Ebook details
Numerical Methods
Piotr Tatjewski
The presented text is an English version of the original textbook written in Polish ("Metody numeryczne"), published in 2013 (first edition) and 2026 (second edition) by OWPW (Oficyna Wydawnicza Politechniki Warszawskiej - the Warsaw University of Technology Publishing House). This second English edition differs a little from the first one published in 2014, the few errors found in the first edition have been only corrected and very minor improvements of the text and figures have been done.
The original version of the textbook (first edition) was written based on the author's long experience as a teacher of several courses on numerical methods, at the Faculty of Electronics and Information Technology, Warsaw University of Technology, mainly the course "Numerical Methods" taught in Polish for ICT students and in English for Electrical and Computer Engineering students. The textbook summed up the experience of a long series of previously prepared Lecture Notes, available for students in electronic form at the web pages of the courses.
The author would like to express his gratitude to all students actively taking part in the courses, for all comments concerning the way of presentation, for indications of editorial errors. All these comments helped the author to improve the quality of this textbook.
Contents
Preface (to the second edition) 9
Chapter 1. Preliminaries 11
1.1. Computer representation of numbers, representation errors 11
1.2. Floating-point arithmetic 14
1.3. Condition number 17
1.4. The algorithm and its numerical realizations 23
1.5. Numerical stability of algorithms 24
Chapter 2. Linear equations, matrix factorizations 29
2.1. Norms of vectors and matrices 29
2.2. Conditioning of a matrix, of a system of linear equations 32
2.3. Gaussian elimination, LU factorization 33
2.3.1. Upper-triangular systems of linear equations 34
2.3.2. Gaussian elimination 35
2.3.3. LU matrix factorization 37
2.3.4. Gaussian elimination with pivoting 40
2.3.5. Residual correction (iterative improvement) 48
2.3.6. Full elimination method (Gauss-Jordan method) 48
2.4. Cholesky-Banachiewicz (LLT) factorization 49
2.4.1. LLT factorization 49
2.4.2. LDLT factorization, relations between triangular factorizations 51
2.5. Calculation of determinants and inverse matrices 53
2.6. Iterative methods for systems of linear equations 56
2.6.1. Jacobi’s method 58
2.6.2. Gauss-Seidel method 59
2.6.3. Stop tests 60
Chapter 3. QR factorization, eigenvalues, singular values 63
3.1. Orthogonal-triangular (QR) matrix factorizations 63
3.2. Eigenvalues 69
3.2.1. Preliminaries 69
3.2.2. The QR method for finding eigenvalues 73
3.3. Singular values, SVD decomposition 79
3.4. Linear least-squares problem 81
3.5. Givens transformation, with applications 85
3.5.1. Givens transformation (rotation) 85
3.5.2. Jacobi’s method for finding eigenvalues of a symmetric matrix 87
3.5.3. The QR matrix factorization using the Givens rotations 89
3.6. Householder transformation, with applications 90
3.6.1. Householder transformation (reflection) 90
3.6.2. The QR matrix factorization using the Householder reflections 92
3.6.3. Transformation of a matrix to the Hessenberg form using the Householder reflections, preserving matrix similarity 93
Chapter 4. Approximation 97
4.1. Discrete least-squares approximation 99
4.1.1. Polynomial approximation 102
4.1.2. Approximation using an orthogonal function basis 105
4.2. Padé approximation 107
Chapter 5. Interpolation 113
5.1. Algebraic polynomial interpolation 114
5.1.1. Lagrange interpolating polynomial 115
5.1.2. Newton’s interpolating polynomial 116
5.2. Spline function interpolation 123
Chapter 6. Nonlinear equations and roots of polynomials 135
6.1. Solving a nonlinear equation 135
6.1.1. Bisection method 136
6.1.2. Regula falsi method 137
6.1.3. Secant method 139
6.1.4. Newton’s method 140
6.1.5. An example realization of an effective algorithm 142
6.2. Systems of nonlinear equations 143
6.2.1. Newton’s method 145
6.2.2. Broyden’s method 146
6.2.3. Fix point method 147
6.3. Roots of polynomials 148
6.3.1. Müller’s method 148
6.3.2. Laguerre’s method 150
6.3.3. Deflation by a linear term 151
6.3.4. Root polishing 152
6.3.5. Bairstow’s algorithm 152
Chapter 7. Ordinary differential equations 157
7.1. Single-step methods 163
7.1.1. Runge-Kutta (RK) methods 165
7.1.2. Runge-Kutta-Fehlberg (RKF) methods 170
7.1.3. Correction of the step size 172
7.2. Multistep methods 175
7.2.1. Adams methods 175
7.2.2. The approximation error 177
7.2.3. Stability and convergence 180
7.2.4. Predictor-corrector methods 183
7.2.5. Predictor-corrector methods with a variable step size 186
7.3. Stiff systems of differential equations 192
Chapter 8. Numerical differentiation and integration 199
8.1. Numerical approximation of derivatives 199
8.2. Numerical integration 206
Bibliography 217
- Title:Numerical Methods
- Author:Piotr Tatjewski
- ISBN:978-83-8156-883-8, 9788381568838
- Date of issue:2026-09-04
- Format:Ebook - PDF
- Item ID: e_53ey
- Publisher: Oficyna Wydawnicza Politechniki Warszawskiej