Numerical Analysis
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Numerical Computation with Rational Functions
NUMERICAL COMPUTATION WITH RATIONAL FUNCTIONS (scalars only) Nick Trefethen, University of Oxford and ENS Lyon + thanks to Silviu Filip, Abi Gopal, Stefan Güttel, Yuji Nakatsukasa, and Olivier Sète 1/26 1. Polynomial vs. rational approximations 2. Four representations of rational functions 2a. Quotient of polynomials 2b. Partial fractions 2c. Quotient of partial fractions (= barycentric) 2d. Transfer function/matrix pencil 3. The AAA algorithm with Nakatsukasa and Sète, to appear in SISC 4. Application: conformal maps with Gopal, submitted to Numer. Math. 5. Application: minimax approximation with Filip, Nakatsukasa, and Beckermann, to appear in SISC 6. Accuracy and noise 2/26 1. Polynomial vs. rational approximation Newman, 1964: approximation of |x| on [−1,1] −휋 푛 퐸푛0 ~ 0.2801. ./푛 , 퐸푛푛 ~ 8푒 Poles and zeros of r: exponentially clustered near x=0, exponentially diminishing residues. Rational approximation is nonlinear, so algorithms are nontrivial. poles There may be nonuniqueness and local minima. type (20,20) 3/26 2. Four representations of rational functions Alpert, Carpenter, Coelho, Gonnet, Greengard, Hagstrom, Koerner, Levy, Quotient of polynomials 푝(푧)/푞(푧) SK, IRF, AGH, ratdisk Pachón, Phillips, Ruttan, Sanathanen, Silantyev, Silveira, Varga, White,… 푎푘 Beylkin, Deschrijver, Dhaene, Drmač, Partial fractions 푧 − 푧 VF, exponential sums Greengard, Gustavsen, Hochman, 푘 Mohlenkamp, Monzón, Semlyen,… Berrut, Filip, Floater, Gopal, Quotient of partial fractions 푛(푧)/푑(푧) Floater-Hormann, AAA Hochman, Hormann, Ionita, Klein, Mittelmann, Nakatsukasa, Salzer, (= barycentric) Schneider, Sète, Trefethen, Werner,… 푇 −1 Antoulas, Beattie, Beckermann, Berljafa, Transfer function/matrix pencil 푐 푧퐵 − 퐴 푏 IRKA, Loewner, RKFIT Druskin, Elsworth, Gugercin, Güttel, Knizhnerman, Meerbergen, Ruhe,… Sometimes the boundaries are blurry! 4/26 2a. -
RESOURCES in NUMERICAL ANALYSIS Kendall E
RESOURCES IN NUMERICAL ANALYSIS Kendall E. Atkinson University of Iowa Introduction I. General Numerical Analysis A. Introductory Sources B. Advanced Introductory Texts with Broad Coverage C. Books With a Sampling of Introductory Topics D. Major Journals and Serial Publications 1. General Surveys 2. Leading journals with a general coverage in numerical analysis. 3. Other journals with a general coverage in numerical analysis. E. Other Printed Resources F. Online Resources II. Numerical Linear Algebra, Nonlinear Algebra, and Optimization A. Numerical Linear Algebra 1. General references 2. Eigenvalue problems 3. Iterative methods 4. Applications on parallel and vector computers 5. Over-determined linear systems. B. Numerical Solution of Nonlinear Systems 1. Single equations 2. Multivariate problems C. Optimization III. Approximation Theory A. Approximation of Functions 1. General references 2. Algorithms and software 3. Special topics 4. Multivariate approximation theory 5. Wavelets B. Interpolation Theory 1. Multivariable interpolation 2. Spline functions C. Numerical Integration and Differentiation 1. General references 2. Multivariate numerical integration IV. Solving Differential and Integral Equations A. Ordinary Differential Equations B. Partial Differential Equations C. Integral Equations V. Miscellaneous Important References VI. History of Numerical Analysis INTRODUCTION Numerical analysis is the area of mathematics and computer science that creates, analyzes, and implements algorithms for solving numerically the problems of continuous mathematics. Such problems originate generally from real-world applications of algebra, geometry, and calculus, and they involve variables that vary continuously; these problems occur throughout the natural sciences, social sciences, engineering, medicine, and business. During the second half of the twentieth century and continuing up to the present day, digital computers have grown in power and availability. -
An Extension of Matlab to Continuous Functions and Operators∗
