There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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The Body in the Mind: The Bodily Basis of Meaning, Imagination, and Reason

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Methods of Numerical Integration

Boundary Value Problems in Physics and Engineering By Frank Chorlton. Pp. 250. (Van Nostrand: London, July 1969.) 70s

Boundary value problems

Preface Acknowledgments Introduction: How Cognitive Science Changes Ethics 1: Reason as Force: The Moral Law Folk Theory 2: Metaphoric Morality 3: The Metaphoric Basis of Moral Theory 4: Beyond Rules 5: The Impoverishment of Reason: Our Enlightenment Legacy 6: What's Wrong with the Objectivist Self 7: The Narrative Context of Self and Action 8: Moral Imagination 9: Living without Absolutes: Objectivity and the Conditions for Criticism 10: Preserving Our Best Enlightenment Moral Ideals Notes Index

Philosophical Applications of Cognitive Science.Moral Imagination: Implications of Cognitive Science for Ethics

Zeros of analytic functions.- Clusters of zeros of analytic functions.- Zeros and poles of meromorphic functions.- Systems of analytic equations.

Computing the Zeros of Analytic Functions

We present a stabilized superfast solver for nonsymmetric Toeplitz systems Tx=b. An explicit formula for T-1 is expressed in such a way that the matrix-vector product T^-1b can be calculated via FFTs and Hadamard products. This inversion formula involves certain polynomials that can be computed by solving two linearized rational interpolation problems on the unit circle. The heart of our Toeplitz solver is a superfast algorithm to solve these interpolation problems. To stabilize the algorithm, i.e., to improve the accuracy, several techniques are used: pivoting, iterative improvement, downdating, and giving "difficult" interpolation points an adequate treatment. We have implemented our algorithm in Fortran 90. Numerical examples illustrate the effectiveness of our approach.

A Stabilized Superfast Solver for Nonsymmetric Toeplitz Systems

Given an analytic function f and a Jordan curve γ that does not pass through any zero of f, we consider the problem of computing all the zeros of f that lie inside γ, together with their respective multiplicities. Our principal means of obtaining information about the location of these zeros is a certain symmetric bilinear form that can be evaluated via numerical integration along γ. If f has one or several clusters of zeros, then the mapping from the ordinary moments associated with this form to the zeros and their respective multiplicities is very ill-conditioned. We present numerical methods to calculate the centre of a cluster and its weight, i.e., the arithmetic mean of the zeros that form a certain cluster and the total number of zeros in this cluster, respectively. Our approach relies on formal orthogonal polynomials and rational interpolation at roots of unity. Numerical examples illustrate the effectiveness of our techniques.

On locating clusters of zeros of analytic functions

Let $f(z)$ be an analytic or meromorphic function in the closed unit disk sampled at the $n$th roots of unity. Based on these data, how can we approximately evaluate $f(z)$ or $f^{(m)}(z)$ at a point $z$ in the disk? How can we calculate the zeros or poles of $f$ in the disk? These questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations, partial differential equations, and large-scale linear algebra. We analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of Cauchy integrals. We then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension. Finally we highlight the power of rational in comparison with polynomial approximations for some of these problems.

/pdf/numerical-algorithms-based-on-analytic-function-values-at-5943irzz6r.pdf

Peter Kravanja

Papers

Computing the Zeros of Analytic Functions

A Stabilized Superfast Solver for Nonsymmetric Toeplitz Systems

On locating clusters of zeros of analytic functions

Numerical Algorithms Based on Analytic Function Values at Roots of Unity

Nonlinear eigenvalue problems and contour integrals