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Numerical Methods Using MathCAD / Edition 1

Numerical Methods Using MathCAD / Edition 1

by Laurene V. Fausett


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Numerical Methods Using MathCAD / Edition 1

This book presents the fundamental numerical techniques used in engineering, applied mathematics, computer science, and the physical and life sciences in a way that is both interesting and understandable. Using a wide range of examples and problems, this book focuses on the use of MathCAD functions and worksheets to illustrate the methods used when discussing the following concepts: solving linear and nonlinear equations, numerical linear algebra, numerical methods for data interpolation and approximation, numerical differentiation and integration, and numerical techniques for solving differential equations. For professionals in the fields of engineering, mathematics, computer science, and physical or life sciences who want to learn MathCAD functions for all major numerical methods.

Product Details

ISBN-13: 2900130610811
Publisher: Pearson
Publication date: 07/28/2001
Edition description: New Edition
Pages: 702
Product dimensions: 6.00(w) x 1.25(h) x 9.00(d)

Table of Contents

Examples/Mathcad Functions/AlgorithmsXIII
1.1Sample Problems and Numerical Methods4
1.1.1Roots of Nonlinear Equations4
1.1.2Fixed-Point Iteration5
1.1.3Linear Systems6
1.1.4Gaussian Elimination7
1.1.5Numerical Integration8
1.1.6Trapezoid Rule8
1.2Some Basic Issues10
1.2.1Key Issues for Iterative Methods10
1.2.2How Good Is the Result?14
1.2.3Getting Better Results22
1.3Getting Started in Mathcad28
1.3.1Overview of the Mathcad Workspace28
1.3.2Mathematical Computations31
1.3.3Operators on the Math Toolbars32
1.3.4Built-In Functions35
1.3.5Programming in Mathcad38
2Solving Equations of One Variable47
2.1Bisection Method50
2.1.1Step-by-Step Computation50
2.1.2Mathcad Function for Bisection52
2.2Regula Falsi and Secant Methods55
2.2.1Step-by-Step Computation for Regula Falsi56
2.2.2Mathcad Function for the Regula Falsi Method58
2.2.3Step-by-Step Computation for the Secant Method60
2.2.4Mathcad Function for the Secant Method62
2.3Newton's Method68
2.3.1Step-by-Step Computation68
2.3.2Mathcad Function for Newton's Method70
2.4Muller's Method75
2.4.1Step-by-Step Computation for Muller's Method76
2.4.2Mathcad Function for Muller's Method78
2.5Mathcad's Methods81
2.5.1Using the Built-In Functions81
2.5.2Understanding the Algorithms84
3Solving Systems of Linear Equations: Direct Methods93
3.1Gaussian Elimination96
3.1.1Using Matrix Notation97
3.1.2Step-by-Step Procedure98
3.1.3Mathcad Function for Basic Gaussian Elimination101
3.2Gaussian Elimination with Row Pivoting106
3.2.1Step-by-Step Computation106
3.2.2Mathcad Function for Gaussian Elimination with Pivoting110
3.3Gaussian Elimination for Tridiagonal Systems113
3.3.1Step-by-Step Procedure116
3.3.2Mathcad Function for the Thomas Method118
3.4Mathcad's Methods122
3.4.1Using the Built-In Functions122
3.4.2Understanding the Algorithms122
4Solving Systems of Linear Equations: Iterative Methods131
4.1Jacobi Method135
4.1.1Step-by-Step Procedure for Jacobi Iteration136
4.1.2Mathcad Function for the Jacobi Method139
4.2Gauss-Seidel Method144
4.2.1Step-by-Step Computation for Gauss-Seidel Method145
4.2.2Mathcad Function for Gauss-Seidel Method148