SIAM J. SCI. COMPUT. c 2004 Society for Industrial and Applied Mathematics Vol. 25, No. 5, pp. 1743–1770 AN EXTENSION OF MATLAB TO CONTINUOUS FUNCTIONS AND OPERATORS∗ † † ZACHARY BATTLES AND LLOYD N. TREFETHEN Abstract. An object-oriented MATLAB system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices. About eighty MATLAB functions from plot and sum to svd and cond have been overloaded so that one can work with our “chebfun” objects using almost exactly the usual MATLAB syntax. All functions live on [−1, 1] and are represented by values at sufficiently many Chebyshev points for the polynomial interpolant to be accurate to close to machine precision. Each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation, differentiation, integration, zerofinding, or transforms. Applications in approximation theory and numerical analysis are explored, and possible extensions for more substantial problems of scientific computing are mentioned. Key words. MATLAB, Chebyshev points, interpolation, barycentric formula, spectral methods, FFT AMS subject classifications. 41-04, 65D05 DOI. 10.1137/S1064827503430126 1. Introduction. Numerical linear algebra and functional analysis are two faces of the same subject, the study of linear mappings from one vector space to another. But it could not be said that mathematicians have settled on a language and notation that blend the discrete and continuous worlds gracefully. Numerical analysts favor a concrete, basis-dependent matrix-vector notation that may be quite foreign to the functional analysts. Sometimes the difference may seem very minor between, say, ex- pressing an inner product as (u, v)orasuTv. -
Efficient Space-Time Sampling with Pixel-Wise Coded Exposure For
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 1 Efficient Space-Time Sampling with Pixel-wise Coded Exposure for High Speed Imaging Dengyu Liu, Jinwei Gu, Yasunobu Hitomi, Mohit Gupta, Tomoo Mitsunaga, Shree K. Nayar Abstract—Cameras face a fundamental tradeoff between spatial and temporal resolution. Digital still cameras can capture images with high spatial resolution, but most high-speed video cameras have relatively low spatial resolution. It is hard to overcome this tradeoff without incurring a significant increase in hardware costs. In this paper, we propose techniques for sampling, representing and reconstructing the space-time volume in order to overcome this tradeoff. Our approach has two important distinctions compared to previous works: (1) we achieve sparse representation of videos by learning an over-complete dictionary on video patches, and (2) we adhere to practical hardware constraints on sampling schemes imposed by architectures of current image sensors, which means that our sampling function can be implemented on CMOS image sensors with modified control units in the future. We evaluate components of our approach - sampling function and sparse representation by comparing them to several existing approaches. We also implement a prototype imaging system with pixel-wise coded exposure control using a Liquid Crystal on Silicon (LCoS) device. System characteristics such as field of view, Modulation Transfer Function (MTF) are evaluated for our imaging system. Both simulations and experiments on a wide range of -
Applied Analysis & Scientific Computing Discrete Mathematics
CENTER FOR NONLINEAR ANALYSIS The CNA provides an environment to enhance and coordinate research and training in applied analysis, including partial differential equations, calculus of Applied Analysis & variations, numerical analysis and scientific computation. It advances research and educational opportunities at the broad interface between mathematics and Scientific Computing physical sciences and engineering. The CNA fosters networks and collaborations within CMU and with US and international institutions. Discrete Mathematics & Operations Research RANKINGS DOCTOR OF PHILOSOPHY IN ALGORITHMS, COMBINATORICS, U.S. News & World Report AND OPTIMIZATION #16 | Applied Mathematics Carnegie Mellon University offers an interdisciplinary Ph.D program in Algorithms, Combinatorics, and #7 | Discrete Mathematics and Combinatorics Optimization. This