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April 2012
The 20 articles with the most full-text downloads during the month, in descending order.
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Impedance Imaging, Inverse Problems, and Harry Potter's Cloak SIAM Rev. 52, pp. 359-377 (19 pages) Online Publication Date: May 06, 2010
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In this article we provide an accessible account of the essential idea behind cloaking, aimed at nonspecialists and undergraduates who have had some vector calculus, Fourier series, and linear algebra. The goal of cloaking is to render an object invisible to detection from electromagnetic energy by surrounding the object with a specially engineered “metamaterial” that redirects electromagnetic waves around the object. We show how to cloak an object against detection from impedance tomography, an imaging technique of much recent interest, though the mathematical ideas apply to much more general forms of imaging. We also include some exercises and ideas for undergraduate research projects. |
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One Can Hear the Composition of a String: Experiments with an Inverse Eigenvalue Problem SIAM Rev. 54, pp. 157-178 (22 pages) Online Publication Date: February 08, 2012
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To what extent do the vibrations of a mechanical system reveal its composition? Despite innumerable applications and mathematical elegance, this question often slips through those cracks that separate courses in mechanics, differential equations, and linear algebra. We address this omission by detailing a classical finite dimensional example: the use of frequencies of vibration to recover positions and masses of beads vibrating on a string. First we derive the equations of motion, then compare the eigenvalues of the resulting linearized model against vibration data measured from our laboratory's monochord. More challenging is the recovery of masses and positions of the beads from spectral data, a problem for which a variety of elegant algorithms exist. After presenting one such method based on orthogonal polynomials in a manner suitable for advanced undergraduates, we confirm its efficacy through physical experiment. We encourage readers to conduct their own explorations using the numerous data sets we provide. |
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Modeling Growth in Biological Materials SIAM Rev. 54, pp. 52-118 (67 pages) Online Publication Date: February 08, 2012
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The biomechanical modeling of growing tissues has recently become an area of intense interest. In particular, the interplay between growth patterns and mechanical stress is of great importance, with possible applications to arterial mechanics, embryo morphogenesis, tumor development, and bone remodeling. This review aims to give an overview of the theories that have been used to model these phenomena, categorized according to whether the tissue is considered as a continuum object or a collection of cells. Among the continuum models discussed is the deformation gradient decomposition method, which allows a residual stress field to develop from an incompatible growth field. The cell-based models are further subdivided into cellular automata, center-dynamics, and vertex-dynamics models. Of these the second two are considered in more detail, especially with regard to their treatment of cell–cell interactions and cell division. The review concludes by assessing the prospects for reconciliation between these two fundamentally different approaches to tissue growth, and by identifying possible avenues for further research. |
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The Structure and Function of Complex Networks SIAM Rev. 45, pp. 167-256 (90 pages) Online Publication Date: August 04, 2006
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Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks. |
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A Survey of Eigenvector Methods for Web Information Retrieval SIAM Rev. 47, pp. 135-161 (27 pages) Online Publication Date: August 04, 2006
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Web information retrieval is significantly more challenging than traditional well-controlled, small document collection information retrieval. One main difference between traditional information retrieval and Web information retrieval is the Web's hyperlink structure. This structure has been exploited by several of today's leading Web search engines, particularly Google and Teoma. In this survey paper, we focus on Web information retrieval methods that use eigenvector computations, presenting the three popular methods of HITS, PageRank, and SALSA. |
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Power-Law Distributions in Empirical Data SIAM Rev. 51, pp. 661-703 (43 pages) Online Publication Date: November 06, 2009
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Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution—the part of the distribution representing large but rare events—and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov–Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out. |
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Matrices, Vector Spaces, and Information Retrieval SIAM Rev. 41, pp. 335-362 (28 pages) Online Publication Date: August 04, 2006
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The evolution of digital libraries and the Internet has dramatically transformed the processing, storage, and retrieval of information. Efforts to digitize text, images, video, and audio now consume a substantial portion of both academic and industrial activity. Even when there is no shortage of textual materials on a particular topic, procedures for indexing or extracting the knowledge or conceptual information contained in them can be lacking. Recently developed information retrieval technologies are based on the concept of a vector space. Data are modeled as a matrix, and a user's query of the database is represented as a vector. Relevant documents in the database are then identified via simple vector operations. Orthogonal factorizations of the matrix provide mechanisms for handling uncertainty in the database itself. The purpose of this paper is to show how such fundamental mathematical concepts from linear algebra can be used to manage and index large text collections. |
