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E-raamat: Elementary Numerical Mathematics for Programmers and Engineers

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Elementary Numerical Mathematics for Programmers and EngineersSuurem pilt
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This book covers the basics of numerical methods, while avoiding the definition-theorem-proof style and instead focusing on numerical examples and simple pseudo-codes.The book is divided into ten chapters. Starting with floating number calculations and continuing up to ordinary differential equations, including "Euler backwards". The final chapter discusses practical error estimations. Exercises (including several in MATLAB) are provided at the end of each chapter. Suitable for readers with minimal mathematical knowledge, the book not only offers an elementary introduction to numerical mathematics for programmers and engineers but also provides supporting material for students and teachers of mathematics.

Floating point arithmetic.- Norms, condition numbers.- Solution of systems of linear equations.- The least squares problem.- Eigenvalue problems.- Interpolation.- Nonlinear equations and systems.- Numerical integration.- Ordinary differential equations.- Practical error estimation.- Bibliography.- Index.

Arvustused

The book is well-written and the typesetting is attractive. Elementary Numerical Mathematics for Programmers and Engineers covers the standard topics well. Results are never simply stated and the main ideas are always fleshed out either through derivations or highly insightful examples. The authors also use clear and consistent notation throughout the book. I would particularly recommend this book to someone who is trying to learn numerical mathematics independently without the help of an instructor. (Jason M. Graham, MAA Reviews, maa.org, February, 2017)