program is the first of its kind in the United States. It is administered jointly #6 | Best Graduate Schools for Logic by the Tepper School of Business (Operations Research group), the Computer Science Department (Algorithms and Complexity group), and the Quantnet Department of Mathematical Sciences (Discrete Mathematics group). #4 | Best Financial Engineering Programs Carnegie Mellon University does not CONTACT discriminate in admission, employment, or Logic administration of its programs or activities on Department of Mathematical Sciences the basis of race, color, national origin, sex, handicap or disability, age, sexual orientation, 5000 Forbes Avenue gender identity, religion, creed, ancestry, -
Metal Complexes of Penicillin and Cephalosporin Antibiotics
I METAL COMPLEXES OF PENICILLIN AND CEPHALOSPORIN ANTIBIOTICS A thesis submitted to THE UNIVERSITY OF CAPE TOWN in fulfilment of the requirement$ forTown the degree of DOCTOR OF PHILOSOPHY Cape of by GRAHAM E. JACKSON University Department of Chernis try, University of Cape Town, Rondebosch, Cape, · South Africa. September 1975. The copyright of th:s the~is is held by the University of C::i~r:: To\vn. Reproduction of i .. c whole or any part \ . may be made for study purposes only, and \; not for publication. The copyright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgementTown of the source. The thesis is to be used for private study or non- commercial research purposes only. Cape Published by the University ofof Cape Town (UCT) in terms of the non-exclusive license granted to UCT by the author. University '· ii ACKNOWLEDGEMENTS I would like to express my sincere thanks to my supervisors: Dr. L.R. Nassimbeni, Dr. P.W. Linder and Dr. G.V. Fazakerley for their invaluable guidance and friendship throughout the course of this work. I would also like to thank my colleagues, Jill Russel, Melanie Wolf and Graham Mortimor for their many useful conrrnents. I am indebted to AE & CI for financial assistance during the course of. this study. iii ABSTRACT The interaction between metal"'.'ions and the penici l)in and cephalosporin antibiotics have been studied in an attempt to determine both the site and mechanism of this interaction. The solution conformation of the Cu(II) and Mn(II) complexes were determined using an n.m.r, line broadening, technique. -
Introduction to Complex Analysis Michael Taylor
Introduction to Complex Analysis Michael Taylor 1 2 Contents Chapter 1. Basic calculus in the complex domain 0. Complex numbers, power series, and exponentials 1. Holomorphic functions, derivatives, and path integrals 2. Holomorphic functions defined by power series 3. Exponential and trigonometric functions: Euler's formula 4. Square roots, logs, and other inverse functions I. π2 is irrational Chapter 2. Going deeper { the Cauchy integral theorem and consequences 5. The Cauchy integral theorem and the Cauchy integral formula 6. The maximum principle, Liouville's theorem, and the fundamental theorem of al- gebra 7. Harmonic functions on planar regions 8. Morera's theorem, the Schwarz reflection principle, and Goursat's theorem 9. Infinite products 10. Uniqueness and analytic continuation 11. Singularities 12. Laurent series C. Green's theorem F. The fundamental theorem of algebra (elementary proof) L. Absolutely convergent series Chapter 3. Fourier analysis and complex function theory 13. Fourier series and the Poisson integral 14. Fourier transforms 15. Laplace transforms and Mellin transforms H. Inner product spaces N. The matrix exponential G. The Weierstrass and Runge approximation theorems Chapter 4. Residue calculus, the argument principle, and two very special functions 16. Residue calculus 17. The argument principle 18. The Gamma function 19. The Riemann zeta function and the prime number theorem J. Euler's constant S. Hadamard's factorization theorem 3 Chapter 5. Conformal maps and geometrical aspects of complex function the- ory 20. Conformal maps 21. Normal families 22. The Riemann sphere (and other Riemann surfaces) 23. The Riemann mapping theorem 24. Boundary behavior of conformal maps 25. Covering maps 26. -
