Some Equivalent Characterizations of the Polynomial Numerical Hull of Degree kk

In this paper we establish several equivalent characterizations, some of which provide a natural distinction between boundary points and interior points, and hence should prove useful for computations. Additionally, we use one of these equivalent definitions to prove that the polynomial numerical hull of any fixed degree $k$ for a Toeplitz matrix whose symbol is piecewise continuous approaches all or most of that of the infinite-dimensional Toeplitz operator, as the matrix goes to infinity

K-Spectral Sets

We use results in [M. Crouzeix and A. Greenbaum,Spectral sets: numerical range and beyond, SIAM Jour. Matrix Anal. Appl., 40 (2019), pp. 1087-1101] to derive a variety of K-spectral sets and show how they can be used in some applications. We compare the K-values derived here to those that can be derived from a straightforward application of the Cauchy integral formula, by replacing the norm of the integral by the integral of the resolvent norm. While, in some cases, the new upper bounds on the optimal K-value are much tighter than those from the Cauchy integral formula, we show that in many cases of interest, the two values are of the same order of magnitude, with the bounds from the Cauchy integral formula actually being slightly smaller. We give a partial explanation of this in terms of the numerical range of the resolvent at points near an ill-conditioned eigenvalue.Comment: 20 pages, 10 figures. For relevant code, see https://github.com/tygris/k-spectral-sets. Abstract refers to arXiv:1803.1090

Error bounds for Lanczos-based matrix function approximation

We analyze the Lanczos method for matrix function approximation (Lanczos-FA), an iterative algorithm for computing $f(\mathbf{A}) \mathbf{b}$ when $\mathbf{A}$ is a Hermitian matrix and $\mathbf{b}$ is a given mathbftor. Assuming that $f : \mathbb{C} \rightarrow \mathbb{C}$ is piecewise analytic, we give a framework, based on the Cauchy integral formula, which can be used to derive {\em a priori} and \emph{a posteriori} error bounds for Lanczos-FA in terms of the error of Lanczos used to solve linear systems. Unlike many error bounds for Lanczos-FA, these bounds account for fine-grained properties of the spectrum of $\mathbf{A}$, such as clustered or isolated eigenvalues. Our results are derived assuming exact arithmetic, but we show that they are easily extended to finite precision computations using existing theory about the Lanczos algorithm in finite precision. We also provide generalized bounds for the Lanczos method used to approximate quadratic forms $\mathbf{b}^\textsf{H} f(\mathbf{A}) \mathbf{b}$, and demonstrate the effectiveness of our bounds with numerical experiments

The undergraduate law curriculum : fitness for purpose?

Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2009.This study reviews the curriculum of the four-year undergraduate Baccalaureus Legum (LLB) degree, introduced in 1998 as part of the transformation agenda in post-apartheid South Africa. Ten years since its inception, the question is whether the vision of the originators has translated into curricula that are producing a representative supply of appropriately-educated graduates for practice as legal professionals. The demand for the transformation of legal education resulted in the introduction of an undergraduate LLB as a single, affordable qualification for entry to legal practice. Law faculties were permitted to develop their own curricula, although there was agreement on core content. Three key principles were to inform curriculum design: (i) South African law exists in and applies to a diverse or pluralistic society; (ii) skills appropriate to the practice of law must be integrated into the degree; and (iii) faculties must strive to inculcate ethical values in students. A decade later, stakeholders are expressing dissatisfaction with the quality of graduates. Few graduates complete the LLB within four years, and a significant proportion of African students, already under-represented in law faculties, do not complete their studies. The attorneys’ profession is still predominantly white-owned. In the first part of the study, phenomenological interviews were conducted with three members of the 1996 Task Group of Law Deans who drafted the proposals for the new degree. The data elicited described the lived experience of curriculum change. Five current Law Deans were also interviewed to develop an understanding of their experience of implementing the law curriculum. The second component of the study was a phenomenographic analysis, in which six graduates, who are now attorneys, were interviewed, to identify their experiences of the law curriculum at one Law faculty. The graduates’ employers were interviewed to ascertain their perceptions of the graduates’ preparedness for professional practice. The study suggests that reactive conservatism on the part of legal academics resulted in law curricula that replicate a cycle of disadvantage, and fail to achieve transformative learning which integrates knowledge, skills and ethical values. A focus on incorporating an ontological component in law curricula, to develop high quality legal professionals is recommended

Teaching legal writing in a South African context : an evaluation of the work of student tutors in assisting with the development of legal writing skills in first year law students at one South African law school.

