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Numerical Methods Using MathCAD,9780130610812
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Numerical Methods Using MathCAD

by
Edition:
1st
ISBN13:

9780130610812

ISBN10:
013061081X
Format:
Paperback
Pub. Date:
7/12/2001
Publisher(s):
Pearson
List Price: $103.00

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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


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