4.3Successive Overrelaxation151
4.3.1Step-by-Step Computation of SOR152
4.3.2Mathcad Function for SOR154
4.4Mathcad's Methods157
4.4.1Using the Built-In Functions157
4.4.2Understanding the Algorithms159
5Systems of Equations and Inequalities171
5.1Newton's Method for Systems of Equations174
5.1.1Matrix-Vector Notation176
5.1.2Mathcad Function for Newton's Method177
5.2Fixed-Point Iteration for Nonlinear Systems181
5.2.1Step-by-Step Computation182
5.2.2Mathcad Function for Fixed-Point Iteration for Nonlinear Systems182
5.3Minimum of a Nonlinear Function187
5.3.1Step-by-Step Computation of Minimization by Gradient Descent187
5.3.2Mathcad Function for Minimization by Gradient Descent188
5.4Mathcad's Methods192
5.4.1Using the Built-In Functions192
5.4.2Understanding the Algorithms193
6LU Factorization201
6.1LU Factorization from Gaussian Elimination203
6.1.1A Step-by-Step Procedure for LU Factorization204
6.1.2Mathcad Function for LU Factorization Using Gaussian Elimination206
6.2LU Factorization of Tridiagonal Matrices207
6.2.1Step-by-Step LU Factorization of a Tridiagonal Matrix207
6.2.2Mathcad Function for LU Factorization of a Tridiagonal Matrix208
6.3LU Factorization with Pivoting209
6.3.1Step-by-Step Computation209
6.3.2Mathcad Function for LU Factorization with Row Pivoting210
6.4Direct LU Factorization215
6.4.1Direct LU Factorization of a General Matrix215
6.4.2LU Factorization of a Symmetric Matrix217
6.5Applications of LU Factorization219
6.5.2Solving a Tridiagonal System Using LU Factorization222
6.5.3Determinant of a Matrix224
6.5.4Inverse of a Matrix224
6.6Mathcad's Methods226
6.6.1Using the Built-In Functions226
6.6.2Understanding the Algorithms226
7Eigenvalues, Eigenvectors, and QR Factorization233
7.1Power Method236
7.1.1Basic Power Method237
7.1.2Inverse Power Method242
7.2QR Factorization248
7.2.1Householder Transformations248
7.2.2Givens Transformations257
7.2.3Basic QR Factorization261
7.3Finding Eigenvalues Using QR Factorization267
7.3.1Basic QR Eigenvalue Method267
7.3.2Better QR Eigenvalue Method268
7.4Mathcad's Methods270
7.4.1Using the Built-In Functions270
7.4.2Understanding the Algorithms271
8.1Polynomial Interpolation286
8.1.1Lagrange Interpolation Polynomials286
8.1.2Newton Interpolation Polynomials295
8.1.3Difficulties with Polynomial Interpolation306
8.2Hermite Interpolation310
8.3Rational Function Interpolation316
8.4Spline Interpolation320
8.4.1Piecewise Linear Interpolation321
8.4.2Piecewise Quadratic Interpolation322
8.4.3Piecewise Cubic Interpolation325
8.5Mathcad's Methods334
8.5.1Using the Built-In Functions334
8.5.2Understanding the Algorithms335
9Function Approximation349
9.1Least Squares Approximation352
9.1.1Linear Least-Squares Approximation352
9.1.2Quadratic Least-Squares Approximation359
9.1.3Cubic Least-Squares Approximation364
9.1.4Least-Squares Approximation for Other Functional Forms369
9.2Continuous Least-Squares Approximation373
9.2.1Continuous Least-Squares with Orthogonal Polynomials376
9.2.2Gram-Schmidt Process376
9.2.3Legendre Polynomials378
9.2.4Least-Squares Approximation with Legendre Polynomials379
9.3Function Approximation at a Point381
9.3.1Taylor Approximation381
9.3.2Pade Function approximation382
9.4Mathcad's Methods385
9.4.1Using the Built-in Functions385
9.4.2Understanding the Algorithms386
10Fourier Methods393
10.1Fourier Approximation and Interpolation396
