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23 results

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  • Mixed precision algorithms in numerical linear algebra
  • Nicholas J. Higham, Theo Mary
  • Journal: Acta Numerica / Volume 31 / May 2022
  • Published online by Cambridge University Press: 09 June 2022, pp. 347-414
  • Print publication: May 2022
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    Today’s floating-point arithmetic landscape is broader than ever. While scientific computing has traditionally used single precision and double precision floating-point arithmetics, half precision is increasingly available in hardware and quadruple precision is supported in software. Lower precision arithmetic brings increased speed and reduced communication and energy costs, but it produces results of correspondingly low accuracy. Higher precisions are more expensive but can potentially provide great benefits, even if used sparingly. A variety of mixed precision algorithms have been developed that combine the superior performance of lower precisions with the better accuracy of higher precisions. Some of these algorithms aim to provide results of the same quality as algorithms running in a fixed precision but at a much lower cost; others use a little higher precision to improve the accuracy of an algorithm. This survey treats a broad range of mixed precision algorithms in numerical linear algebra, both direct and iterative, for problems including matrix multiplication, matrix factorization, linear systems, least squares, eigenvalue decomposition and singular value decomposition. We identify key algorithmic ideas, such as iterative refinement, adapting the precision to the data, and exploiting mixed precision block fused multiply–add operations. We also describe the possible performance benefits and explain what is known about the numerical stability of the algorithms. This survey should be useful to a wide community of researchers and practitioners who wish to develop or benefit from mixed precision numerical linear algebra algorithms.

  • 1 - Bede and the Early English Church
  • Edited by Alexander R. Rumble
  • Book: Leaders of the Anglo-Saxon Church
  • Published by: Boydell & Brewer
  • Published online: 05 February 2013
  • Print publication: 15 April 2012, pp 25-40
  • View extract

    Summary

    IN 731, Bede completed his Historia Ecclesiastica nostrae insulae ac igentis in libri v – his ‘Ecclesiastical History of our island and people in five books’, a work which he entitled internally the Historia Ecclesiastica gentis Anglorum (in the opening line of the Preface and in the headings of the contents lists which open each book). The core of the title is, however, the same in each case, the Historia Ecclesiastica. Unsurprisingly, Bede made frequent use of the word ‘ecclesia’ meaning ‘church’ across this work; it occurs 378 times. This term has several different significances, however, which we need to distinguish before we can really discuss Bede's conception of, and attitude towards, the early English church.

    ‘Ecclesia’ is frequently used in the Historia Ecclesiastica to refer to the universal Christian church, occurring in such phrases as ‘catholica et apostolica ecclesia’ (‘the catholic and apostolic church’), ‘ecclesia sua’ (‘his church’, i.e. ‘Christ's church’), ‘ecclesia catholica’ (‘the catholic church’) and ‘omnis ecclesia’ (‘the whole church’), all taken from V. 21, which features Abbot Ceolfrith's letter to King Nechtan. This document was, of course, urging unity on the Picts as regards the calculation of Easter, so made frequent use of the terminologies of universality and conformity, of which such phrases form a central part. This was a terminology with which Bede was very familiar, appearing as it does in the New Testament, most famously in Matt. XVI.18, in a passage which re appears in HE III. 25, but elsewhere also in the Epistles of St Paul.

Place-names, Language and the Anglo-Saxon Landscape
  • Place-names, Language and the Anglo-Saxon Landscape
  • Edited by Nicholas J. Higham, Martin J. Ryan
  • Published by: Boydell & Brewer
  • Published online: 05 February 2013
  • Print publication: 17 March 2011
  • View description
    An exploration of the landscape of Anglo-Saxon England, particularly through the prism of place-names and what they can reveal. The landscape of modern England still bears the imprint of its Anglo-Saxon past. Villages and towns, fields, woods and forests, parishes and shires, all shed light on the enduring impact of the Anglo-Saxons. The essays in this volume explore the richness of the interactions between the Anglo-Saxons and their landscape: how they understood, described, and exploited the environments of which they were a part. Ranging from the earliest settlement period through to the urban expansion of late Anglo-Saxon England, this book draws on evidence from place-names, written sources, and the landscape itself to provide fresh insights into the topic. Subjects explored include the history of the study of place-names and the Anglo-Saxon landscape; landscapes of particular regions and the exploitation of particular landscape types; the mechanisms of the transmission and survival of written sources; and the problems and potentials of interdisciplinary research into the Anglo-Saxon landscape. Nicholas J. Higham is Professor of Early Medieval and Landscape History at the University of Manchester; Martin Ryan lectures in Medieval History at the University of Manchester. Contributors: Ann Cole, Linda M. Corrigan, Dorn Van Dommelen, Simon Draper, Gillian Fellows-Jensen, Della Hooke, Duncan Probert, Alexander R. Rumble, Martin J. Ryan, Peter A. Stokes, Richard Watson.
The Landscape Archaeology of Anglo-Saxon England

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