiplies by a scalar to bring the largest element within a factor θ ≤ 1 of the overflow level, and finally rounds to half precision. The second step ensures that fulluse is made of the limited range of half precision arithmetic, and θ must be chosen to allow sufficientheadroom for subsequent computations. We apply the new algorithm to GMRES-based iterative refinement(GMRES-IR), which solves a linear system Ax = b with single or double precision data byLU factorizing A in half precision and carrying out iterative refinement with the correction equationssolved by GMRES preconditioned with the low precision LU factors. Previous implementations ofthis algorithm have used a crude conversion to half precision that our experiments show can causeslow convergence of GMRES-IR for badly scaled matrices or failure to converge at all. The newconversion algorithm computes ≤-norms of rows and columns of the matrix and its cost is negligiblein the context of LU factorization. We show that it leads to faster convergence of GMRES-IR forbadly scaled matrices and thereby allows a much wider class of problems to be solved. [ABSTRACT FROM AUTHOR]

Published
2019
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3. Accurate calculation of Euclidean Norms using Double-word arithmetic

Author

Vincent Lefèvre, Nicolas Louvet, Jean-Michel Muller, Joris Picot, Laurence Rideau, Arithmétiques des ordinateurs, méthodes formelles, génération de code (ARIC), Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Inria Lyon, Institut National de Recherche en Informatique et en Automatique (Inria), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon, Centre National de la Recherche Scientifique (CNRS), École normale supérieure de Lyon (ENS de Lyon), Sûreté du logiciel et Preuves Mathématiques Formalisées (STAMP), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-20-CE48-0014,NuSCAP,Sûreté numérique pour les preuves assistées par ordinateur(2020), Arithmetic and Computing (ARIC), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), École normale supérieure - Lyon (ENS Lyon), Muller, Jean-Michel, and Sûreté numérique pour les preuves assistées par ordinateur - - NuSCAP2020 - ANR-20-CE48-0014 - AAPG2020 - VALID

Subjects
Formalization, Euclidean norms, Applied Mathematics, [INFO.INFO-AO]Computer Science [cs]/Computer Arithmetic, Square-root, [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH], Floating-Point arithmetic, [INFO.INFO-OH] Computer Science [cs]/Other [cs.OH], Underflow, [INFO.INFO-MS] Computer Science [cs]/Mathematical Software [cs.MS], Overflow, Coq, Double-double arithmetic, [INFO.INFO-AO] Computer Science [cs]/Computer Arithmetic, Hardware_ARITHMETICANDLOGICSTRUCTURES, Proof assistant, Software, Double-word arithmetic, [INFO.INFO-MS]Computer Science [cs]/Mathematical Software [cs.MS]
Abstract

We consider the computation of the Euclidean (or L2) norm of an n -dimensional vector in floating-point arithmetic. We review the classical solutions used to avoid spurious overflow or underflow and/or to obtain very accurate results. We modify a recently published algorithm (that uses double-word arithmetic) to allow for a very accurate solution, free of spurious overflows and underflows. To that purpose, we use a double-word square-root algorithm of which we provide a tight error analysis. The returned L2 norm will be within very slightly more than 0.5 ulp from the exact result, which means that we will almost always provide correct rounding.

Published
2023
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4. Efficient Calculations of Faithfully Rounded l2-Norms of n-Vectors.

Author

Graillat, Stef, Lauter, Christoph, Tang, PING Tak Peter, Yamanaka, Naoya, and Oishi, Shin’ichi

Subjects
*FLOATING-point arithmetic, *ALGORITHMS, *MATRIX norms, *VECTOR analysis, *PRODUCTION possibility curve, *T-matrix
Abstract

In this article, we present an efficient algorithm to compute the faithful rounding of the l2-norm of a floating-point vector. This means that the result is accurate to within 1 bit of the underlying floating-point type. This algorithm does not generate overflows or underflows spuriously, but does so when the final result calls for such a numerical exception to be raised. Moreover, the algorithm is well suited for parallel implementation and vectorization. The implementation runs up to 3 times faster than the netlib version on current processors. [ABSTRACT FROM AUTHOR]

Published
2015
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5. Efficient Calculations of Faithfully Rounded l2-Norms of n-Vectors

Author

Christoph Lauter, Ping Tak Peter Tang, Naoya Yamanaka, Stef Graillat, Shin'ichi Oishi, Performance et Qualité des Algorithmes Numériques (PEQUAN), Laboratoire d'Informatique de Paris 6 (LIP6), Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS), Intel Corporation [USA], Teikyo Heisei University/CREST, JST, Tokyo, Japan, Waseda University [Tokyo, Japan], and Waseda University

Subjects
Netlib, faithful rounding, Arithmetic underflow, Floating point, Current (mathematics), Design, Efficient algorithm, Applied Mathematics, Rounding, Performance, [INFO.INFO-AO]Computer Science [cs]/Computer Arithmetic, underflow, Floating-point arithmetic, error-free transformations, overflow, Type (model theory), Vectorization (mathematics), 2-norm, Algorithm, Software, Algorithms, Mathematics
Abstract

In this article, we present an efficient algorithm to compute the faithful rounding of the l 2 -norm of a floating-point vector. This means that the result is accurate to within 1 bit of the underlying floating-point type. This algorithm does not generate overflows or underflows spuriously, but does so when the final result calls for such a numerical exception to be raised. Moreover, the algorithm is well suited for parallel implementation and vectorization. The implementation runs up to 3 times faster than the netlib version on current processors.

Published
2015
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6. Scaling Newton-Raphson division iterations to avoid double rounding

Author

Muller, Jean-Michel, Computer arithmetic (ARENAIRE), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), and Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL)

Subjects
Underflow, Subnormal numbers, [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH], MathematicsofComputing_NUMERICALANALYSIS, Floating-point arithmetic, Hardware_ARITHMETICANDLOGICSTRUCTURES, Newton-raphson iteration, Double rounding, Division
Abstract

When performing divisions using Newton-Raphson (or similar) iterations on a processor with a floating-point fused multiply-add instruction, one must sometimes scale the iterations, to avoid over/underflow and/or loss of accuracy. This may lead to double-roundings, resulting in output values that may not be correctly rounded when the quotient is in the subnormal range. We show how to avoid this problem.

Published
2010

7. Some notes on the possible under/overflow of the most common elementary functions

Author

Muller, Jean-Michel, Lefèvre, Vincent, Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS), Muller, Jean-Michel, and École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL)

Subjects
[INFO.INFO-OH] Computer Science [cs]/Other [cs.OH], elementary functions, floating-point arithmetic, [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH], underflow, ACM B.2.4, G.1.0, overflow
Abstract

The purpose of this short note is not to describe when underflow or overflow must be signalled (it is quite clear that the rules are the same as for the basic arithmetic operations). We just want to show that for some of the most common functions and floating-point formats, in many cases, we can know in advance that the results will always lie in the range of the numbers that are representable by normal floating-point numbers, so that in these cases there is no need to worry about underflow or overflow. Note that when it is not the case, an implementation is still possible using a run-time test.

Published
2007

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