Numerical Methods Using MathCAD

by Fausett, Laurene V.

• ISBN13: 9780130610812
• ISBN10: 013061081X
• Edition: 1st
• Format: Paperback
• Copyright: 2001-07-12
• Publisher: Pearson

Summary

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.

Table of Contents

Preface	xi
Examples/Mathcad Functions/Algorithms	xiii
Foundations	1	(46)
Sample Problems and Numerical Methods	4	(6)
Roots of Nonlinear Equations	4	(1)
Fixed-Point Iteration	5	(1)
Linear Systems	6	(1)
Gaussian Elimination	7	(1)
Numerical Integration	8	(1)
Trapezoid Rule	8	(2)
Some Basic Issues	10	(18)
Key Issues for Iterative Methods	10	(4)
How Good Is the Result?	14	(8)
Getting Better Results	22	(6)
Getting Started in Mathcad	28	(19)
Overview of the Mathcad Workspace	28	(3)
Mathematical Computations	31	(1)
Operators on the Math Toolbars	32	(3)
Built-In Functions	35	(3)
Programming in Mathcad	38	(9)
Solving Equations of One Variable	47	(46)
Bisection Method	50	(5)
Step-by-Step Computation	50	(2)
Mathcad Function for Bisection	52	(2)
Discussion	54	(1)
Regula Falsi and Secant Methods	55	(13)
Step-by-Step Computation for Regula Falsi	56	(2)
Mathcad Function for the Regula Falsi Method	58	(2)
Step-by-Step Computation for the Secant Method	60	(2)
Mathcad Function for the Secant Method	62	(2)
Discussion	64	(4)
Newton's Method	68	(7)
Step-by-Step Computation	68	(2)
Mathcad Function for Newton's Method	70	(2)
Discussion	72	(3)
Muller's Method	75	(6)
Step-by-Step Computation for Muller's Method	76	(2)
Mathcad Function for Muller's Method	78	(2)
Discussion	80	(1)
Mathcad's Methods	81	(12)
Using the Built-In Functions	81	(3)
Understanding the Algorithms	84	(9)
Solving Systems of Linear Equations: Direct Methods	93	(38)
Gaussian Elimination	96	(10)
Using Matrix Notation	97	(1)
Step-by-Step Procedure	98	(3)
Mathcad Function for Basic Gaussian Elimination	101	(2)
Discussion	103	(3)
Gaussian Elimination with Row Pivoting	106	(7)
Step-by-Step Computation	106	(4)
Mathcad Function for Gaussian Elimination with Pivoting	110	(3)
Discussion	113	(1)
Gaussian Elimination for Tridiagonal Systems	113	(9)
Step-by-Step Procedure	116	(2)
Mathcad Function for the Thomas Method	118	(1)
Discussion	119	(3)
Mathcad's Methods	122	(9)
Using the Built-In Functions	122	(1)
Understanding the Algorithms	122	(9)
Solving Systems of Linear Equations: Iterative Methods	131	(40)
Jacobi Method	135	(9)
Step-by-Step Procedure for Jacobi Iteration	136	(3)
Mathcad Function for the Jacobi Method	139	(3)
Discussion	142	(2)
Gauss-Seidel Method	144	(7)
Step-by-Step Computation for Gauss-Seidel Method	145	(3)
Mathcad Function for Gauss-Seidel Method	148	(2)
Discussion	150	(1)
Successive Overrelaxation	151	(6)
Step-by-Step Computation of SOR	152	(2)
Mathcad Function for SOR	154	(1)
Discussion	155	(2)
Mathcad's Methods	157	(14)
Using the Built-In Functions	157	(2)
Understanding the Algorithms	159	(12)
Systems of Equations and Inequalities	171	(30)
Newton's Method for Systems of Equations	174	(7)
Matrix-Vector Notation	176	(1)
Mathcad Function for Newton's Method	177	(4)
Fixed-Point Iteration for Nonlinear Systems	181	(6)
Step-by-Step Computation	182	(1)
Mathcad Function for Fixed-Point Iteration for Nonlinear Systems	182	(4)
Discussion	186	(1)
Minimum of Nonlinear Function	187	(5)