10.2Fast Fourier Transforms for n = 2[superscript r]407
10.2.1Discrete Fourier Transform407
10.2.2Fast Fourier Transform408
10.3Fast Fourier Transforms for General n415
10.4Mathcad's Methods423
10.4.1Using the Built-In Functions423
10.4.2Understanding the Algorithms424
11Numerical Differentiation and Integration436
11.1.1First Derivatives436
11.1.2Higher Derivatives440
11.1.3Partial Derivatives441
11.1.4Richardson Extrapolation442
11.2Basic Numerical Integration445
11.2.1Trapezoid Rule446
11.2.2Simpson Rule448
11.2.3The Midpoint Formula450
11.2.4Other Newton-Cotes Open Formulas452
11.3Better Numerical Integration452
11.3.1Composite Trapezoid Rule453
11.3.2Composite Simpson's Rule455
11.3.3Extrapolation Methods for Quadrature458
11.4Gaussian Quadrature462
11.4.1Gaussian Quadrature on [-1,1]462
11.4.2Gaussian Quadrature on [a,b]464
11.5Mathcad's Methods468
11.5.1Using the Operators468
11.5.2Understanding the Algorithms469
12Ordinary Differential Equations: Initial-Value Problems477
12.1Taylor Methods479
12.1.1Euler's Method479
12.1.2Higher-Order Taylor Methods484
12.2Runge-Kutta Methods487
12.2.1Midpoint Method487
12.2.2Other Second-Order Runge-Kutta Methods492
12.2.3Third-Order Runge-Kutta Methods494
12.2.4Classic Runge-Kutta Method495
12.2.5Other Runge-Kutta Methods499
12.2.6Runge-Kutta-Fehlberg Methods501
12.3Multistep Methods502
12.3.1Adams-Bashforth Methods502
12.3.2Adams-Moulton Methods508
12.3.3Predictor-Corrector Methods509
12.5Mathcad's Methods517
12.5.1Using the Built-In Functions517
12.5.2Understanding the Algorithms520
13Systems of Ordinary Differential Equations529
13.1Higher-Order ODEs532
13.2Systems of Two First-Order ODE534
13.2.1Euler's Method for Solving Two ODE-IVPs534
13.2.2Midpoint Method for Solving Two ODE-IVPs537
13.3Systems of First-Order ODE-IVP541
13.3.1Euler's Method for Solving Systems of ODEs542
13.3.2Runge-Kutta Methods for Solving Systems of ODEs544
13.3.3Multistep Methods for Systems552
13.4Stiff ODE and Ill-Conditioned Problems557
13.5Mathcad's Methods559
13.5.1Using the Built-In Functions559
13.5.2Understanding the Algorithms562
14Ordinary Differential Equations--Boundary Value Problems575
14.1Shooting Method for Solving Linear BVP578
14.1.1Simple Boundary Conditions578
14.1.2General Boundary Condition at x = b583
14.1.3General Boundary Conditions at Both Ends of the Interval584
14.2Shooting Method for Solving Nonlinear BVP585
14.2.1Nonlinear Shooting Based on the Secant Method585
14.2.2Nonlinear Shooting Using Newton's Method588
14.3Finite-Difference Method for Solving Linear BVP592
14.4Finite-Difference Method for Nonlinear BVP599
14.5Mathcad's Methods602
14.5.1Using the Built-In Functions602
14.5.2Understanding the Algorithms604
15Partial Differential Equations609
15.1Classification of PDE613
15.2Heat Equation: Parabolic PDE614
15.2.1Explicit Method for Solving the Heat Equation615
15.2.2Implicit Method for Solving the Heat Equation623
15.2.3Crank-Nicolson Method for Solving the Heat Equation628
15.2.4Heat Equation with Insulated Boundary632
15.3Wave Equation: Hyperbolic PDE633
15.3.1Explicit Method for Solving Wave Equations634
15.3.2Implicit Method for Solving Wave Equation638
15.4Poisson Equation: Elliptic PDE640
15.5Finite-Element Method for Solving an Elliptic PDE645
15.6Mathcad's Methods658
15.6.1Using the Built-In Functions658
15.6.2Understanding the Algorithms659
Answers to Selected Problems673


