Approximation Techniques for Engineers: Second Edition, 2nd Edition (Hardback) book cover

Approximation Techniques for Engineers

Second Edition, 2nd Edition

By Louis Komzsik

CRC Press

366 pages

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This second edition includes eleven new sections based on the approximation of matrix functions, deflating the solution space and improving the accuracy of approximate solutions, iterative solution of initial value problems of systems of ordinary differential equations, and the method of trial functions for boundary value problems. The topics of the two new chapters are integral equations and mathematical optimization. The book provides alternative solutions to software tools amenable to hand computations to validate the results obtained by "black box" solvers. It also offers an insight into the mathematics behind many CAD, CAE tools of the industry. The book aims to provide a working knowledge of the various approximation techniques for engineering practice.


"Dr. Komzsik's book is a rare combination of the basic knowledge in the areas of approximation and computational mechanics which is essential for an engineer. Exhaustive coverage of all the topics along with lucid presentation of the concepts and a wide variety of illustrative problems make this book a valuable source for advanced users. Logical sequence of the topics and nomenclature even allows beginners to cope with the complex field of approximation methods in science and technology. This book contains a large amount of information not found in standard textbooks. Written for the engineer involved in complex issues, it combines the modern mathematical standards of numerical analysis with an understanding on how to effectively use these abstract methods.

The new edition features integral equations and mathematical optimization methods toward their practical usage in the focus of finite and boundary element methods in solitary and coupled formulation as well."

—Thomas Flöck, R&D Engineer, Siemens Digital Factory/Motion Control, Erlangen, Germany

"The first edition of Louis Komzsik' s book was an extremely valuable source of numerical routines for the last decade pushing the envelope in automotive applications. We awaited this second edition book for years and looking forward utilizing it in our work.

The BMW i8 model in the picture on the front depicts the various stages of approximations during the development of a car, ranging from sketches through finite element models to the final geometry. In this process engineers use many of the methods described in this book."

—Daniel Heiserer, BMW, Munich, Germany

"This book features an approachable blend of theoretical motivation with illustrative examples. Many of the methods described lend themselves to a matrix formulation, making them suitable for implementation using MATLAB®. The second edition is further improved by the inclusion of solution methods for integral equations and modern optimization applications. Almost any scientist or engineer will be able to find a topic of interest in this volume."

—Leonard Hoffnung, MathWorks Inc., USA

"This book collects a wide variety of numerical approximation techniques for easy reference, with detailed examples. There is, happily, sufficient exposition for the mathematically mature reader to use "Approximation Techniques" as an introductory text. Notation is clear and consistent, and the style is elegant yet comfortable; proof computations are included when they provide valuable context and intuition, and left out when the details are too cumbersome. The new chapters describe techniques which have applications as diverse as operations research and mathematical physics, but the new edition continues to provide ample engineering context and illustration from the perspective of Dr. Komzsik's extensive experience in the field."

—Gillian Grindstaff, The University of Texas at Austin, USA

"For the electronics engineer who wants to better understand numerical analysis, this book is a good fit, especially if you want to improve your matrix math ability. It has an intermediate amount of detail and the most important methods that will be found and used in power electronics, and for this reason, it can be a good book to acquire."

Dennis Feucht, How2Power Newsletter, September 2017

"The reader can find many examples of numerical analyses of technical problems. This book is especially useful in those situations when engineers need a quantitative estimation of the phenomenon under study. The book may also be useful to graduate engineering students and to all who are interested in applied numerical calculations."

