This introduction to numerical analysis shows how the mathematics of calculus and linear algebra are implemented in computer algorithms. It develops a deep understanding of why numerical methods work and exactly what their limitations are.

This book grew out of a conference organized by the German Historical Institute (GHI) on May 23-5, 1996, iii Berlin. For three days, thirty-five scholars from n\ne countries became acht-und-sechziger (sixty-eighters), not on the streets and barricades but in lively debate over the events and significance of the momentous year 1968. This book consists of revised versions of many of the original papers as well as entirely new essays that complement our overall theme. I thank all of the authors and also those who participated in the conference but whose work could not be included here. I am grateful to Carl-Ludwig Holtfrerich (Free University Berlin), Luisa Passerini (European University Institute, Florence), Keith A. Reader (University of Newcastle), and Dieter Rucht (Wissenschaftszentrum Berlin) for chairing the various sessions and for guiding us through spirited discussions. In Berlin, the Wissenschaftszentrum afforded a comfortable setting for three days of intensive debate. I thank Friedheim Neidhardt, the center'sdirector, and his dedicated staff - Birgit Hahn, Britta Heinrich, Ilse Kischlat, Dietmar Kremser, and Burckhard Wiebe - for hosting the conference and helping with its organization. Dieter Senoner of the mayor's office kindly welcomed us to Berlin at a reception at the Rotes Rathaus. Anneke de Rudder, who took us on a "1968 Walking Tour" through the German capital, gave us the opportunity to visit the key sites of that historic year.At the GHI, I thank Barbel Bernhardt, Christa Brown, Dieter Schneider, and Barbel Thomas - all of whom were involved in the organization before, during, and after the conference. Pamela Abraham tracked down the copyrights for the illustrations and assisted in assembling the manuscript. Daniel S. Mattern, the series editor at the Institute, deserves a special note of appreciation for guiding us through the difficult process of publishing this collection and for preparing a readable, clear, and concise manuscript. Frank Smith, the executive editor for social sciences at Cambridge University Press, took great interest in this project from the begin-IX X Preface ning and helped bring about its final shape. Further, I thank the two anonymous readers for their very useful critical evaluations of the manuscript. Finally, I thank my two coeditors. Carole Fink provided the original plan, which she presented to me at another GHI conference, in Berkeley, California, in May 1994. In subsequent discussions, we discovered that "1968" is a particularly well-suited topic for understanding the intellectual challenges that increasing global interdependence poses to historians of the twentieth century. We were later joined by Philipp Gassert, who worked with us to organize the conference and edit the book. I shall miss these last three and a half years of spirited collaboration among three very engaged historians. The topic of this book dovetails nicely with the efforts of the GHI in Washington to promote comparative research. In fact, most of the four to five international conferences the Institute rganizes annually are based on this perspective, comparing events, ideas, and structures in the United States and Europe. At times we have even taken a global perspective, with 1968: The World Traniformed being a case in point. In this age of globalization, historians should also become global players - at least mentally.