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Thin-Layer Solutions of the Helmholtz and Related Equations SIAM Rev. 54, pp. 3-51 (49 pages) Online Publication Date: February 08, 2012
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This paper concerns a certain class of two-dimensional solutions to four generic partial differential equations—the Helmholtz, modified Helmholtz, and convection–diffusion equations, and the heat conduction equation in the frequency domain—and the connections between these equations for this particular class of solutions. Specifically, we consider “thin-layer” solutions, valid in narrow regions across which there is rapid variation, in the singularly perturbed limit as the coefficient of the Laplacian tends to zero. For the well-studied Helmholtz equation, this is the high-frequency limit and the solutions in question underpin the conventional ray theory/WKB approach in that they provide descriptions valid in some of the regions where these classical techniques fail. Examples are caustics, shadow boundaries, whispering gallery, and creeping waves and focusing and bouncing ball modes. It transpires that virtually all such thin-layer models reduce to a class of generalized parabolic wave equations, of which the heat conduction equation is a special case. Moreover, in most situations, we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the Helmholtz equation. We also show how reasonably well-understood thin-layer phenomena associated with any one of the four generic equations may translate into less well-known effects associated with the others. In addition, our considerations also shed some light on the relationship between the methods of matched asymptotic, WKB, and multiple-scales expansions. |
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SIAM Rev. 54, pp. 1-1 (1 page)
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The first article of this section, “Thin-Layer Solutions of the Helmholtz and Related Equations,” by Ockendon and Tew, is firmly rooted in the customs of applied mathematical analysis. The survey considers four intimately related elliptic PDEs: Helmholtz, modified Helmholtz, convection-diffusion, and complex Helmholtz. The focus is on singularly perturbed solutions in two dimensions that arise in the asymptotic limit where the Laplacian term vanishes. This work was “compiled over the past six years” but reflects many more years of experience gained by the authors in the arts of matched asymptotic expansions, the Wentzel–Kramers–Brillouin (WKB) method, and multiple scales. Here you will find a catalogue of interrelated asymptotic thin-layer solutions that apply when there are rapid transverse variations in one dependent variable. The authors comment authoritatively on the relative merits of the asymptotic tools of the trade and show how phenomena associated with one PDE can be translated across to the others, creating a valuable resource in the best traditions of applied mathematics. The second article, “Modeling Growth in Biological Materials,” by Jones and Chapman, echoes the first in (a) having an author from the Oxford Mathematical Institute and (b) exploiting the presence of small parameters wherever possible. But it has a different, application-oriented flavor, focusing on mathematical models of tissue growth. This is a field where mathematical advances can considerably improve both biological knowledge and therapeutics. The survey covers a great deal of ground, quoting experimental results on tennis players' bones and rabbits' lungs, touching on some classic work of D'Arcy Thompson and John Conway, and drawing together a range of complementary modeling approaches. Using surprisingly few equations, it gives equal weight to the two broad themes of macro and micro. The continuum, macroscale models reviewed by Jones and Chapman use the familiar applied mathematical tools of kinematics, mechanics, and constitutive laws to produce PDEs for bulk behavior. At the other end of the scale, they look at the idea of starting with individual cells and introducing rules that govern their interactions. Such cell-based models can be more flexible and easier to calibrate and evaluate, but are less amenable to analysis. Their use is also limited by the availability of computing power. It is natural to ask whether these two distinct approaches could be combined into a single unified theory. Is there a modeling journey that begins with precise micro-level interactions, moves through one or more multiscale layers, and emerges with a macroscale description? This philosophy of putting together rather than taking apart, which perhaps comes more naturally to applied mathematicians than experimental life scientists, has been a driving force behind a range of high-profile activities in the area of systems biology. However, the authors explain here that even in the well-defined field of tissue growth, rigorous application of the underlying mathematical principles of homogenization and localization are far from trivial; many inconsistencies need to be ironed out and fundamental challenges remain. |
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Tensor Decompositions and Applications SIAM Rev. 51, pp. 455-500 (46 pages) Online Publication Date: August 05, 2009
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This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors. |
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Theory and Applications of Robust Optimization SIAM Rev. 53, pp. 464-501 (38 pages) Online Publication Date: August 05, 2011
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In this paper we survey the primary research, both theoretical and applied, in the area of robust optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multistage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering. |
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SIAM Rev. 54, pp. 181-208 (28 pages)
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Even long-time readers of SIAM Review may wonder what a featured review is. I, too, sometimes wonder. Occasionally, two or three related books are reviewed simultaneously. More often, the objective is to highlight an important new book or topic. It takes planning to arrange such a review for each issue. Sometimes other deserving books appear in the meantime or a lovely review or two will be otherwise submitted. For this issue, I asked Jim Simmonds to write a featured review of Audoly and Pomeau's Elasticity and Geometry: From Hair Curls to the Non-Linear Response of Shells. As reported, it features very novel applications and a remarkable combination of physical insight and analytical adeptness. Roy's brilliant new monograph on Sources in the Development of Mathematics appears to be so central to our understanding of calculus that I couldn't pass up the opportunity to highlight it as well. The issue also includes reviews by long-time favorite authors on such diverse subjects as boundary elements, combinatorics, classical analysis, discrete integrable systems, dynamical systems, estimation, fluid mechanics, quantum mechanics, and probability and stochastics. As always, our expert reviewers deserve our thanks and attention. |