Zentralblatt MATH, 2017

Table of Contents

I Data approximations

1 Classical interpolation methods

1.1 Newton interpolation

1.2 Lagrange interpolation

1.3 Hermite interpolation

1.3.1 Computational example

1.4 Interpolation of functions of two variables with polynomials


2 Approximation with splines

2.1 Natural cubic splines

2.2 Bezier splines

2.3 Approximation with B-splines

2.4 Surface spline approximation


3 Least squares approximation

3.1 The least squares principle

3.2 Linear least squares approximation

3.3 Polynomial least squares approximation

3.4 Computational example

3.5 Exponential and logarithmic least squares approximations

3.6 Nonlinear least squares approximation

3.7 Trigonometric least squares approximation

3.8 Directional least squares approximation

3.9 Weighted least squares approximation


4 Approximation of functions

4.1 Least squares approximation of functions

4.2 Approximation with Legendre polynomials

4.3 Chebyshev approximation

4.4 Fourier approximation

4.5 Pad´e approximation

4.6 Approximating matrix functions


5 Numerical differentiation

5.1 Finite difference formulae

5.2 Higher order derivatives

5.3 Richardson’s extrapolation

5.4 Multipoint finite difference formulae


6 Numerical integration

6.1 The Newton-Cotes class

6.2 Advanced Newton-Cotes methods

6.3 Gaussian quadrature

6.4 Integration of functions of multiple variables

6.5 Chebyshev quadrature

6.6 Numerical integration of periodic functions


II Approximate solutions

7 Nonlinear equations in one variable

7.1 General equations

7.2 Newton’s method

7.3 Solution of algebraic equations

7.4 Aitken’s acceleration


8 Systems of nonlinear equations

8.1 The generalized fixed point method

8.2 The method of steepest descent

8.3 The generalization of Newton’s method

8.4 Quasi-Newton method

8.5 Nonlinear static analysis application


9 Iterative solution of linear systems

9.1 Iterative solution of linear systems

9.2 Splitting methods

9.3 Ritz-Galerkin method

9.4 Conjugate gradient method

9.5 Preconditioning techniques

9.6 Biconjugate gradient method

9.7 Least squares systems

9.8 The minimum residual approach

9.9 Algebraic multigrid method

9.10 Linear static analysis application


10 Approximate solution of eigenvalue problems

10.1 Classical iterations

10.2 The Rayleigh-Ritz procedure

10.3 The Lanczos method

10.4 The solution of the tridiagonal eigenvalue problem

10.5 The biorthogonal Lanczos method

10.6 The Arnoldi method

10.7 The block Lanczos method

10.7.1 Preconditioned block Lanczos method

10.8 Normal modes analysis application


11 Initial value problems

11.1 Solution of initial value problems

11.2 Single-step methods

11.3 Multistep methods

11.4 Initial value problems of systems of ordinary differential equations

11.5 Initial value problems of higher order ordinary differential equations

11.6 Linearization of second order initial value problems

11.7 Transient response analysis application


12 Boundary value problems

12.1 Boundary value problems of ordinary differential equations

12.2 The finite difference method for boundary value problems of

ordinary differential equations

12.3 Boundary value problems of partial differential equations

12.4 The finite difference method for boundary value problems of

partial differential equations

12.5 The finite element method

12.6 Finite element analysis of three-dimensional continuum

12.7 Fluid-structure interaction application


13 Integral equations

13.1 Converting initial value problems to integral equations

13.2 Converting boundary value problems to integral equations

13.3 Classification of integral equations

13.4 Fredholm solution

13.5 Neumann approximation

13.6 Nystrom method

13.7 Nonlinear integral equations

13.8 Integro-differential equations

13.8.1 Computational example

13.9 Boundary integral method application


14 Mathematical optimization

14.1 Existence of solution

14.2 Penalty method

14.3 Quadratic optimization

14.4 Gradient based methods

14.5 Global optimization

14.6 Topology optimization

14.7 Structural compliance application


List of figures

List of tables



Closing remarks

About the Author


Dr. Komzsik is a graduate of the Technical University and the Eötvös University of Sciences, both in Budapest, Hungary. He worked for the Hungarian Shipyards in Budapest during the 1970s as an engineering analyst. After immigrating to the U.S., he worked for the McDonnell-Douglas Corporation between 1981-1982 as a senior analyst. Following that, he spent two decades as Chief Numerical Analyst at the MacNeal-Schwendler (now MSC Software) Corporation. After another decade and a half, he recently retired as Principal Key Expert from Siemens PLM Software.

He is the author of the original NASTRAN Numerical Methods Handbook, and a widely read book on The Lanczos Method that was also published in Chinese, Japanese, and Hungarian. His books titled "Computational Techniques of Finite Element Analysis" and "Applied Calculus of Variations for Engineers and Approximation Techniques for Engineers" are already in their second editions. He is also the co-author of the book, "Rotor Dynamics with Finite Elements."

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