• Notation
• Scientific computing
• Systems of linear equations
• Linear least squares
• Eigenvalue problems
• Nonlinear equations
• Optimization
• Interpolation
• Numerical integration and differentiation
• Initial value problems for ODEs
• Boundary value problems for ODEs
• Partial differential equations
• Fast fourier transform
• Random numbers and simulations

This book differs from traditional numerical analysis texts in that it focuses on the motivation and ideas behind the algorithms presented rather than on detailed analyses of them. It presents a broad overview of methods and software for solving mathematical problems arising in computational modeling and data analysis, including proper problem formulation, selection of effective solution algorithms, and interpretation of results. In the 20 years since its original publication, the modern, fundamental perspective of this book has aged well, and it continues to be used in the classroom. This Classics edition has been updated to include pointers to Python software and the Chebfun package, expansions on barycentric formulation for Lagrange polynomial interpretation and stochastic methods, and the availability of about 100 interactive educational modules that dynamically illustrate the concepts and algorithms in the book. Scientific Computing: An Introductory Survey, Second Edition is intended as both a textbook and a reference for computationally oriented disciplines that need to solve mathematical problems.
(source: Nielsen Book Data)

• 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 Muller'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.
• (source: Nielsen Book Data)

This well-respected book introduces readers to the theory and application of modern numerical approximation techniques. 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 readers' intuition, and demonstrate the subject's practical applications to important everyday problems in math, computing, engineering, and physical science disciplines. Three decades after it was first published, Burden, Faires, and Burden's NUMERICAL ANALYSIS remains the definitive introduction to a vital and practical subject.
(source: Nielsen Book Data)

• Numerical algorithms
• Roundoff errors
• Nonlinear equations in one variable
• Linear algebra background
• Linear systems : direct methods
• Linear least squares problems
• Linear systems : iterative methods
• Eigenvalues and singular values
• Nonlinear systems and optimization
• Polynomial interpolation
• Piecewise polynomial interpolation
• Best approximation
• Fourier transform
• Numerical differentiation
• Numerical integration
• Differential equations.

• Numerical algorithms
• Roundoff errors
• Nonlinear equations in one variable
• Linear algebra background
• Linear systems: direct methods
• Linear least squares problems
• Linear systems : iterative methods
• Eigenvalues and singular values
• Nonlinear systems and optimization
• Polynomial interpolation
• Piecewise polynomial interpolation
• Best approximation
• Fourier Transform
• Numerical differentiation
• Numerical integration
• Differential equations.

This book is designed for students and researchers who seek practical knowledge of modern techniques in scientific computing. Avoiding encyclopedic and heavily theoretical exposition, the book provides an in-depth treatment of fundamental issues and methods, the reasons behind the success and failure of numerical software, and fresh and easy-to-follow approaches and techniques. The authors focus on current methods, issues, and software while providing a comprehensive theoretical foundation, enabling those who need to apply the techniques to successfully design solutions to nonstandard problems. The book also illustrates algorithms using the programming environment of MATLAB℗®, with the expectation that the reader will gradually become proficient in it while learning the material covered in the book. A variety of exercises are provided within each chapter along with review questions aimed at self-testing. The book takes an algorithmic approach, focusing on techniques that have a high level of applicability to engineering, computer science, and industrial mathematics.