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The Mathematics of Atmospheric Dispersion Modeling SIAM Rev. 53, pp. 349-372 (24 pages) Online Publication Date: May 05, 2011
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The Gaussian plume model is a standard approach for studying the transport of airborne contaminants due to turbulent diffusion and advection by the wind. This paper reviews the assumptions underlying the model, its derivation from the advection-diffusion equation, and the key properties of the plume solution. The results are then applied to solving an inverse problem in which emission source rates are determined from a given set of ground-level contaminant measurements. This source identification problem can be formulated as an overdetermined linear system of equations that is most easily solved using the method of least squares. Various generalizations of this problem are discussed, and we illustrate our results with an application to the study of zinc emissions from a smelting operation. |
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An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations SIAM Rev. 43, pp. 525-546 (22 pages) Online Publication Date: August 04, 2006
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A practical and accessible introduction to numerical methods for stochastic differential equations is given. The reader is assumed to be familiar with Euler's method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable; however, no knowledge of advanced probability theory or stochastic processes is assumed. The article is built around $10$ MATLAB programs, and the topics covered include stochastic integration, the Euler--Maruyama method, Milstein's method, strong and weak convergence, linear stability, and the stochastic chain rule. |
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SIAM Rev. 53, pp. 217-288 (72 pages) Online Publication Date: May 05, 2011
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Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and/or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the $k$ dominant components of the singular value decomposition of an $m \times n$ matrix. (i) For a dense input matrix, randomized algorithms require $\bigO(mn \log(k))$ floating-point operations (flops) in contrast to $ \bigO(mnk)$ for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to $\bigO(k)$ passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data. |
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SIAM Rev. 54, pp. 119-119 (1 page)
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A new section with a new direction. Research Spotlights replaced Expository Research Papers on January 31, 2012. Papers in Research Spotlights should cover topics in applied and computational mathematics of particularly wide interest and importance. Contributions must also be accessible to the broad and diverse SIAM Review readership. The principal theme of Research Spotlights is author flexibility with the intent that an expanded format promotes creativity and spurs innovative articles. Authors now have latitude in terms of considering a standard research paper or else a nontraditional article such as a mini-survey, a timely communication, a software description, or a new mathematical perspective within an application area. Prospective authors are encouraged to first consult with the section editor Ray Tuminaro (rstumin@sandia.gov) about potential contributions—especially those that lie outside the scope of traditional SIAM articles. While they can have a more relaxed format, ultimately articles must be of broad interest, must be accessible to the community, and must meet the technical review standards of SIAM journals. Focus groups provide feedback on potential new products: Which juice is more appealing to you, the slimy green or the yucky yellow one? Systems of sensors process signals to decide: Is there an intruder or not? These are examples of “group decision making,” a process where individuals work together to make a collective decision. The problem of figuring out how the collective arrives at a decision occurs in areas as varied as cognitive psychology, economics, political science, and signal processing. Margot Kimura and Jeff Moehlis in their paper “Group Decision-Making Models for Sequential Tasks” consider the “two-alternative forced-choice test,” where one must choose between two hypotheses: slimy green or yucky yellow; intruder or no intruder. Decisions must be made quickly and can only tolerate certain error rates. This means that there are limits on how often the sensors are allowed to miss an intruder, or signal a false alarm. The model in this paper assumes $N$ independent decision makers, each of whom receives observations, sequentially, one at a time. Each decision maker continues to process the observations until s/he is able to make a decision. The incoming observations are represented by independent random variables, with known prior probabilities for each decision. The processing consists of applying the “sequential probability ratio test” to each new observation. Based on the prespecified error rates for the number of misses and false alarms, this test either reports a decision or continues to process the next observation. The authors also consider a continuous version of this test, which becomes a drift-diffusion model as the time between observations goes to zero. Once a decision maker has come up with a decision, s/he reports it to the “fusion center,” which is responsible for arriving at a collective decision. The fusion center can operate in one of three modes: race (report only the very first decision that arrives), majority-total (wait until all $N$ decisions have arrived and then report the majority), and majority-first (wait until $N/2$ identical decisions have arrived, and then report this smallest possible majority). For each such mode, the authors derive probability distribution functions for the collective error