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About Nicholas J. Higham
Nick Higham is the Richardson Professor of Applied Mathematics at the University of Manchester. His research interests focus on numerical analysis, ranging from theory to the development of algorithms and software. He is a keen expositor and blogs at http://nickhigham.wordpress.com. Nick was elected a Fellow of the Royal Society in 2007.
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Blog postA vector norm measures the size, or length, of a vector. For complex vectors, a vector norm is a function satisfying
with equality if and only if , for all , , for all (the triangle inequality). An important class of norms is the Hölder norms
The norm is interpreted as and is given by
Other important special cases are
A useful concept is that of the dual norm associated with a given vector norm, which is defined by
The maximum is attained5 days ago Read more 
Blog postThe largest dense linear systems being solved today are of order , and future exascale computer systems will be able to tackle even larger problems. Rounding error analysis shows that the computed solution satisfies a componentwise backward error bound that, under favorable assumptions, is of order , where is the unit roundoff of the floatingpoint … Continue reading Can We Solve Linear Algebra Problems at Extreme Scale and Low Precisions?1 week ago Read more

Blog postIn July 2021, Sven Hammarling, Françoise Tisseur and I organized an online workshop New Directions in Numerical Linear Algebra and High Performance Computing. The workshop brought together researchers working in numerical linear algebra and high performance computing to discuss current developments and challenges in the light of evolving computer hardware. It was held to honour … Continue reading Videos from New Directions in Numerical Linear Algebra and High Performance Computing Wo1 month ago Read more


Blog postImage ViridisDragonEye10 courtesy of Rob Corless. The twopart minisymposium Bohemian Matrices and Applications, organized by Rob Corless and I, took place at the SIAM Annual Meeting, July 22 and 23, 2021. This page makes available slides from some of the talks.
The minisymposium followed a twopart minisymposium on Bohemian matrices at the 2019 ICIAM meeting in Valencia and a 3day workshop on Bohemian matrices in Manchester in 2018.
For more on Bohemian matrices see the Boh2 months ago Read more 
Blog postThe determinant of a square submatrix of a matrix is called a minor. A matrix is totally positive if every minor is positive. It is totally nonnegative if every minor is nonnegative. These definitions require, in particular, that all the matrix elements must be nonnegative or positive, as must .
An important property is that total nonnegativity is preserved under matrix multiplication and hence under taking positive integer powers.
Theorem 1. If are totally nonnegative then so2 months ago Read more 
Blog postA real matrix is nonnegative if all its elements are nonnegative and it is positive if all its elements are positive. Nonnegative matrices arise in a wide variety of applications, for example as matrices of probabilities in Markov processes and as adjacency matrices of graphs. Information about the eigensystem is often essential in these applications.
Perron (1907) proved results about the eigensystem of a positive matrix and Frobenius (1912) extended them to nonnegative matrices.2 months ago Read more 
Blog postThe Kac–Murdock–Szegö matrix is the symmetric Toeplitz matrix
It was considered by Kac, Murdock, and Szegö (1953), who investigated its spectral properties. It arises in the autoregressive AR(1) model in statistics and signal processing.
The matrix is singular for , as is the rank matrix , and it is also rank for , as in this case every column is a multiple of the vector with alternating elements . The determinant . For , is nonsingular and the inverse is the trid3 months ago Read more 
Blog postBy Len Freeman, Nick Higham and Jim Nagy.
Ian Gladwell giving talk “Software for the Numerical Solution of ODEs—a University of Manchester and NAG Library Perspective” at Numerical Analysis and Computers—50 Years of Progress, University of Manchester, June 16–17, 1998. Ian Gladwell passed away on May 23, 2021 at the age of 76. He was born in Bolton, Lancashire in 1944. He did his secondary education at Thornleigh College, Bolton and was an undergraduate at Hertford College, University3 months ago Read more 
Blog postThe determinant of an matrix is defined by
where the sum is over all permutations ) of the sequence and is the number of inversions in , that is, the number of pairs with . Each term in the sum is a signed product of entries of and the product contains one entry taken from each row and one from each column.
The determinant is sometimes written with vertical bars, as .
Three fundamental properties are
The first property is immediate, the second can be proved u3 months ago Read more 
Blog postA Vandermonde matrix is defined in terms of scalars , , …, by
The are called points or nodes. Note that while we have indexed the nodes from , they are usually indexed from in papers concerned with algorithms for solving Vandermonde systems.
Vandermonde matrices arise in polynomial interpolation. Suppose we wish to find a polynomial of degree at most that interpolates to the data , that is, , . These equations are equivalent to
where is the vector of coefficients. Th3 months ago Read more
The musthave compendium on applied mathematics
This is the most authoritative and accessible singlevolume reference book on applied mathematics. Featuring numerous entries by leading experts and organized thematically, it introduces readers to applied mathematics and its uses; explains key concepts; describes important equations, laws, and functions; looks at exciting areas of research; covers modeling and simulation; explores areas of application; and more.
Modeled on the popular Princeton Companion to Mathematics, this volume is an indispensable resource for undergraduate and graduate students, researchers, and practitioners in other disciplines seeking a userfriendly reference book on applied mathematics.
 Features nearly 200 entries organized thematically and written by an international team of distinguished contributors
 Presents the major ideas and branches of applied mathematics in a clear and accessible way
 Explains important mathematical concepts, methods, equations, and applications
 Introduces the language of applied mathematics and the goals of applied mathematical research
 Gives a wide range of examples of mathematical modeling
 Covers continuum mechanics, dynamical systems, numerical analysis, discrete and combinatorial mathematics, mathematical physics, and much more
 Explores the connections between applied mathematics and other disciplines
 Includes suggestions for further reading, crossreferences, and a comprehensive index