• (NOTE: Each chapter begins with An Overview.)
• 1. Getting Started. Algorithms. Convergence. Floating Point Numbers. Floating Point Arithmetic.
• 2. Rootfinding. Bisection Method. Method of False Position. Fixed Point Iteration. Newton's Method. The Secant Method and Muller's Method. Accelerating Convergence. Roots of Polynomials.
• 3. Systems of Equations. Gaussian Elimination. Pivoting Strategies. Norms. Error Estimates. LU Decomposition. Direct Factorization. Special Matrices. Iterative Techniques for Linear Systems: Basic Concepts and Methods. Iterative Techniques for Linear Systems: Conjugate-Gradient Method. Nonlinear Systems.
• 4. Eigenvalues and Eigenvectors. The Power Method. The Inverse Power Method. Deflation. Reduction to Tridiagonal Form. Eigenvalues of Tridiagonal and Hessenberg Matrices.
• 5. Interpolation and Curve Fitting. Lagrange Form of the Interpolating Polynomial. Neville's Algorithm. The Newton Form of the Interpolating Polynomial and Divided Differences. Optimal Interpolating Points. Piecewise Linear Interpolation. Hermite and Hermite Cubic Interpolation. Regression.
• 6. Numerical Differentiation and Integration. Continuous Theory and Key Numerical Concepts. Euler's Method. Higher-Order One-Step Methods. Multistep Methods. Convergence Analysis. Error Control and Variable Step Size Algorithms. Systems of Equations and Higher-Order Equations. Absolute Stability and Stiff Equations.
• 7. Numerical Methods for Initial Value Problems of Ordinary Differential Equations. Continuous Theory and Key Numerical Concepts. Euler's Method. Higher-Order One-Step Methods. Multistep Methods. Convergence Analysis. Error Control and Variable Step Size Algorithms. Systems of Equations and Higher-Order Equations. Absolute Stability and Stiff Equations.
• 8. Second-Order One-Dimensional Two-Point Boundary Value Problems. Finite Difference Method, Part I: The Linear Problem with Dirichlet Boundary Conditions. Finite Difference Method, Part II: The Linear Problem with Non-Dirichlet Boundary Conditions. Finite Difference Method, Part III: Nonlinear Problems. The Shooting Method, Part I: Linear Boundary Value Problems. The Shooting Method, Part II: Nonlinear Boundary Value Problems.
• 9. Finite Difference Method for Elliptic Partial Differential Equations. The Poisson Equation on a Rectangular Domain, I: Dirichlet Boundary Conditions. The Poisson Equation on a Rectangular Domain, II: Non-Dirichlet Boundary Conditions. Solving the Discrete Equations: Relaxation Schemes. Local Mode Analysis of Relaxation and the Multigrid Method. Irregular Domains.
• 10. Finite Difference Method for Parabolic Partial Differential Equations. The Heat Equation with Dirichlet Boundary Conditions. Stability. More General Parabolic Equations. Non-Dirichlet Boundary Conditions. Polar Coordinates. Problems in Two Space Dimensions.
• 11. Finite Difference Method for Hyperbolic Partial Differential Equations and the Convection-Diffusion Equation. Advection Equation, I: Upwind Differencing. Advection Equation, II: MacCormack Method. Convection-Diffusion Equation. The Wave Equation. Appendices. Appendix A. Important Theorems from Calculus. Appendix B. Algorithm for Solving a Tridiagonal System of Linear Equations. References. Index. Answers to Selected Problems.
• (source: Nielsen Book Data)

For one or two-semester undergraduate/graduate-level courses in Numerical Analysis/Methods in mathematics departments, CS departments, and all engineering departments. This student-friendly text develops concepts and techniques in a clear, concise, easy-to-read manner, followed by fully-worked examples. Application problems drawn from the literature of many different fields prepares students to use the techniques covered to solve a wide variety of practical problems.
(source: Nielsen Book Data)

• 1 Scientific Computing2 Systems of Linear Equations3 Linear Least Squares4 Eigenvalues Problems5 Nonlinear Equations6 Optimization7 Interpolation8 Numerical Integration and Differentiation9 Initial Value Problems for ODEs10 Boundary Value Problems for ODEs11 Partial Differential Equations12 Fast Fourier Transform13 Random Numbers and Simulation.
• (source: Nielsen Book Data)

"Scientific Computing, 2/e", presents a broad overview of numerical methods for solving all the major problems in scientific computing, including linear and nonlinear equations, least squares, eigenvalues, optimization, interpolation, integration, ordinary and partial differential equations, fast Fourier transforms, and random number generators. The treatment is comprehensive yet concise, software-oriented yet compatible with a variety of software packages and programming languages. The book features more than 160 examples, 500 review questions, 240 exercises, and 200 computer problems. Changes for the second edition include: expanded motivational discussions and examples; formal statements of all major algorithms; expanded discussions of existence, uniqueness, and conditioning for each type of problem so that students can recognize 'good' and 'bad' problem formulations and understand the corresponding quality of results produced; and, expanded coverage of several topics, particularly eigenvalues and constrained optimization. The book contains a wealth of material and can be used in a variety of one- or two-term courses in computer science, mathematics, or engineering. Its comprehensiveness and modern perspective, as well as the software pointers provided, also make it a highly useful reference for practicing professionals who need to solve computational problems.
(source: Nielsen Book Data)