Solutions Manual to accompany An Introduction to Numerical Methods and Analysis

Description

A solutions manual to accompany An Introduction to Numerical Methods and Analysis, Third Edition An Introduction to Numerical Methods and Analysis helps students gain a solid understanding of a wide range of numerical approximation methods for solving problems of mathematical analysis. Designed for entry-level courses on the subject, this popular textbook maximizes teaching flexibility by first covering basic topics before gradually moving to more advanced material in each chapter and section. Throughout the text, students are provided clear and accessible guidance on a wide range of numerical methods and analysis techniques, including root-finding, numerical integration, interpolation, solution of systems of equations, and many others. This fully revised third edition contains new sections on higher-order difference methods, the bisection and inertia method for computing eigenvalues of a symmetric matrix, a completely re-written section on different methods for Poisson equations, and spectral methods for higher-dimensional problems. New problem sets—ranging in difficulty from simple computations to challenging derivations and proofs—are complemented by computer programming exercises, illustrative examples, and sample code. This acclaimed textbook: Explains how to both construct and evaluate approximations for accuracy and performance Covers both elementary concepts and tools and higher-level methods and solutions Features new and updated material reflecting new trends and applications in the field Contains an introduction to key concepts, a calculus review, an updated primer on computer arithmetic, a brief history of scientific computing, a survey of computer languages and software, and a revised literature review Includes an appendix of proofs of selected theorems and author-hosted companion website with additional exercises, application models, and supplemental resources Cover Title Page Copyright Preface to the Solutions Manual for the Third Edition CHAPTER 1: INTRODUCTORY CONCEPTS AND CALCULUS REVIEW 1.1 BASIC TOOLS OF CALCULUS 1.2 ERROR, APPROXIMATE EQUALITY, AND ASYMPTOTIC ORDER NOTATION 1.3 A PRIMER ON COMPUTER ARITHMETIC 1.4 A WORD ON COMPUTER LANGUAGES AND SOFTWARE 1.5 A BRIEF HISTORY OF SCIENTIFIC COMPUTING Chapter 2: A SURVEY OF SIMPLE METHODS AND TOOLS 2.1 HORNER'S RULE AND NESTED MULTIPLICATION 2.2 DIFFERENCE APPROXIMATIONS TO THE DERIVATIVE 2.3 APPLICATION: EULER'S METHOD FOR INITIAL VALUE PROBLEMS 2.4 LINEAR INTERPOLATION 2.5 APPLICATION — THE TRAPEZOID RULE 2.6 SOLUTION OF TRIDIAGONAL LINEAR SYSTEMS 2.7 APPLICATION: SIMPLE TWO‐POINT BOUNDARY VALUE PROBLEMS CHAPTER 3: ROOT‐FINDING 3.1 THE BISECTION METHOD 3.2 NEWTON'S METHOD: DERIVATION AND EXAMPLES 3.3 HOW TO STOP NEWTON'S METHOD 3.4 APPLICATION: DIVISION USING NEWTON'S METHOD 3.5 THE NEWTON ERROR FORMULA 3.6 NEWTON'S METHOD: THEORY AND CONVERGENCE 3.7 APPLICATION: COMPUTATION OF THE SQUARE ROOT 3.8 THE SECANT METHOD: DERIVATION AND EXAMPLES 3.9 FIXED POINT ITERATION 3.10 ROOTS OF POLYNOMIALS (PART 1) 3.11 SPECIAL TOPICS IN ROOT‐FINDING METHODS 3.12 VERY HIGH‐ORDER METHODS AND THE EFFICIENCY INDEX NOTES CHAPTER 4: INTERPOLATION AND APPROXIMATION 4.1 LAGRANGE INTERPOLATION 4.2 NEWTON INTERPOLATION AND DIVIDED DIFFERENCES 4.3 INTERPOLATION ERROR 4.4 APPLICATION: MULLER'S METHOD AND INVERSE QUADRATIC INTERPOLATION 4.5 APPLICATION: MORE APPROXIMATIONS TO THE DERIVATIVE 4.6 HERMITE INTERPOLATION 4.7 PIECEWISE POLYNOMIAL INTERPOLATION 4.8 AN INTRODUCTION TO SPLINES 4.9 TENSION SPLINES 4.10 LEAST SQUARES CONCEPTS IN APPROXIMATION 4.11 ADVANCED TOPICS IN INTERPOLATION ERROR NOTES CHAPTER 5: NUMERICAL INTEGRATION 5.1 A REVIEW OF THE DEFINITE INTEGRAL 5.2 IMPROVING THE TRAPEZOID RULE 5.3 SIMPSON'S RULE AND DEGREE OF PRECISION 5.4 THE MIDPOINT RULE 5.5 APPLICATION: STIRLING'S FORMULA 5.6 GAUSSIAN QUADRATURE 5.7 EXTRAPOLATION METHODS 5.8 SPECIAL TOPICS IN NUMERICAL INTEGRATION CHAPTER 6: NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 6.1 The Initial Value Problem—Background 6.2 Euler's Method 6.3 Analysis of Euler's Method 6.4 Variants of Euler's Method 6.5 Single Step Methods—Runge‐Kutta 6.6 Multistep Methods 6.7 Stability Issues 6.8 Application to Systems of Equations 6.9 Adaptive Solvers 6.10 Boundary Value Problems NOTE CHAPTER 7: NUMERICAL METHODS FOR THE SOLUTION OF SYSTEMS OF EQUATIONS 7.1 LINEAR ALGEBRA REVIEW 7.2 LINEAR SYSTEMS AND GAUSSIAN ELIMINATION 7.3 OPERATION COUNTS 7.4 THE FACTORIZATION 7.5 PERTURBATION, CONDITIONING AND STABILITY 7.6 SPD MATRICES AND THE CHOLESKY DECOMPOSITION 7.7 APPLICATION: NUMERICAL SOLUTION OF LINEAR LEAST SQUARES PROBLEMS 7.8 SPARSE AND STRUCTURED MATRICES 7.9 ITERATIVE METHODS FOR LINEAR SYSTEMS – A BRIEF SURVEY 7.10 NONLINEAR SYSTEMS: NEWTON'S METHOD AND RELATED IDEAS 7.11 APPLICATION: NUMERICAL SOLUTION OF NONLINEAR BVP's CHAPTER 8: APPROXIMATE SOLUTION OF THE ALGEBRAIC EIGENVALUE PROBLEM 8.1 EIGENVALUE REVIEW 8.2 REDUCTION TO HESSENBERG FORM 8.3 POWER METHODS 8.4 BISECTION AND INERTIA TO COMPUTE EIGENVALUES OF SYMMETRIC MATRICES 8.5 AN OVERVIEW OF THE ITERATION 8.6 APPLICATION: ROOTS OF POLYNOMIALS, II 8.7 APPLICATION: COMPUTATION OF GAUSSIAN QUADRATURE RULES NOTE CHAPTER 9: A SURVEY OF NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS 9.1 DIFFERENCE METHODS FOR THE DIFFUSION EQUATION 9.2 FINITE ELEMENT METHODS FOR THE DIFFUSION EQUATION 9.3 DIFFERENCE METHODS FOR POISSON EQUATIONS NOTE CHAPTER 10: AN INTRODUCTION TO SPECTRAL METHODS 10.1 SPECTRAL METHODS FOR TWO‐POINT BOUNDARY VALUE PROBLEMS 10.2 SPECTRAL METHODS IN TWO DIMENSIONS 10.3 SPECTRAL METHODS FOR TIME‐DEPENDENT PROBLEMS 10.4 CLENSHAW‐CURTIS QUADRATURE End User License Agreement