Quality Improvement of Compressed Color Images Using a Probabilistic Approach
QUALITY IMPROVEMENT OF COMPRESSED COLOR IMAGES USING A PROBABILISTIC APPROACH Nobuteru Takao, Shun Haraguchi, Hideki Noda, Michiharu Niimi Kyushu Institute of Technology, Dept. of Systems Design and Informatics, 680-4 Kawazu, Iizuka, 820-8502 Japan E-mail: {takao, haraguchi, noda, niimi}@mip.ces.kyutech.ac.jp ABSTRACT method using information on luminance component which is not downsampled. In compressed color images, colors are usually represented by luminance and chrominance (YCbCr) components. Con- The interpolation aims to recover only resolution of chromi- sidering characteristics of human vision system, chromi- nance components lost by downsampling. Alternatively, we nance (CbCr) components are generally represented more aim to recover not only resolution lost by downsampling coarsely than luminance component. Aiming at possible re- but also precision lost by a coarser quantization, if possi- covery of chrominance components, we propose a model- ble. Aiming at such recovery of chrominance components, based chrominance estimation algorithm where color im- we propose a model-based method where color images are ages are modeled by a Markov random field (MRF). A sim- modeled by a Markov random field (MRF). A simple MRF ple MRF model is here used whose local conditional proba- model is here used whose local conditional probability den- bility density function (pdf) for a color vector of a pixel is a sity function (pdf) for a color vector of a pixel is a Gaussian Gaussian pdf depending on color vectors of its neighboring pdf depending on color vectors of its neighboring pixels. pixels. Chrominance components of a pixel are estimated Chrominance components of a pixel are estimated by max- by maximizing the conditional pdf given its luminance com- imizing the conditional pdf given its luminance component ponent and its neighboring color vectors. -
Numerical Analysis to Quantum Computing
Credit: evv/Shutterstock.com and D-Wave, Inc. and D-Wave, Credit: evv/Shutterstock.com How NASA-USRA collaborations have advanced knowledge in and with the use of new computing technologies. When USRA was created in 1969, later its first task was the management the Chief Scientist at the FROM of the Lunar Science Institute near Center. In February of 1972, Duberg NASA’s Manned Spacecraft Center wrote a memorandum to the senior NUMERICAL (now the Johnson Space Center). management of LaRC, expressing A little more than three years his view that: TO later, USRA began to manage the ANALYSIS Institute for Computer Applications The field of computers and their in Science and Engineering (ICASE) application in the scientific QUANTUM at NASA’s Langley Research Center community has had a profound (LaRC). The rationale for creating effect on the progress of ICASE was developed by Dr. John aerospace research as well as COMPUTING E. Duberg (1917-2002), who was technology in general for the the Associate Director at LaRC and past 15 years. With the advent of “super computers,” based on parallel and pipeline techniques, the potentials for research and problem solving in the future seem even more promising and challenging. The only question is how long will it take to identify the potentials, harness the power, and develop the disciplines necessary to employ such tools effectively.1 Twenty years later, Duberg reflected on the creation of ICASE: By the 1970s, Langley’s computing capabilities had kept pace with the rapidly developing John E. Duberg, Chief Scientist, LaRC; George M. -
Householder Triangularization of a Quasimatrix
