Numerical Methods

The presented text is an English version of the original Polish textbook Metody numeryczne, published by OWPW — Oficyna Wydawnicza Politechniki Warszawskiej, the Warsaw University of Technology Publishing House. The Polish editions appeared in 2013 and 2026, while the English edition was first published in 2014. The present second English edition differs only slightly from the first one: a few errors have been corrected and minor improvements have been made to the text and figures.

The original textbook was written on the basis of the author’s long-standing teaching experience in numerical methods at the Faculty of Electronics and Information Technology of the Warsaw University of Technology. It was prepared mainly for the course Numerical Methods, taught in Polish to ICT students and in English to students of Electrical and Computer Engineering. The textbook summarizes and expands a long series of lecture notes previously prepared and made available to students electronically on the course websites.

The author expresses his gratitude to all students who actively participated in the courses, especially for their comments on the presentation of the material and for pointing out editorial errors. These remarks helped improve the quality of the 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

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