1 Floating Point Arithmetic
1(14)
1.1 Integers
1(1)
1.2 Floating Point Numbers
2(2)
1.3 Floating Point Arithmetic, Rounding
4(5)
1.4 Accumulation of Errors
9(3)
1.5 Conclusions
12(1)
1.6 Exercises
13(2)
2 Norms, Condition Numbers
15(22)
2.1 Norms and Their Elementary Properties
16(3)
2.2 The Induced Matrix Norm
19(7)
2.2.1 Definition and Properties
19(1)
2.2.2 Computation of the Induced Matrix Norm for the Vector p-Norms, p = 1, ∞
20(3)
2.2.3 Computation of the Induced Matrix Norm (p = 2)
23(3)
2.3 Error Estimations
26(7)
2.3.1 The Right-Hand Side of the Linear System is Perturbed
26(1)
2.3.2 Condition Number
27(4)
2.3.3 The Matrix of the Linear System is Perturbed
31(2)
2.4 Exercises
33(4)
3 Solution of Systems of Linear Equations
37(32)
3.1 Gaussian Elimination
37(3)
3.2 When Can Gaussian Elimination be Performed?
40(2)
3.3 The LU Factorization
42(4)
3.4 Algorithms, Cost of Solving
46(4)
3.5 Influence of Rounding Errors
50(1)
3.6 LU Factorization for General Matrices
51(5)
3.6.1 Algorithm of the LDU Factorization, Test Examples
54(2)
3.7 Cholesky Factorization
56(5)
3.7.1 Algorithm of the LDLT Factorization, Test Examples
60(1)
3.8 Band Matrices
61(4)
3.8.1 Tridiagonal Systems of Equations
62(2)
3.8.2 The Tridiagonal Algorithm, Test Examples
64(1)
3.9 Exercises
65(4)
4 The Least Squares Problem
69(16)
4.1 Linear Regression
70(6)
4.1.1 Algebraic Description
71(1)
4.1.2 The Method of Least Squares
72(4)
4.2 Normal Equations
76(4)
4.3 Solution Algorithm, Test Examples
80(2)
4.4 Exercises
82(3)
5 Eigenvalue Problems
85(26)
5.1 Fundamental Properties
85(7)
5.1.1 Normal Matrices
87(2)
5.1.2 The Characteristic Polynomial
89(1)
5.1.3 Localization of the Eigenvalues
90(2)
5.2 Power Iteration
92(9)
5.2.1 Conditions of Convergence
93(4)
5.2.2 The Rayleigh Quotient
97(1)
5.2.3 Algorithm of the Power Iteration, Test Examples
98(2)
5.2.4 The Shift
100(1)
5.3 The Inverse Iteration
101(4)
5.3.1 Conditions of Convergence
102(1)
5.3.2 Algorithm of the Inverse Iteration, Test Examples
103(2)
5.4 The QR Algorithm
105(2)
5.5 Exercises
107(4)
6 Interpolation
111(24)
6.1 Interpolation Problems
112(1)
6.2 Lagrangian Interpolation
112(13)
6.2.1 Lagrange Interpolation Problem
112(3)
6.2.2 Newton's Divided Differences Method
115(2)
6.2.3 The Difference Scheme
117(5)
6.2.4 Algorithm of Lagrangian Interpolation, Test Examples
122(1)
6.2.5 Error Estimations
123(2)
6.3 Hermite Interpolation
125(4)
6.4 Piecewise Polynomial Interpolation
129(4)
6.5 Exercises
133(2)
7 Nonlinear Equations and Systems
135(26)
7.1 Bisection Method, Fixed Point Iterations
136(2)
7.2 Newton's Method
138(10)
7.2.1 Damped Newton Method
143(2)
7.2.2 Secant Method
145(3)
7.3 Solution of Systems of Equations
148(7)
7.3.1 Newton's Method
148(2)
7.3.2 Algorithm of Damped Newton Method, Test Examples
150(2)
7.3.3 Approximation of the Jacobian Matrix
152(1)
7.3.4 Broyden's Method
153(2)
7.4 Gauss-Newton Method
155(4)
7.4.1 Description
155(1)
7.4.2 Algorithm of the Gauss-Newton Method, Test Examples
156(3)
7.5 Exercises
159(2)
8 Numerical Integration
161(24)
8.1 Elementary Quadrature Formulae
162(3)
8.2 Interpolational Quadrature Formulae
165(3)
8.3 Composite Quadrature Rules
168(6)
8.3.1 Construction of Composite Formulae
168(3)
8.3.2 Convergence of Composite Formulae
171(3)
8.4 Practical Points of View
174(3)
8.5 Calculation of Multiple Integrals
177(6)
8.5.1 Reduction to Integration of Functions of One Variable
177(2)
8.5.2 Approximation of the Domain
179(3)
8.5.3 Algorithm of the Two-Dimensional Simpson Rule, Tests
182(1)
8.6 Exercises
183(2)
9 Numerical Solution of Ordinary Differential Equations
185(26)
9.1 Motivation
185(3)
9.2 Initial Value Problems
188(3)
9.3 Euler's Method
191(2)
9.3.1 Algorithm of Euler's Method, Test Examples
192(1)
9.4 Error Analysis of Euler's Method
193(3)
9.5 The Modified Euler Method, Runge-Kutta Methods
196(5)
9.6 The Implicit Euler Method
201(7)
9.6.1 The Implicit Euler Method for Linear Systems
201(3)
9.6.2 Nonlinear Systems
204(2)
9.6.3 Algorithm of the Implicit Euler Method, Test Examples
206(2)
9.7 Exercises
208(3)
10 Practical Error Estimation
211(6)
Bibliography 217(2)
Index 219
Gisbert Stoyan worked for more than 10 years on industrial problems at the WIAS in Berlin and was teaching numerical mathematics at ELTE University (Budapest, Hungary) for over 30 years. He has written research papers mostly on the numerical solution of partial differential equations. A three-volume textbook published in Hungarian brings together his experiences in areas including (along with the basic topics like numerical linear algebra, nonlinear equations) strongly stable methods for ODEs, multigrid algorithms, finite element praxis and theory, finite elements for Navier-Stokes equations, and methods for first-order hyperbolic equations.



Agnes Baran received her PhD in Mathematics in 2008 at the University of Debrecen. Her doctoral thesis on high-order finite element methods for Stokes equations was supervised by Gisbert Stoyan. Since 2008, she works as an Assistant Professor at the Faculty of Informatics at the the University of Debrecen where she teaches courses on numerical analysis for students of Mathematics and Computer Science.
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