IMA Journal of Numerical Analysis (2010) 30, 887–897 doi:10.1093/imanum/drp018 Advance Access publication on August 21, 2009 Householder triangularization of a quasimatrix LLOYD N.TREFETHEN† Oxford University Computing Laboratory, Wolfson Building, Parks Road, Downloaded from Oxford OX1 3QD, UK [Received on 4 July 2008] A standard algorithm for computing the QR factorization of a matrix A is Householder triangulariza- tion. Here this idea is generalized to the situation in which A is a quasimatrix, that is, a ‘matrix’ whose http://imajna.oxfordjournals.org/ ‘columns’ are functions defined on an intervala [ , b]. Applications are mentioned to quasimatrix least squares fitting, singular value decomposition and determination of ranks, norms and condition numbers, and numerical illustrations are presented using the chebfun system. Keywords: Householder triangularization; QR factorization; chebfun; quasimatrix; singular value decom- position. 1. QR factorization and Householder triangularization at Radcliffe Science Library, Bodleian Library on April 18, 2012 Let A be an m n matrix, where m n. A (reduced) QR factorization of A is a factorization × > x x x x q q q q r r r r x x x x q q q q r r r A QR, x x x x q q q q r r , (1.1) = x x x x q q q q r x x x x q q q q x x x x q q q q where Q is m n with orthonormal columns and R is upper triangular. Here and in subsequent equations we illustrate our× matrix operations by schematic pictures in which x denotes an arbitrary entry, r denotes an entry of an upper triangular matrix, q denotes an entry of an orthonormal column and a blank denotes a zero. -
On Algebraic Functions
On Algebraic Functions N.D. Bagis Stenimahou 5 Edessa Pella 58200, Greece [email protected] Abstract In this article we consider functions with Moebius-periodic rational coefficients. These functions under some conditions take algebraic values and can be recovered by theta functions and the Dedekind eta function. Special cases are the elliptic singular moduli, the Rogers- Ramanujan continued fraction, Eisenstein series and functions asso- ciated with Jacobi symbol coefficients. Keywords: Theta functions; Algebraic functions; Special functions; Peri- odicity; 1 Known Results on Algebraic Functions The elliptic singular moduli kr is the solution x of the equation 1 1 2 2F1 , ;1;1 x 2 2 − = √r (1) 1 1 2 2F1 2 , 2 ; 1; x where 1 2 π/2 ∞ 1 1 2 2 n 2n 2 2 dφ 2F1 , ; 1; x = 2 x = K(x)= (2) 2 2 (n!) π π 2 n=0 Z0 1 x2 sin (φ) X − q The 5th degree modular equation which connects k25r and kr is (see [13]): 5/3 1/3 krk25r + kr′ k25′ r +2 (krk25rkr′ k25′ r) = 1 (3) arXiv:1305.1591v3 [math.GM] 27 Mar 2014 The problem of solving (3) and find k25r reduces to that solving the depressed equation after named by Hermite (see [3]): u6 v6 +5u2v2(u2 v2)+4uv(1 u4v4) = 0 (4) − − − 1/4 1/4 where u = kr and v = k25r . The function kr is also connected to theta functions from the relations 2 θ (q) ∞ 2 ∞ 2 k = 2 , where θ (q)= θ = q(n+1/2) and θ (q)= θ = qn r θ2(q) 2 2 3 3 3 n= n= X−∞ X−∞ (5) 1 π√r q = e− . -
Symmetries and Polynomials
Symmetries and Polynomials Aaron Landesman and Apurva Nakade June 30, 2018 Introduction In this class we’ll learn how to solve a cubic. We’ll also sketch how to solve a quartic. We’ll explore the connections between these solutions and group theory. Towards the end, we’ll hint towards the deep connection between polynomials and groups via Galois Theory. This connection allows one to convert the classical question of whether a general quintic can be solved by radicals to a group theory problem, which can be relatively easily answered. The 5 day outline is as follows: 1. The first two days are dedicated to studying polynomials. On the first day, we’ll study a simple invariant associated to every polynomial, the discriminant. 2. On the second day, we’ll solve the cubic equation by a method motivated by Galois theory. 3. Days 3 and 4 are a “practical” introduction to group theory. On day 3 we’ll learn about groups as symmetries of geometric objects. 4. On day 4, we’ll learn about the commutator subgroup of a group. 5. Finally, on the last day we’ll connect the two theories and explain the Galois corre- spondence for the cubic and the quartic. There are plenty of optional problems along the way for those who wish to explore more. Tricky optional problems are marked with ∗, and especially tricky optional problems are marked with ∗∗. 1 1 The Discriminant Today we’ll introduce the discriminant of a polynomial. The discriminant of a polynomial P is another polynomial Q which tells you whether P has any repeated roots over C.