Step-by-Step Computation of Minimization by Gradient Descent	187	(1)
Mathcad Function for Minimization by Gradient Descent	188	(4)
Mathcad's Methods	192	(9)
Using the Built-In Functions	192	(1)
Understanding the Algorithms	193	(8)
LU Factorization	201	(32)
LU Factorization from Gaussian Elimination	203	(4)
A Step-by-Step Procedure for LU Factorization	204	(2)
Mathcad Function for LU Factorization Using Gaussian Elimination	206	(1)
LU Factorization of Tridiagonal Matrices	207	(2)
Step-by-Step LU Factorization of a Tridiagonal Matrix	207	(1)
Mathcad Function for LU Factorization of a Tridiagonal Matrix	208	(1)
LU Factorization with Pivoting	209	(6)
Step-by-Step Computation	209	(1)
Mathcad Function for LU Factorization with Row Pivoting	210	(2)
Discussion	212	(3)
Direct LU Factorization	215	(4)
Direct LU Factorization of a General Matrix	215	(2)
LU Factorization of a Symmetric Matrix	217	(2)
Applications of LU Factorization	219	(7)
Solving a Tridiagonal System Using LU Factorization	222	(2)
Determinant of a Matrix	224	(1)
Inverse of a Matrix	224	(2)
Mathcad's Methods	226	(7)
Using the Built-In Functions	226	(1)
Understanding the Algorithms	226	(7)
Eigenvalues, Eigenvectors, and QR Factorization	233	(50)
Power Method	236	(12)
Basic Power Method	237	(5)
Inverse Power Method	242	(5)
Discussion	247	(1)
QR Factorization	248	(19)
Householder Transformations	248	(9)
Givens Transformations	257	(4)
Basic QR Factorization	261	(6)
Finding Eigenvalues Using QR Factorization	267	(3)
Basic QR Eigenvalue Method	267	(1)
Better QR Eigenvalue Method	268	(2)
Discussion	270	(1)
Mathcad's Methods	270	(13)
Using the Built-In Functions	270	(1)
Understanding the Algorithms	271	(12)
Interpolation	283	(66)
Polynomial Interpolation	286	(24)
Lagrange Interpolation Polynomials	286	(9)
Newton Interpolation Polynomials	295	(11)
Difficulties with Polynomial Interpolation	306	(4)
Hermite Interpolation	310	(6)
Rational Function Interpolation	316	(4)
Spline Interpolation	320	(14)
Piecewise Linear Interpolation	321	(1)
Piecewise Quadratic Interpolation	322	(3)
Piecewise Cubic Interpolation	325	(9)
Mathcad's Methods	334	(15)
Using the Built-In Functions	334	(1)
Understanding the Algorithms	335	(14)
Function Approximation	349	(44)
Least Squares Approximation	352	(21)
Linear Least-Squares Approximation	352	(7)
Quadratic Least-Squares Approximation	359	(5)
Cubic Least-Squares Approximation	364	(5)
Least-Squares Approximation for Other Functional Forms	369	(4)
Continuous Least-Squares Approximation	373	(8)
Continuous Least-Squares with Orthogonal Polynomials	376	(1)
Gram-Schmidt Process	376	(2)
Legendre Polynomials	378	(1)
Least-Squares Approximation with Legendre Polynomials	379	(2)
Function Approximation at a Point	381	(4)
Taylor Approximation	381	(1)
Pade Function approximation	382	(3)
Mathcad's Methods	385	(8)
Using the Built-in Functions	385	(1)
Understanding the Algorithms	386	(7)
Fourier Methods	393	(43)
Fourier Approximation and Interpolation	396	(11)
Fast Fourier Transforms for n = 2r	407	(8)
Discrete Fourier Transform	407	(1)
Fast Fourier Transform	408	(7)
Fast Fourier Transforms for General n	415	(8)
Mathcad's Methods	423	(13)
Using the Built-In Functions	423	(1)
Understanding the Algorithms	424	(12)
Numerical Differentiation and Integration	436	(41)
Differentiation	436	(9)
