Applied Numerical Methods Using MATLAB
- Length: 950 pages
- Edition: 1
- Language: English
- Publisher: Mercury Learning and Information
- Publication Date: 2023-03-30
- ISBN-10: 1683928687
- ISBN-13: 9781683928683
- Sales Rank: #0 (See Top 100 Books)
The book is designed to cover all major aspects of applied numerical methods, including numerical computations, solution of algebraic and transcendental equations, finite differences and interpolation, curve fitting, correlation and regression, numerical differentiation and integration, matrices and linear system of equations, numerical solution of ordinary differential equations, and numerical solution of partial differential equations. MATLAB is incorporated throughout the text and most of the problems are executed in MATLAB code. It uses a numerical problem-solving orientation with numerous examples, figures, and end of chapter exercises. Presentations are limited to very basic topics to serve as an introduction to more advanced topics.
FEATURES:
- Integrates MATLAB throughout the text
- Includes over 600 fully-solved problems with step-by-step solutions
- Limits presentations to basic concepts of solving numerical methods
Cover Half-Title Title Copyright Contents Preface Chapter 1: Numerical Computations 1.1 Taylor’s Theorem 1.2 Number Representation 1.3 Error Considerations 1.3.1 Absolute and Relative Errors 1.3.2 Inherent Errors 1.3.3 Round-off Errors 1.3.4 Truncation Errors 1.3.5 Machine Epsilon 1.3.6 Error Propagation 1.4 Error Estimation 1.5 General Error Formula 1.5.1 Function Approximation 1.5.2 Stability and Condition 1.5.3 Uncertainty in Data or Noise 1.6 Sequences 1.6.1 Linear Convergence 1.6.2 Quadratic Convergence 1.6.3 Aitken’s Acceleration Formula 1.7 Summary Exercises Chapter 2: Linear System of Equations 2.1 Introduction 2.2 Methods of Solution 2.3 The Inverse of a Matrix 2.4 Matrix Inversion Method 2.4.1 Augmented Matrix 2.5 Gauss Elimination Method 2.5.1 MATLAB Program for the Gauss Elimination Method 2.6 Gauss-Jordan Method 2.6.1 MATLAB Program for the Gauss Jordan Method 2.7 Cholesky’s Triangularization Method 2.8 Crout’s Method 2.8.1 MATLAB Program for Crout’s Method 2.9 Thomas Algorithm for Tridiagonal System 2.9.1 MATLAB Program for the Thomas Method for Tridiagonal Systems 2.10 Jacobi’s Iteration Method 2.10.1 MATLAB Program for the Jacobi Iteration Method 2.11 Gauss-Seidel Iteration Method 2.11.1 MATLAB Program for the Gauss Seidel Method 2.12 Symmetric Matrix Eigenvalue Problems 2.12.1 The Jacobi Method 2.12.2 MATLAB Function for the Jacobi Method 2.12.3 Householder Reduction to Tridiagonal Form 2.12.4 Gerschgorin’s Circle Theorem 2.12.5 Sturm Sequence 2.12.6 QR Method 2.12.7 Power Method 2.12.8 Inverse Power Method 2.13 Summary Exercises Chapter 3: Solution of Algebraic and Transcendental Equations 3.1 Introduction 3.2 Bisection Method 3.2.1 Error Bounds 3.3 Method of False Position 3.3.1 MATLAB Program for the False Position Method 3.4 Newton-Raphson Method 3.4.1 Convergence of the Newton-Raphson Method 3.4.2 Rate of Convergence of the Newton-Raphson Method 3.4.3 MATLAB Program for the Newton Raphson Method 3.4.4 Modified Newton-Raphson Method 3.4.5 Rate of Convergence of Modified Newton-Raphson Method 3.5 Successive Approximation Method 3.5.1 Error Estimate in the Successive Approximation Method 3.6 Secant Method 3.6.1 Convergence of the Secant Method 3.6.2 MATLAB Program to Search for a Root of the Function f(x) in the Interval (a,b) 3.6.3 MATLAB Program for Secant Method 3.7 Muller’s Method 3.7.1 MATLAB Program for Muller’s Method 3.8 Chebyshev Method 