rates and decision times, from the error rates and decision times of the individual decision makers. Simulations are presented to compare the different modes. Which mode turns out to be the most efficient is not at all obvious and depends on the scenario at hand, whether decision makers can have different error rates or can malfunction. Finally, the authors extend their analysis to more general modes, where the fusion center makes a collective decision based on the first $\eta$ decisions that arrive. The approach presented here has many advantages. It is general and applies to many situations that require collective decision making based on sequential observations, including even “cybernetic groups” with human observers and nonhuman detectors. Furthermore it is systematic and elegant, because it provides a clear path for deriving the efficiency of collective decisions from those of individual decision makers. Especially appreciated is a list of acronyms thoughtfully included by the authors at the beginning of the paper, which makes it easy to decipher the many acronyms in this area. |
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Group Decision-Making Models for Sequential Tasks SIAM Rev. 54, pp. 121-138 (18 pages) Online Publication Date: February 08, 2012
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The sequential probability ratio test (SPRT) and related drift-diffusion model (DDM) are optimal for choosing between two hypotheses using the minimal (average) number of samples and relevant for modeling the decision-making process in human observers. This work extends these models to group decision making. Previous works have focused almost exclusively on group accuracy; here, we explicitly address group decision time. First, we derive explicit solutions for the error rate and probability distribution function of decision times for a group of independent, (possibly) nonidentical decision makers using one of three simple rules: Race, Majority Total, and Majority First. We illustrate our solutions with a group of $N$ i.i.d. decision makers who each make an individual decision using the SPRT-based DDM, then compare the performance of each group rule under different constraints. We then generalize these group rules to the $\eta$-Total and $\eta$-First schemes, to demonstrate the flexibility and power of our approach in characterizing the performance of a group, given the performance of its individual members. |
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Numerical Methods for Electronic Structure Calculations of Materials SIAM Rev. 52, pp. 3-54 (52 pages) Online Publication Date: February 05, 2010
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The goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems. The paper is intended for a diverse scientific computing audience. For this reason, we assume the reader does not have an extensive background in the related physics. Our overview focuses on the nature of the numerical problems to be solved, their origin, and the methods used to solve the resulting linear algebra or nonlinear optimization problems. It is common knowledge that the behavior of matter at the nanoscale is, in principle, entirely determined by the Schrödinger equation. In practice, this equation in its original form is not tractable. Successful but approximate versions of this equation, which allow one to study nontrivial systems, took about five or six decades to develop. In particular, the last two decades saw a flurry of activity in developing effective software. One of the main practical variants of the Schrödinger equation is based on what is referred to as density functional theory (DFT). The combination of DFT with pseudopotentials allows one to obtain in an efficient way the ground state configuration for many materials. This article will emphasize pseudopotential-density functional theory, but other techniques will be discussed as well. |
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Modeling and Simulating Chemical Reactions SIAM Rev. 50, pp. 347-368 (22 pages) Online Publication Date: May 05, 2008
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Many students are familiar with the idea of modeling chemical reactions in terms of ordinary differential equations. However, these deterministic reaction rate equations are really a certain large-scale limit of a sequence of finer-scale probabilistic models. In studying this hierarchy of models, students can be exposed to a range of modern ideas in applied and computational mathematics. This article introduces some of the basic concepts in an accessible manner and points to some challenges that currently occupy researchers in this area. Short, downloadable MATLAB codes are listed and described. |
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Introduction to Automatic Differentiation and MATLAB Object-Oriented Programming SIAM Rev. 52, pp. 545-563 (19 pages) Online Publication Date: August 05, 2010
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An introduction to both automatic differentiation and object-oriented programming can enrich a numerical analysis course that typically incorporates numerical differentiation and basic MATLAB computation. Automatic differentiation consists of exact algorithms on floating-point arguments. This implementation overloads standard elementary operators and functions in MATLAB with a derivative rule in addition to the function value; for example, $\sin u$ will also compute $(\cos u)\ast u^{\prime}$, where $u$ and $u^{\prime }$ are numerical values. These methods are mostly one-line programs that operate on a class of value-and-derivative objects, providing a simple example of object-oriented programming in MATLAB using the new (as of release 2008a) class definition structure. The resulting powerful tool computes derivative values and multivariable gradients, and is applied to Newton's method for root-finding in both single and multivariable settings. To compute higher-order derivatives of a single-variable function, another class of series objects keeps Taylor polynomial coefficients up to some order. Overloading multiplication on series objects is a combination (discrete convolution) of coefficients. This idea leads to algorithms for other operations and functions on series objects. A survey of more advanced topics in automatic differentiation includes an introduction to the reverse mode (our implementation is forward mode) and considerations in arbitrary-order multivariable series computation. |
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