Thesis (M.Ed.)-University of Natal, 2001.In this study, an analysis and an evaluation of the feedback comments made by fourth year law students on legal writing assignments of first year law students is made. The purpose is to formulate a sense of the student tutors' shared capacity to critique legal writing, and thereby assist in developing the writing skills of first year students at one South African law school. A review of the literature on legal writing from the United States of America suggests that there has been a shift away from the current-traditional paradigm, which focussed on formal features of legal writing. The 'new legal rhetoric' approach, based on research and theoretical understandings which view writing as a recursive process , has now been widely adopted. This approach has been further extended by the 'social perspective,' which acknowledges writing as a social practice. which novice writers can learn from experts within the legal discourse community. Finally, critical reading and writing theorists propose a politicised approach to writing, encouraging critiques of alienating discourse practices. Legal writing pedagogy in the United States has reflected these changing theoretical perspectives. The 'process' approach to teaching writing requires intensive writing instruction and practice by novices. Several drafts of assignments are submitted, and instructors respond with appropriate written (and verbal) feedback comments, which are intended to motivate revisions. The implications of this approach are that increased numbers of trained writing instructors are required to implement such a labour-intensive pedagogy. Resource constraints, and the difficulties of staffing such programmes, have resulted in innovative models being devised. The use of student tutors to assist in teaching legal writing inspired the introduction of a comparable tutor-training course at the University of Natal, Durban Law School, in 1999. In this study, eighteen pieces of writing: three different examples of first year law students' writing, on which six tutors had each written feedback comments, were analysed. The number, accuracy and type of comments were tabulated, and the tone and quality of the responses were evaluated against the theoretical frameworks reviewed above. A descriptive, qualitative interpretation of their commenting practice develops a detailed sense of their successes and deficiencies. The conclusions which emerged suggest that modifications to the tutors' education and training , and closer supervision/monitoring procedures would enhance the tutors' theoretical understandings, as well as their commenting practice. The value and viability of such a programme is confirmed by the empirical information, and indicates that student tutors can extend teaching resources, to provide the assistance necessary to implement intensive legal writing instruction. In a South African context, where academic literacy skills are so often deficient in first year students, a programme which builds capacity and extends limited teaching resources can be extremely beneficial

Near-Optimal Approximation of Matrix Functions by the Lanczos Method

We study the widely used Lanczos method for approximating the action of a matrix function $f(\mathbf{A})$ on a vector $\mathbf{b}$ (Lanczos-FA). For the function $f(\mathbf{A})=\mathbf{A}^{-1}$, it is known that, when $\mathbf{A}$ is positive definite, the $\mathbf{A}$-norm error of Lanczos-FA after $k$ iterations matches the optimal approximation from the Krylov subspace of degree $k$ generated by $\mathbf A$ and $\mathbf b$. In this work, we ask whether Lanczos-FA also obtains similarly strong optimality guarantees for other functions $f$. We partially answer this question by showing that, up to a multiplicative approximation factor, Lanczos-FA also matches the optimal approximation from the Krylov subspace for any rational function with real poles outside the interval containing $\mathbf{A}$'s eigenvalues. The approximation factor depends on the degree of $f$'s denominator and the condition number of $\mathbf{A}$, but not on the number of iterations $k$. This result provides theoretical justification for the excellent performance of Lanczos-FA on functions that are well approximated by rationals. Additionally, using different techniques, we prove that Lanczos-FA achieves a weaker notion of optimality for the functions $\mathbf A^{1/2}$ and $\mathbf A^{-1/2}$. Experiments confirm that our new bounds more accurately predict the convergence of Lanczos-FA than existing bounds, in particular those based on uniform polynomial approximation