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E-raamat: Scientific Computing: A Historical Perspective

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This book explores the most significant computational methods and the history of their development. It begins with the earliest mathematical / numerical achievements made by the Babylonians and the Greeks, followed by the period beginning in the 16th century. For several centuries the main scientific challenge concerned the mechanics of planetary dynamics, and the book describes the basic numerical methods of that time. 

In turn, at the end of the Second World War scientific computing took a giant step forward with the advent of electronic computers, which greatly accelerated the development of numerical methods. As a result, scientific computing became established as a third scientific method in addition to the two traditional branches: theory and experimentation.









The book traces numerical methods journey back to their origins and to the people who invented them, while also briefly examining the development of electronic computers over the years. Featuring 163 references and more than 100 figures, many of them portraits or photos of key historical figures, the book provides a unique historical perspective on the general field of scientific computing making it a valuable resource for all students and professionals interested in the history of numerical analysis and computing, and for a broader readership alike.

Arvustused

The book evolves like a numerical course where the historical aspects are somewhat expanded at the expense of the numerical issues, but the main formulas and ideas are there. The numerics are however too thin to be useful as lecture notes for a course on numerical analysis. Therefore the book can only serve as additional literature to accompany a true course about the numerics. (Adhemar Bultheel, European Mathematical Society, euro-math-soc.eu, March 20, 2019)

1 Scientific Computing: An Introduction
1(4)
2 Computation Far Back in Time
5(12)
2.1 The Babylonians
6(4)
2.2 Archimedes and Iterative Methods
10(5)
2.3 Chinese Mathematics
15(2)
3 The Centuries Before Computers
17(72)
3.1 Nonlinear Algebraic Equations
19(6)
3.1.1 The Fixed Point Method
19(1)
3.1.2 Newton Methods
20(5)
3.2 Interpolation
25(7)
3.3 Integrals
32(4)
3.4 The Least Squares Method
36(6)
3.5 Gauss and Linear Systems of Equations
42(4)
3.6 Series Expansion
46(12)
3.6.1 Taylor Series
47(1)
3.6.2 Orthogonal Polynomial Expansions
48(3)
3.6.3 Fourier Series
51(4)
3.6.4 The Gibbs Phenomenon
55(1)
3.6.5 The Fourier Transform
56(2)
3.7 Ordinary Differential Equations (ODE)
58(10)
3.7.1 The Euler Method
58(3)
3.7.2 Adams Methods
61(3)
3.7.3 Runge-Kutta Methods
64(1)
3.7.4 Richardson Extrapolation
65(3)
3.8 Partial Differential Equations (PDE)
68(13)
3.8.1 The Ritz-Galerkin Method
69(3)
3.8.2 Courant's Article on FEM
72(3)
3.8.3 Richardson's First Paper on Difference Methods
75(2)
3.8.4 Weather Prediction; A First Attempt
77(2)
3.8.5 The CFL-Article
79(2)
3.9 Optimization
81(8)
3.9.1 The Fourier-Motzkin Algorithm
82(3)
3.9.2 Linear Programming and the War Effort
85(2)
3.9.3 Nonlinear Optimization
87(2)
4 The Beginning of the Computer Era
89(44)
4.1 The First Electronic Computers
89(5)
4.2 Further Developments of Difference Methods for ODE
94(5)
4.2.1 Stability and Convergence
94(2)
4.2.2 Stiff ODE
96(3)
4.3 PDE and Difference Methods
99(22)
4.3.1 Fourier Analysis and the von Neumann Condition
99(6)
4.3.2 Implicit Methods and Operator Splitting
105(3)
4.3.3 Stability and Convergence
108(1)
4.3.4 The Matrix Theorem
109(2)
4.3.5 Fluid Dynamics and Shocks
111(8)
4.3.6 Stability for Dissipative Methods
119(2)
4.4 Progress in Linear Algebra
121(12)
4.4.1 Orthogonalization
121(4)
4.4.2 Iterative Methods for Linear Systems of Equations
125(3)
4.4.3 Eigenvalues
128(5)
5 The Establishment of Scientific Computing as a New Discipline
133(100)
5.1 Faster and Faster Computers
133(8)
5.1.1 New Hardware
134(5)
5.1.2 Programming Computers
139(2)
5.2 Mathematical Aspects of Nonlinear Problems
141(2)
5.3 Refinement of ODE Methods
143(5)
5.4 Initial Boundary Value Problems for PDE and Difference Methods
148(5)
5.4.1 The Godunov-Ryabenkii Condition
148(2)
5.4.2 Kreiss' Stability Theory
150(3)
5.5 New Types of Functions
153(12)
5.5.1 Piecewise Polynomials and Splines
154(4)
5.5.2 Wavelets
158(5)
5.5.3 Radial Basis Functions
163(2)
5.6 Finite Element Methods
165(10)
5.6.1 Structural Mechanics
165(1)
5.6.2 General Stationary Problems
166(3)
5.6.3 Time-Dependent Problems
169(2)
5.6.4 Discontinuous Galerkin Methods
171(3)
5.6.5 Grid Generation
174(1)
5.7 The Fast Fourier Transform
175(8)
5.7.1 The Algorithm
175(3)
5.7.2 An Application of FFT: Computed Tomography
178(5)
5.8 Spectral Methods
183(5)
5.8.1 Pseudo-Spectral Methods
183(4)
5.8.2 Spectral Element Methods
187(1)
5.9 Multiscale Methods
188(11)
5.9.1 Multigrid Methods
188(7)
5.9.2 Fast Multipole Methods (FMM)
195(2)
5.9.3 Heterogeneous Multiscale Methods (HMM)
197(2)
5.10 Singular Value Decomposition (SVD)
199(4)
5.11 Krylov Space Methods for Linear Systems of Equations
203(7)
5.12 Domain Decomposition
210(5)
5.13 Wave Propagation and Open Boundaries
215(6)
5.14 Stochastic Procedures for Deterministic Problems
221(8)
5.14.1 Monte Carlo Methods
221(7)
5.14.2 An Example from Financial Mathematics
228(1)
5.15 Level Set Methods
229(4)
6 Impact of Numerical Analysis and Scientific Computing
233(8)
6.1 Numerical Methods
233(2)
6.2 Analysis Techniques
235(1)
6.3 Examples from Computational Physics, Chemistry and Engineering
236(5)
6.3.1 Computational Physics
236(1)
6.3.2 Computational Chemistry
237(1)
6.3.3 Computational Engineering
238(3)
References 241(8)
Index 249
Bertil Gustafsson is Professor at Uppsala University and a member of the Royal Swedish Academy of Sciences. His main area of research is the numerical solution of partial differential equations.

Earlier books:



B. Gustafsson, H.-O. Kreiss, J. Oliger: Time-Dependent Problems and Difference Methods, Whiley (1995).

B. Gustafsson: High Order Difference Methods for Time Dependent PDE. Springer (2008).









B.Gustafsson: Fundamentals of Scientific Computing. Springer (2011).

B. Gustafsson, H.-O. Kreiss, J. Oliger: Time-Dependent Problems and Difference Methods, 2nd ed., Whiley (2013).
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