Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

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Iterative Methods for Sparse Linear Systems

Determining optical flow

The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization. To overcome this limitation, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. It is more accurate, easier to implement, and uses the same order of calculations as linearization. This paper reviews the motivation, development, use, and implications of the UT.

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Unscented filtering and nonlinear estimation

Excess CO2 is generated when lactate is increased during exercise because its [H+] is buffered primarily by HCO-3 (22 ml for each meq of lactic acid). We developed a method to detect the anaerobic threshold (AT), using computerized regression analysis of the slopes of the CO2 uptake (VCO2) vs. O2 uptake (VO2) plot, which detects the beginning of the excess CO2 output generated from the buffering of [H+], termed the V-slope method. From incremental exercise tests on 10 subjects, the point of excess CO2 output (AT) predicted closely the lactate and HCO-3 thresholds. The mean gas exchange AT was found to correspond to a small increment of lactate above the mathematically defined lactate threshold [0.50 +/- 0.34 (SD) meq/l] and not to differ significantly from the estimated HCO-3 threshold. The mean VO2 at AT computed by the V-slope analysis did not differ significantly from the mean value determined by a panel of six experienced reviewers using traditional visual methods, but the AT could be more reliably determined by the V-slope method. The respiratory compensation point, detected separately by examining the minute ventilation vs. VCO2 plot, was consistently higher than the AT (2.51 +/- 0.42 vs. 1.83 +/- 0.30 l/min of VO2). This method for determining the AT has significant advantages over others that depend on regular breathing pattern and respiratory chemosensitivity.

A new method for detecting anaerobic threshold by gas exchange

The goals of this chapter are (1) to outline and substantiate a broad conceptualization of what it means to think mathematically, (2) to summarize the literature relevant to understanding mathematical thinking and problem solving, and (3) to point to new directions in research, development, and assessment consonant with an emerging understanding of mathematical thinking and the goals for instruction outlined here. The use of the phrase “learning to think mathematically” in this chapter’s title is deliberately broad. Although the original charter for this chapter was to review the literature on problem solving and metacognition, the literature itself is somewhat ill defined and poorly grounded. As the literature summary will make clear, problem solving has been used with multiple meanings that range from “working rote exercises” to “doing mathematics as a professional”; metacognition has multiple and almost disjoint meanings (from knowledge about one’s thought processes to self-regulation during problem solving) that make it difficult to use as a concept. This chapter outlines the various meanings that have been ascribed to these terms and discusses their role in mathematical thinking. The discussion will not have the character of a classic literature review, which is typically encyclopedic in its references and telegraphic in its discussions of individual papers or results. It will, instead, be selective and illustrative, with main points illustrated by extended discussions of pertinent examples. Problem solving has, as predicted in the 1980 Yearbook of the National Council of Teachers of Mathematics (Krulik, 1980, p. xiv), been the theme of the 1980s. The decade began with NCTM’s widely heralded statement, in its Agenda for Action, that “problem solving must be the focus of school mathematics” (NCTM, 1980, p. 1). It concluded with the publication of Everybody Counts (National Research Council, 1989) and the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989), both of which emphasize problem solving. One might infer, then, that there is general acceptance of the idea that the primary goal of mathematics instruction should be to have students become competent problem solvers. Yet, given the multiple interpretations of the term, the goal is hardly clear. Equally unclear is the role that problem solving, once adequately characterized, should play in the larger context of school mathematics. What are the goals for mathematics instruction, and how does problem solving fit within those goals? Such questions are complex. Goals for mathematics instruction depend on one’s conceptualization of what mathematics is, and what it means to understand mathematics. Such conceptualizations vary widely. At one end of the spectrum, mathematical knowledge is seen as a body of facts and procedures dealing with quantities, magnitudes, and forms, and the relationships among them; knowing mathematics is seen as having mastered these facts and procedures. At the other end of the spectrum, mathematics is conceptualized as the “science of patterns,” an (almost) empirical discipline closely akin to the sciences in its emphasis on pattern-seeking on the basis of empirical evidence. The author’s view is that the former perspective trivializes mathematics; that a curriculum based on mastering a corpus of mathematical facts and procedures is severely impoverished—in much the same way that an English curriculum would be considered impoverished if it focused largely, if not exclusively, on issues of grammar. The author characterizes the mathematical enterprise as follows:

Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics (Reprint):

Methods of Numerical Integration.

Methods of Numerical Integration

Numerical Methods for Scientists and Engineers.

Winner of the 1983 National Book Award! \"...a perfectly marvelous book about the Queen of Sciences, from which one will get a real feeling for what mathematicians do and who they are. The exposition is clear and full of wit and humor...\" - The New Yorker (1983 National Book Award edition) Mathematics has been a human activity for thousands of years. Yet only a few people from the vast population of users are professional mathematicians, who create, teach, foster, and apply it in a variety of situations. The authors of this book believe that it should be possible for these professional mathematicians to explain to non-professionals what they do, what they say they are doing, and why the world should support them at it. They also believe that mathematics should be taught to non-mathematics majors in such a way as to instill an appreciation of the power and beauty of mathematics. Many people from around the world have told the authors that they have done precisely that with the first edition and they have encouraged publication of this revised edition complete with exercises for helping students to demonstrate their understanding. This edition of the book should find a new generation of general readers and students who would like to know what mathematics is all about. It will prove invaluable as a course text for a general mathematics appreciation course, one in which the student can combine an appreciation for the esthetics with some satisfying and revealing applications. The text is ideal for 1) a GE course for Liberal Arts students 2) a Capstone course for perspective teachers 3) a writing course for mathematics teachers. A wealth of customizable online course materials for the book can be obtained from Elena Anne Marchisotto (elena.marchisotto@csun.edu) upon request.

The Mathematical Experience.

Winner of the 1983 National Book Award! "...a perfectly marvelous book about the Queen of Sciences, from which one will get a real feeling for what mathematicians do and who they are. The exposition is clear and full of wit and humor..." - The New Yorker (1983 National Book Award edition) Mathematics has been a human activity for thousands of years. Yet only a few people from the vast population of users are professional mathematicians, who create, teach, foster, and apply it in a variety of situations. The authors of this book believe that it should be possible for these professional mathematicians to explain to non-professionals what they do, what they say they are doing, and why the world should support them at it. They also believe that mathematics should be taught to non-mathematics majors in such a way as to instill an appreciation of the power and beauty of mathematics. Many people from around the world have told the authors that they have done precisely that with the first edition and they have encouraged publication of this revised edition complete with exercises for helping students to demonstrate their understanding. This edition of the book should find a new generation of general readers and students who would like to know what mathematics is all about. It will prove invaluable as a course text for a general mathematics appreciation course, one in which the student can combine an appreciation for the esthetics with some satisfying and revealing applications. The text is ideal for 1) a GE course for Liberal Arts students 2) a Capstone course for perspective teachers 3) a writing course for mathematics teachers. A wealth of customizable online course materials for the book can be obtained from Elena Anne Marchisotto (elena.marchisotto@csun.edu) upon request.

Philip J. Davis

Papers

Methods of Numerical Integration.

Methods of Numerical Integration

Numerical Methods for Scientists and Engineers.

The Mathematical Experience.

The Mathematical Experience