Format:
Online-Ressource (XII, 468 p, digital)
ISBN:
9783642112997
Series Statement:
Texts in Computational Science and Engineering 7
Content:
Computing Integrals.- Differential Equations: The First Steps -- Systems of Ordinary Differential Equations -- Nonlinear Algebraic Equations.- The Method of Least Squares.- About Scientific Software -- The Diffusion Equation -- Analysis of the Diffusion Equation -- Parameter Estimation and Inverse Problems. - A Glimpse of Parallel Computing -- Index.
Content:
Science used to be experiments and theory, now it is experiments, theory and computations. The computational approach to understanding nature and technology is currently flowering in many fields such as physics, geophysics, astrophysics, chemistry, biology, and most engineering disciplines. This book is a gentle introduction to such computational methods where the techniques are explained through examples. It is our goal to teach principles and ideas that carry over from field to field. You will learn basic methods and how to implement them. In order to gain the most from this text, you will need prior knowledge of calculus, basic linear algebra and elementary programming.
Note:
Includes bibliographical references (p. 455-456) and index
,
""Elements of ScientificComputing""; ""Preface""; ""Contents""; ""1 Computing Integrals""; ""1.1 Background""; ""1.2 The Computational Problem""; ""1.3 The Trapezoidal Method""; ""1.3.1 Approximating the Integral Using One Trapezoid""; ""1.3.2 Approximating the Integral Using Two Trapezoids""; ""1.3.3 Approximating the Integral Using n Trapezoids""; ""1.4 Computational Analysis""; ""1.5 Back to the Bagels""; ""1.6 Exercises""; ""1.7 Projects""; ""1.7.1 Show that the Error is O(h2)""; ""1.7.2 Derive Other Methods for Numerical Integration""; ""1.7.3 Compute the Length of a Cable""
,
""2 Differential Equations: The First Steps""""2.1 Modeling Changes""; ""2.1.1 The Cultivation of Rabbits""; ""2.1.2 The Simplest Possible Case""; ""2.1.3 Exponential Growth""; ""2.1.4 Logistic Growth""; ""2.2 Numerical Solution""; ""2.2.1 The Simplest Possible Model""; ""2.2.2 Numerical Approximation of Exponential Growth ""; ""2.2.3 Numerical Stability""; ""2.2.4 An Implicit Scheme""; ""2.2.5 Explicit and Implicit Schemes""; ""2.2.6 Numerical Solution of the Logistic Equation""; ""2.3 Exercises""; ""2.4 Projects""; ""2.4.1 More on Stability""; ""2.4.2 More on Accuracy""
,
""3 Systems of Ordinary Differential Equations""""3.1 Rabbits and Foxes; Fish and Sharks""; ""3.2 A Numerical Method: Unlimited Resources""; ""3.3 A Numerical Method: Limited Resources""; ""3.4 Phase Plane Analysis""; ""3.4.1 A Simplified Model""; ""3.4.2 The Phase Plane""; ""3.4.3 Circles in the Analytical Phase Plane""; ""3.4.4 Alternative Analysis""; ""3.4.5 Circles in the Numerical Phase Plane""; ""3.4.6 More on Numerics""; ""3.5 Exercises""; ""3.6 Project: Analysis of a Simple System""; ""4 Nonlinear Algebraic Equations""; ""4.1 The Bisection Method""; ""4.2 Efficiency Consideration""
,
""4.3 Newton's Method""""4.4 The Secant Method""; ""4.5 Fixed-Point Iterations""; ""4.5.1 Convergence of Fixed-Point Iterations""; ""4.5.2 Speed of Convergence""; ""4.5.3 Existence and Uniqueness of a Solution""; ""4.5.4 Uniqueness""; ""4.5.5 Existence""; ""4.6 Systems of Nonlinear Equations""; ""4.6.1 A Linear System""; ""4.6.2 A Nonlinear System""; ""4.6.3 Newton's Method""; ""4.6.4 A Nonlinear Example""; ""4.6.5 The Nonlinear System Revisited""; ""4.7 Exercises""; ""4.8 Project: Convergence of Newton's Method""; ""5 The Method of Least Squares""
,
""5.1 Building Models of Discrete Data Sets""""5.1.1 Approximation by a Constant""; ""5.1.2 Approximation by a Linear Function""; ""5.1.3 Approximation by a Quadratic Function""; ""5.1.4 Large Data Sets""; ""5.2 Application to Population Models""; ""5.2.1 Exponential Growth of the World Population?""; ""5.2.2 Logistic Growth of the World Population""; ""5.3 Least Squares Approximations of Functions""; ""5.3.1 Approximating Functions by a Constant""; ""5.3.2 Approximation Using Linear Functions""; ""5.3.3 Approximation Using Quadratic Functions""; ""5.3.4 Summary of the Examples""
,
""5.4 Exercises""
Additional Edition:
ISBN 9783642112980
Additional Edition:
Buchausg. u.d.T. Tveito, Aslak, 1961 - Elements of scientific computing Berlin : Springer, 2010 ISBN 9783642112980
Additional Edition:
ISBN 9783642265198
Language:
English
Subjects:
Computer Science
,
Mathematics
Keywords:
Wissenschaftliches Rechnen
;
Wissenschaftliches Rechnen
DOI:
10.1007/978-3-642-11299-7
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Author information:
Langtangen, Hans Petter 1962-
Author information:
Cai, Xing
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