First Derivatives	436	(4)
Higher Derivatives	440	(1)
Partial Derivatives	441	(1)
Richardson Extrapolation	442	(3)
Basic Numerical Integration	445	(7)
Trapezoid Rule	446	(2)
Simpson Rule	448	(2)
The Midpoint Formula	450	(2)
Other Newton-Cotes Open Formulas	452	(1)
Better Numerical Integration	452	(10)
Composite Trapezoid Rule	453	(2)
Composite Simpson's Rule	455	(3)
Extrapolation Methods for Quadrature	458	(4)
Gaussian Quadrature	462	(6)
Gaussian Quadrature on [-1,1]	462	(2)
Gaussian Quadrature on [a,b]	464	(4)
Mathcad's Methods	468	(9)
Using the Operators	468	(1)
Understanding the Algorithms	469	(8)
Ordinary Differential Equations: Initial-Value Problems	477	(52)
Taylor Methods	479	(8)
Euler's Method	479	(5)
Higher-Order Taylor Methods	484	(3)
Runge-Kutta Methods	487	(15)
Midpoint Method	487	(5)
Other Second-Order Runge-Kutta Methods	492	(2)
Third-Order Runge-Kutta Methods	494	(1)
Classic Runge-Kutta Method	495	(4)
Other Runge-Kutta Methods	499	(2)
Runge-Kutta-Fehlberg Methods	501	(1)
Multistep Methods	502	(12)
Adams-Bashforth Methods	502	(6)
Adams-Moulton Methods	508	(1)
Predictor-Corrector Methods	509	(5)
Stability	514	(3)
Mathcad's Methods	517	(12)
Using the Built-In Functions	517	(3)
Understanding the Algorithms	520	(9)
Systems of Ordinary Differential Equations	529	(46)
Higher-Order ODEs	532	(2)
Systems of Two First-Order ODE	534	(7)
Euler's Method for Solving Two ODE-IVPs	534	(3)
Midpoint Method for Solving Two ODE-IVPs	537	(4)
Systems of First-Order ODE-IVP	541	(16)
Euler's Method for Solving Systems of ODEs	542	(2)
Runge-Kutta Methods for Solving Systems of ODEs	544	(8)
Multistep Methods for Systems	552	(5)
Stiff ODE and Ill-Conditioned Problems	557	(2)
Mathcad's Methods	559	(16)
Using the Built-In Functions	559	(3)
Understanding the Algorithms	562	(13)
Ordinary Differential Equations-Boundary Value Problems	575	(34)
Shooting Method for Solving Linear BVP	578	(7)
Simple Boundary Conditions	578	(5)
General Boundary Condition at x = b	583	(1)
General Boundary Conditions at Both Ends of the Interval	584	(1)
Shooting Method for Solving Nonlinear BVP	585	(7)
Nonlinear Shooting Based on the Secant Method	585	(3)
Nonlinear Shooting Using Newton's Method	588	(4)
Finite-Difference Method for Solving Linear BVP	592	(7)
Finite-Difference Method for Nonlinear BVP	599	(3)
Mathcad's Methods	602	(7)
Using the Built-In Functions	602	(2)
Understanding the Algorithms	604	(5)
Partial Differential Equations	609	(58)
Classification of PDE	613	(1)
Heat Equation: Parabolic PDE	614	(19)
Explicit Method for Solving the Heat Equation	615	(8)
Implicit Method for Solving the Heat Equation	623	(5)
Crank-Nicolson Method for Solving the Heat Equation	628	(4)
Heat Equation with Insulated Boundary	632	(1)
Wave Equation: Hyperbolic PDE	633	(7)
Explicit Method for Solving Wave Equations	634	(4)
Implicit Method for Solving Wave Equation	638	(2)
Poisson Equation: Elliptic PDE	640	(5)
Finite-Element Method for Solving an Elliptic PDE	645	(13)
Mathcad's Methods	658	(9)
Using the Built-In Functions	658	(1)
Understanding the Algorithms	659	(8)
Bibliography	667	(6)
Answers to Selected Problems	673	(22)
Index	695

Excerpts

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 important formulas, 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