This well-respected text introduces the theory and application of modern numerical approximation techniques to students taking a one- or two-semester course in numerical analysis. Providing an accessible treatment that only requires a calculus prerequisite, the authors explain how, why, and when approximation techniques can be expected to work-and why, in some situations, they fail. A wealth of examples and exercises develop students' intuition, and demonstrate the subject's practical applications to important everyday problems in math, computing, engineering, and physical science disciplines. The first book of its kind when crafted more than 30 years ago to serve a diverse undergraduate audience, Burden, Faires, and Burden's NUMERICAL ANALYSIS remains the definitive introduction to a vital and practical subject.
1. MATHEMATICAL PRELIMINARIES AND ERROR ANALYSIS.
Review of Calculus. Round-off Errors and Computer Arithmetic. Algorithms and Convergence. Numerical Software and Chapter Summary.
2. SOLUTIONS OF EQUATIONS IN ONE VARIABLE.
The Bisection Method. Fixed-Point Iteration. Newton's Method and Its Extensions. Error Analysis for Iterative Methods. Accelerating Convergence. Zeros of Polynomials and Müller's Method. Numerical Software and Chapter Summary.
3. INTERPOLATION AND POLYNOMIAL APPROXIMATION.
Interpolation and the Lagrange Polynomial. Data Approximation and Neville's Method. Divided Differences. Hermite Interpolation. Cubic Spline Interpolation. Parametric Curves. Numerical Software and Chapter Summary.
4. NUMERICAL DIFFERENTIATION AND INTEGRATION.
Numerical Differentiation. Richardson's Extrapolation. Elements of Numerical Integration. Composite Numerical Integration. Romberg Integration. Adaptive Quadrature Methods. Gaussian Quadrature. Multiple Integrals. Improper Integrals. Numerical Software and Chapter Summary.
5. INITIAL-VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS.
The Elementary Theory of Initial-Value Problems. Euler's Method. Higher-Order Taylor Methods. Runge-Kutta Methods. Error Control and the Runge-Kutta-Fehlberg Method. Multistep Methods. Variable Step-Size Multistep Methods. Extrapolation Methods. Higher-Order Equations and Systems of Differential Equations. Stability. Stiff Differential Equations. Numerical Software and Chapter Summary.
6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS.
Linear Systems of Equations. Pivoting Strategies. Linear Algebra and Matrix Inversion. The Determinant of a Matrix. Matrix Factorization. Special Types of Matrices. Numerical Software and Chapter Summary.
7. ITERATIVE TECHNIQUES IN MATRIX ALGEBRA.
Norms of Vectors and Matrices. Eigenvalues and Eigenvectors. The Jacobi and Gauss-Siedel Iterative Techniques. Relaxation Techniques for Solving Linear Systems. Error Bounds and Iterative Refinement. The Conjugate Gradient Method. Numerical Software and Chapter Summary.
8. APPROXIMATION THEORY.
Discrete Least Squares Approximation. Orthogonal Polynomials and Least Squares Approximation. Chebyshev Polynomials and Economization of Power Series. Rational Function Approximation. Trigonometric Polynomial Approximation. Fast Fourier Transforms. Numerical Software and Chapter Summary.
9. APPROXIMATING EIGENVALUES.
Linear Algebra and Eigenvalues. Orthogonal Matrices and Similarity Transformations. The Power Method. Householder's Method. The QR Algorithm. Singular Value Decomposition. Numerical Software and Chapter Summary.
10. NUMERICAL SOLUTIONS OF NONLINEAR SYSTEMS OF EQUATIONS.
Fixed Points for Functions of Several Variables. Newton's Method. Quasi-Newton Methods. Steepest Descent Techniques. Homotopy and Continuation Methods. Numerical Software and Chapter Summary.
