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"Higham, N. J"
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King Arthur : the making of the legend
According to legend, King Arthur saved Britain from the Saxons and reigned over it gloriously sometime around A.D. 500. Whether or not there was a \"real\" King Arthur has all too often been neglected by scholars; most period specialists today declare themselves agnostic on this important matter. In this erudite volume, Nick Higham sets out to solve the puzzle, drawing on his original research and expertise to determine precisely when, and why, the legend began. Higham surveys all the major attempts to prove the origins of Arthur, weighing up and debunking hitherto claimed connections with classical Greece, Roman Dalmatia, Sarmatia, and the Caucasus. He then explores Arthur's emergence in Wales - up to his rise to fame at the hands of Geoffrey of Monmouth. Certain to arouse heated debate among those committed to defending any particular Arthur, Higham's book is an essential study for anyone seeking to understand how Arthur's story began.
Book
Bede's Agenda in Book IV of the ‘Ecclesiastical History of the English People’: A Tricky Matter of Advising the King
Bede's preoccupations in the later years of his life have recently come under close scrutiny. This article will set out the argument to this point, then explore how the Ecclesiastical history conforms to more general perceptions of Bede's purposes. It will conclude that this work was designed to address just one part of his wider reform agenda, as that pertained to the Northumbrian king of the day, Ceolwulf. To this end, Bede painted a picture of the current situation within the Church which is far more optimistic than that on offer in the Letter to Ecgberht just a few years later. It must be concluded that his specific purposes as regards any particular work, and the audience at which that work was aimed, exercised a considerable influence over his strategy, which varies enormously from one part of his output to another.
Journal Article
The Anglo-Saxon world
In this authoritative work, the authors re-examine Anglo-Saxon England in light of new research in disciplines as wide-ranging as historical genetics, paleobotany, archaeology, literary studies, art history, and numanistics. The result is a definitive introduction to the Anglo-Saxon world, enhanced with a rich array of photographs, maps, genealogies, and other illustrations.
Book
King Arthur
This seminal new study explores how and why historians and writers from the Middle Ages to the present day have constructed different accounts of this well-loved figure.
N. J Higham offers an in-depth examintaion of the first two Arthurian texts: the History of the Britons and the Welsh Annals. He argues that historians have often been more influenced by what the idea of Arthur means in their present context than by such primary sources
King Arthur: Myth-making and History illuminates and discusses some central points of debate:
What role was Arthur intended to perform in the political and cultural worlds that constructed him?
How did the idea of King Arthur evolve?
What did the myth of Arthur mean to both authors and their audiences?
King Arthur: Myth-making and History is fascinating reading for anyone interested in the origins and evolution of the Arthurian legend.
eBook
A Survey of Condition Number Estimation for Triangular Matrices
We survey and compare a wide variety of techniques for estimating the condition number of a triangular matrix, and make recommendations concerning the use of the estimates in applications. Each of the methods is shown to bound the condition number; the bounds can broadly be categorised as upper bounds from matrix theory and lower bounds from heuristic or probabilistic algorithms. For each bound we examine by how much, at worst, it can overestimate or underestimate the condition number. Numerical experiments are presented in order to illustrate and compare the practical performance of the condition estimators.
Journal Article
Optimization by Direct Search in Matrix Computations
A direct search method attempts to maximize a function $f :{\\bf R}^n \\to {\\bf R}$ using function values only. Many questions about the stability and accuracy of algorithms in matrix computations can be expressed in terms of the maximum value of some easily computable function $f$. For a variety of algorithms it is shown that direct search is capable of revealing instability or poor performance, even when such failure is difficult to discover using theoretical analysis or numerical tests with random or nonrandom data. Informative numerical examples generated by direct search provide the impetus for further analysis and improvement of an algorithm. The direct search methods used are the method of alternating directions and the multi-directional search method of Dennis and Torczon. The problems examined include the reliability of matrix condition number estimators and the stability of Strassen's fast matrix inversion method.
Journal Article
The Scaling and Squaring Method for the Matrix Exponential Revisited
The scaling and squaring method is the most widely used method for computing the matrix exponential, not least because it is the method implemented in MATLAB's {\\tt expm} function. The method scales the matrix by a power of 2 to reduce the norm to order 1, computes a Pade approximant to the matrix exponential, and then repeatedly squares to undo the effect of the scaling. We give a new backward error analysis of the method (in exact arithmetic) that employs sharp bounds for the truncation errors and leads to an implementation of essentially optimal efficiency. We also give new rounding error analysis that shows the computed Pade approximant of the scaled matrix to be highly accurate. For IEEE double precision arithmetic the best choice of degree of Pade approximant turns out to be 13, rather than the 6 or 8 used by previous authors. Our implementation of the scaling and squaring method always requires at least two fewer matrix multiplications than {\\tt expm} when the matrix norm exceeds 1, which can amount to a 37% saving in the number of multiplications, and it is typically more accurate, owing to the fewer required squarings. We also investigate a different scaling and squaring algorithm proposed by Najfeld and Havel that employs a Pade approximation to the function $x \\coth(x)$. This method is found to be essentially a variation of the standard one with weaker supporting error analysis.
Journal Article
The Accuracy of Floating Point Summation
The usual recursive summation technique is just one of several ways of computing the sum of $n$ floating point numbers. Five summation methods and their variations are analyzed here. The accuracy of the methods is compared using rounding error analysis and numerical experiments. Four of the methods are shown to be special cases of a general class of methods, and an error analysis is given for this class. No one method is uniformly more accurate than the others, but some guidelines are given on the choice of method in particular cases.
Journal Article
Factorizing complex symmetric matrices with positive definite real and imaginary parts
Complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for LU factorization. This result adds to the classes of matrix for which it is known to be safe not to pivot in LU factorization. Block LDLT\\mathrm {LDL^T} factorization with the pivoting strategy of Bunch and Kaufman is also considered, and it is shown that for such matrices only 1×11\\times 1 pivots are used and the same growth factor bound of 2 holds, but that interchanges that destroy band structure may be made. The latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in LINPACK and LAPACK.
Journal Article
Computing the Polar Decomposition—with Applications
A quadratically convergent Newton method for computing the polar decomposition of a full-rank matrix is presented and analysed. Acceleration parameters are introduced so as to enhance the initial rate of convergence and it is shown how reliable estimates of the optimal parameters may be computed in practice. To add to the known best approximation property of the unitary polar factor, the Hermitian polar factor $H$ of a nonsingular Hermitian matrix $A$ is shown to be a good positive definite approximation to $A$and $\\frac{1}{2}(A + H)$ is shown to be a best Hermitian positive semi-definite approximation to $A$. Perturbation bounds for the polar factors are derived. Applications of the polar decomposition to factor analysis, aerospace computations and optimisation are outlined; and a new method is derived for computing the square root of a symmetric positive definite matrix.
Journal Article