3.9 Aitken’s Δ2 Method 3.10 Brent’s Method 3.10.1 MATLAB Program for Brent’s Method 3.11 Newton Method for a System of Nonlinear Equations 3.12 Comparison of Iterative Methods 3.13 MATLAB Built-in Function: fzero 3.14 Summary Exercises Chapter 4: Numerical Differentiation 4.1 Introduction 4.2 Derivatives Based on Newton’s Forward Integration Formula 4.2.1 MATLAB Program for Derivatives Based on Newton’s Forward Integration Formula—Equally Spaced Points 4.3 Derivatives Based on Newton’s Backward Interpolation Formula 4.4 Derivatives Based on Stirling’s Interpolation Formula 4.5 Maxima and Minima of a Tabulated Function 4.6 Cubic Spline Method 4.7 Richardson Extrapolation 4.8 Differentiation of Unequally Spaced Data 4.9 MATLAB Built-in Functions: diff and gradient 4.10 Summary Exercises Chapter 5: Finite Differences and Interpolation 5.1 Introduction 5.2 Finite Difference Operators 5.2.1 Forward Differences 5.2.2 Backward Differences 5.2.3 Central Differences 5.2.4 Error Propagation in a Difference Table 5.2.5 Properties of the Operator Δ 5.2.6 Difference Operators 5.2.7 Relation Among the Operators 5.2.8 Representation of a Polynomial using Factorial Notation 5.3 Interpolation with Equal Intervals 5.3.1 Missing Values 5.3.2 Newton’s Binomial Expansion Formula 5.3.3 Newton’s Forward Interpolation Formula 5.3.4 MATLAB M-file: Newtonint 5.3.5 Newton’s Backward Interpolation Formula 5.3.6 Error in the Interpolation Formula 5.4 Interpolation with Unequal Intervals 5.4.1 Lagrange’s Interpolating Polynomial for Equal Intervals 5.4.2 function yint = Lagrangeint (x,y,xx) 5.4.3 Lagrange’s Formula for Unequal Intervals 5.4.4 Hermite’s Interpolation Formula 5.4.5 Inverse Interpolation 5.4.6 Lagrange’s Formula for Inverse Interpolation 5.5 Central Difference Interpolation Formulae 5.5.1 Gauss’s Forward Interpolation Formula 5.5.2 Gauss Backward Interpolation Formula 5.5.3 Bessel’s Formula 5.5.4 Stirling’s Formula 5.5.5 Laplace-Everett’s Formula 5.5.6 Selection of an Interpolation Formula 5.6 Divided Differences 5.6.1 Newton’s Divided Difference Interpolation Formula 5.7 Cubic Spline Interpolation 5.8 Generalized Spline Method 5.8.1 Splines 5.8.2 Linear Splines 5.8.3 Quadratic Splines 5.8.4 Cubic Splines 5.8.5 End Conditions 5.8.6 MATLAB Built-in Function: spline 5.8.7 Multidimensional Interpolation 5.8.8 MATLAB Built-in Function: interpl 5.9 Summary Exercises Chapter 6: Curve Fitting, Regression, and Correlation Approximating Curves Linear Regression 6.1 Linear Equation 6.2 Curve Fitting With a Linear Equation 6.3 Criteria for a Best Fit 6.4 Linear Least-Squares Regression 6.5 Linear Regression Analysis 6.5.1 MATLAB built-in function: polyfit 6.5.2 MATLAB built-in function: polyval 6.6 Interpretation of a and b Assumptions in the Regression Model 6.7 Standard Deviation of Random Errors 6.8 Coefficient of Determination 6.9 Linear Correlation Properties of the Linear Correlation Coefficient r Explained and Unexplained Variation 6.10 Linearization of Nonlinear Relationships 6.11 Polynomial Regression 6.11.1 Polynomial Fit 6.11.2 MATLAB Built-in Functions for Polynomial Fit 6.12 Quantification of Error of Linear Regression 6.13 Multiple Linear Regression 6.14 Weighted Least-Squares Method 6.15 Orthogonal Polynomials and Least-Squares Approximation 6.16 Least-Squares Method for Continuous Data 6.17 Approximation Using Orthogonal Polynomials 6.18 Gram-Schmidt Orthogonalization Process 6.19 Fitting a Function Having a Specified Power 6.20 Fitting a Cubic Spring Model 6.21 Additional Example Problems and Solutions 6.22 Summary Exercises Chapter 7: Numerical Integration 7.1 Introduction 7.1.1 Relative Error 7.2 Newton-Cotes Closed Quadrature Formula 7.3 Trapezoidal Rule 7.3.1 Error Estimate in Trapezoidal Rule 7.3.2 MATLAB Functions: trapz and cumtrapz 7.4 Simpson’s 1/3 Rule 7.4.1 Error Estimate in Simpson’s 1/3 Rule 7.4.2 MATLAB Program for Simpson’s Integration: simpsonint 7.4.3 MATLAB Built-in Functions: quad and quad1 7.5 Simpson’s 3/8 Rule 7.6 Boole’s and Weddle’s Rules 7.6.1 Boole’s Rule 7.6.2 Weddle’s Rule 7.7 Romberg’s Integration 7.7.1 Richardson’s Extrapolation 7.7.2 Romberg Integration Formula 7.7.3 MATLAB Program for Romberg Integration: Romberg 7.8 Gaussian Quadrature 7.8.1 Gaussian Integration Formulas 7.8.2 Orthogonal Polynomials 7.8.3 Gauss-Lagendre Quadrature 7.8.4 Gauss-Chebyshev Quadrature Method 7.8.5 Gauss-Laguerre Quadrature 7.8.6 Gauss-Hermite Quadrature 7.8.7 MATLAB Programs for Gaussian Quadrature: gaussnodes and gaussquad 7.9 Double Integration 7.9.1 Trapezoidal Method 7.9.2 Simpson’s 1/3 Rule 7.9.3 MATLAB Built-in Function for Double Integration: dblquad 7.10 Summary Exercises Chapter 8: Numerical Solution of Ordinary Differential Equations 8.1 Introduction 8.2 One-Step Methods or Single-Step Methods 8.2.1 Picard’s Method of Successive Approximation 8.2.2 The Taylor’s Series Method 8.3 Step-by-Step Methods or Marching Methods 8.3.1 Euler’s Method 8.3.2 MATLAB Program for Euler’s Method: euler 8.3.3 Modified Euler’s Method 8.3.4 MATLAB Program for the Modified Euler’s Method: modeuler 8.3.5 Runge-Kutta Methods 8.3.6 Predictor-Corrector Methods 8.4 MATLAB Functions for Ordinary Differential Equations: ode45, ode23, ode113, ode15s, ode23s, ode23t, ode23tb 8.5 System of First-order Ordinary Differential Equations 8.6 Initial Value Problems 8.6.1 The Taylor Series Method 8.6.2 Picard’s Method 8.6.3 Second-Order Runge-Kutta Method 8.6.4 Fourth-Order Runge-Kutta Method 8.6.5 Euler’s Formula 8.6.6 Modified Euler’s Formula 8.6.7 Burlirsch-Stoer Method (Mid-Point Method) 8.6.8 The Runge-Kutta-Fehlberg Method 8.6.9 The Runge-Kutta-Butcher Method 8.7 Two-Point Boundary Value Problems 8.7.1 Finite Difference Method 8.7.2 Second-Order Differential Equations 8.7.3 The Shooting Method 8.8 Second-Order Initial Value Problem (IVP) 8.9 Second-Order Boundary Value Problem (BVP) 8.10 MATLAB Built-in Functions 8.11 Summary Exercises Chapter 9: Direct Numerical Integration Methods 9.1 Introduction 9.2 Single Degree of Freedom System 9.2.1 Finite Difference Method 9.2.2 Central Difference Method 9.2.3 The Runge-Kutta Method 9.3 Multi-degree of Freedom Systems 9.4 Explicit Schemes 9.4.1 Central Difference Method 9.4.2 Two-Cycle Iteration with Trapezoidal Rule 9.4.3 Fourth-Order Runge-Kutta Method 9.5 Implicit Schemes 9.5.1 The Houbolt Method 9.5.2 Wilson Theta Method 9.5.3 The Newmark Beta Method 9.5.4 Park Stiffly Stable Method 9.6 Example Problems and Solutions Using MATLAB 9.7 Summary Exercises Additional Exercises Chapter 10: Matlab Basics 10.1 Introduction 10.1.1 Starting and Quitting MATLAB 10.1.2 Display Windows 10.1.3 Entering Commands 10.1.4 MATLAB Expo 10.1.5 Abort 10.1.6 The Semicolon (;) 10.1.7 Typing % 10.1.8 The clc Command 10.1.9 Help 10.1.10 Statements and Variables 10.2 Arithmetic Operations 10.3 Display Formats 10.4 Elementary Math Built-In Functions 10.5 Variable Names 10.6 Predefined Variables 10.7 Commands For Managing Variables 10.8 General Commands 10.9 Arrays 10.9.1 Row Vector 10.9.2 Column Vector 10.9.3 Matrix 10.9.4 Addressing Arrays 10.9.5 Adding Elements to a Vector or a Matrix 10.9.6 Deleting Elements 10.9.7 Built-in Functions 10.10 Operations with