The purpose of this text is to present the fundamental numerical techniques used in engineering, applied mathematics, computer science, and the physical and life sciences in a manner that is both interesting and understandable to undergraduate and beginning graduate students in those fields. The organization of the chapters, and of the material within each chapter, the use of Mathcad worksheets and functions to illustrate the methods, and the exercises provided are all designed with student learning as the primary objective.

The first chapter sets the stage for the material in the rest of the text, by giving a brief introduction to the long history of numerical techniques, and a "preview of coming attractions" for some of the recurring themes of the remainder of the text. It also presents enough description of Mathcad to allow students to use the Mathcad functions presented for each of the numerical methods discussed in the other chapters. An algorithmic statement of each method is also included; the algorithm may be used as the basis for computations using a variety of types of technological support, ranging from paper and pencil, to calculators, Mathcad worksheets or developing computer programs.

Each of the subsequent chapters begins with a one-page overview of the subject matter, together with an indication as to how the topics presented in the chapter are related to those in previous and subsequent chapters. Introductory examples are presented to suggest a few of the types of problems for which the topics of the chapter may be used. Following the sections in which the methods are presented, each chapter concludes with a summary of the most importantformulas, a selection of suggestions for further reading, and an extensive set of exercises. The first group of problems provide fairly routine practice of the techniques; the second group are applications adapted from a variety of fields, and the final group of problems encourage students to extend their understanding of either the theoretical or the computational aspects of the methods.

The presentation of each numerical technique is based on the successful teaching methodology of providing examples and geometric motivation for a method, and a concise statement of the steps to carry out the computation, before giving a mathematical derivation of the process or a discussion of the more theoretical issues that are relevant to the use and understanding of the topic. Each topic is illustrated by examples that range in complexity from very simple to moderate.

Geometrical or graphical illustrations are included whenever they are appropriate. A simple Mathcad function is presented for each method, which also serves as a clear step-by-step description of the process; discussion of theoretical considerations is placed at the conclusion of the section. The last section of each chapter gives a brief discussion of Mathcad's built-in functions for solving the kinds of problems covered in the chapter.

The chapters are arranged according to the following general areas:

  • Chapters 2-5 deal with solving linear and nonlinear equations.
  • Chapters 6 and 7 treat topics from numerical linear algebra.
  • Chapters 8-10 cover numerical methods for data interpolation and approximation.
  • Chapters 11 presents numerical differentiation and integration.
  • Chapters 12-15 introduce numerical techniques for solving differential equations.

For much of the material, a calculus sequence that includes an introduction to differential equations and linear algebra provides adequate background. For more in depth coverage of the topics from linear algebra (especially the QR method for eigenvalues) a linear algebra course would be an appropriate prerequisite. The coverage of Fourier approximation and FFT (Chapter 10) and partial differential equations (Chapter 15) also assumes that the students have somewhat more mathematical maturity than the other chapters, since the material in intrinsically more challenging. The subject matter included is suitable for a two-semester sequence of classes, or for any of several different one-term courses, depending on the desired emphasis, student background, and selection of topics.

Many people have contributed to the development of this text. My colleagues at Florida Institute of Technology, the Naval Postgraduate School, the University of South Carolina Aiken, and Georgia Southern University have provided support, encouragement, and suggestions. I especially want to thank Jacalyn Huband for the development of the Mathcad functions and examples. I also wish to thank three other colleagues for their particular contributions: Jane Lybrand for the data from classroom experiments used in several examples and exercises in Ch 9. Jack Leifer for providing data, as well as helpful discussions on engineering applications and the use of Mathcad in engineering; Pierre Larochelle for the example of robot motion in Ch 13.1 also appreciate the many contributions my students have made to this text, which was after all written with them in mind. The comments made by the reviewers of the text have helped greatly in the fine-tuning of the final presentation. The editorial and production staff at Prentice Hall, as well as Patty Donovan, and the rest of the staff at Pinetree Composition, have my heartfelt gratitude for their efforts in insuring that the text is as accurate and as well designed as possible. And, saving the most important for last, I thank my husband and colleague, Don Fausett, for his patience and support.

Laurene Fausett

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