11. BOUNDARY-VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS.
The Linear Shooting Method. The Shooting Method for Nonlinear Problems. Finite-Difference Methods for Linear Problems. Finite-Difference Methods for Nonlinear Problems. The Rayleigh-Ritz Method. Numerical Software and Chapter Summary.
12. NUMERICAL SOLUTIONS TO PARTIAL DIFFERENTIAL EQUATIONS.
Elliptic Partial Differential Equations. Parabolic Partial Differential Equations. Hyperbolic Partial Differential Equations. An Introduction to the Finite-Element Method. Numerical Software and Chapter Summary.
Bibliography.
Answers to Selected Exercises.
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Richard L. Burden
Richard L. Burden is Emeritus Professor of Mathematics at Youngstown State University. His master's degree in mathematics and doctoral degree in mathematics, with a specialization in numerical analysis, were both awarded by Case Western Reserve University. He also earned a masters degree in computer science from the University of Pittsburgh. His mathematical interests include numerical analysis, numerical linear algebra, and mathematical statistics. Dr. Burden has been named a distinguished professor for teaching and service three times at Youngstown State University. He was also named a distinguished chair as the chair of the Department of Mathematical and Computer Sciences. He wrote the Actuarial Examinations in Numerical Analysis from 1990 until 1999.
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J. Douglas Faires
J. Douglas Faires, late of Youngstown State University, pursued mathematical interests in analysis, numerical analysis, mathematics history, and problem solving. Dr. Faires won numerous awards, including the Outstanding College-University Teacher of Mathematics by the Ohio Section of MAA and five Distinguished Faculty awards from Youngstown State University, which also awarded him an Honorary Doctor of Science award in 2006.
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Annette M. Burden
Annette M. Burden is a Professor of Mathematics at Youngstown State University (YSU) and for four years served as YSU Interim Distance Education Director. Her master's degree in mathematics was awarded by Youngstown State University and her doctoral degree in mathematics educational technology with a specialization in numerical analysis was awarded by Union Institute & University. Dr. Burden worked under Carnegie Mellon Professor Werner C. Rheinboldt from the University of Pittsburgh for several years. She is past President of the International Society of Technology in Education's Technology Coordinators, was appointed to the MAPLE Academic Advisory Board, and served as co-chair of Ohio's Distance Education Advisory Group. She has also developed numerous upper-level online courses including courses in Numerical Analysis and Numerical Methods. Dr. Burden has been named a distinguished professor for teaching and service three times at Youngstown State University.
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The algorithms in the text are designed to work with a wide variety of software packages and programming languages, allowing maximum flexibility for users to harness computing power to perform approximations. The book's companion website offers Maple, Mathematica, and MATLAB worksheets, as well as C, FORTRAN, Java, and Pascal programs.
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The design of the text gives instructors flexibility in choosing topics they wish to cover, selecting the level of theoretical rigor desired, and deciding which applications are most appropriate or interesting for their classes.
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Virtually every concept in the text is illustrated by examples. In addition, concepts and examples are reinforced by more than 2500 class-tested exercises ranging from elementary applications of methods and algorithms to generalizations and extensions of the theory.
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The exercise sets include many applied problems from diverse areas of engineering, as well as from the physical, computer, biological, and social sciences.
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The algorithms in the text are designed to work with a wide variety of software packages and programming languages, allowing maximum flexibility for users to harness computing power to perform approximations. The book's companion website offers Maple, Mathematica, and MATLAB worksheets, as well as C, FORTRAN, Java, and Pascal programs.
-
The design of the text gives instructors flexibility in choosing topics they wish to cover, selecting the level of theoretical rigor desired, and deciding which applications are most appropriate or interesting for their classes.
-
Virtually every concept in the text is illustrated by examples. In addition, concepts and examples are reinforced by more than 2500 class-tested exercises ranging from elementary applications of methods and algorithms to generalizations and extensions of the theory.
-
The exercise sets include many applied problems from diverse areas of engineering, as well as from the physical, computer, biological, and social sciences.