Arrays 10.10.1 Addition and Subtraction of Matrices 10.10.2 Dot Product 10.10.3 Array Multiplication 10.10.4 Array Division 10.10.5 Identity Matrix 10.10.6 Inverse of a Matrix 10.10.7 Transpose 10.10.8 Determinant 10.10.9 Array Division 10.10.10 Left Division 10.10.11 Right Division 10.11 Element-By-Element Operations 10.11.1 Built-In Functions For Arrays 10.12 Random Numbers Generation 10.12.1 The Random Command 10.13 Polynomials 10.14 System of Linear Equations 10.14.1 Matrix Division 10.14.2 Matrix Inverse 10.15 Script Files 10.15.1 Creating and Saving a Script File 10.15.2 Running a Script File 10.15.3 Input to a Script File 10.15.4 Output Commands 10.16 Programming in Matlab 10.16.1 Relational and Logical Operators 10.16.2 Order of Precedence 10.16.3 Built-in Logical Functions 10.16.4 Conditional Statements 10.16.5 Nested if Statements 10.16.6 else AND elseif Clauses 10.16.7 MATLAB while Structures 10.17 Graphics 10.17.1 Basic 2-D Plots 10.17.2 Specialized 2-D Plots 10.17.3 3-D Plots 10.17.4 Saving and Printing Graphs 10.18 Input/Output In Matlab 10.18.1 The fopen Statement 10.19 Symbolic Mathematics 10.19.1 Symbolic Expressions 10.19.2 Solution to Differential Equations 10.19.3 Calculus 10.23 Summary References Exercises Chapter 11: Optimization 11.1 Introduction 11.2 Unconstrained Minimization of Functions 11.3 Minimization with Constraints Using Lagrange Multipliers 11.4 Numerical Optimization 11.4.1 Optimization Involving Single Variables 11.4.2 Local and Global Optima 11.4.3 Bracketing 11.4.4 Golden-Section Search 11.4.5 MATLAB Program for Bracketing Method 11.4.6 MATLAB Program for Golden-Section Search Method 11.5 Multidimensional Optimization 11.6 Gradient Methods 11.7 Newton’s Method 11.7.1 MATLAB Program for Newton’s Method 11.8 Methods Based on the Concept of Quadratic Convergence 11.8.1 Conjugate Directions for a Quadratic Function 11.9 Powell’s Method 11.9.1 MATLAB Program for Powell’s Optimization Method 11.10 Fletcher-Reeves Method 11.10.1 MATLAB Program for Fletcher-Reeves Optimization Method 11.11 The Hooks and Jeeves Method 11.12 Method of Successive Linear Approximation 11.13 Interior Penalty Function Method 11.14 MATLAB Built-in Functions 11.14.1 MATLAB Function: fminbnd 11.14.2 MATLAB Function: fminsearch 11.15 Additional Example Problems and Solutions 11.16 Summary References Exercises Chapter 12: Partial Differential Equations 12.1 Introduction 12.2 Classification of Linear Second-Order Partial Differential Equation 12.3 Types of Problems 12.4 Finite-Difference Approximation to Partial Derivatives 12.5 Physical Phenomena 12.5.1 Laplace’s Equation 12.5.2 Heat Equation 12.5.3 Wave Equation 12.5.4 Equation Classification 12.6 Elliptic Equations 12.6.1 Central Difference Method 12.6.2 Boundary Conditions 12.6.3 Iterative Solution Methods 12.6.4 The Jacobi Method 12.6.5 Gauss-Seidel Method 12.6.6 Successive Over-Relaxation or S.O.R. Method 12.7 One-Dimensional Parabolic Equations 12.7.1 Explicit Forward Euler Method 12.7.2 Implicit Backward Euler Method 12.7.3 The Crank-Nicolson Implicit Method 12.7.4 function [t,x,U] =Heatone(T,a,m,n,beta,c,f,g) 12.7.5 function [x,y,U] = Heattwo(T,a,b,m,n,p,beta,f,g) 12.7.6 function [t,x,U] = Waveone(T,a,m,n,beta,f,g) 12.7.7 function [x,y,U] = Wavetwo (T,a,b,m,n,p,beta,f,g) 12.7.8 function [alpha,r,x,y,U] = Poisson (a,b,m,n,q,tol,f,g) 12.8 Two-Dimensional Parabolic Equations 12.9 One-Dimensional Hyperbolic Equations 12.9.1 D’Alembert’s Solution 12.9.2 Explicit Central Difference Method 12.10 Two-Dimensional Hyperbolic Equations 12.10.1 Explicit Central Difference Method 12.11 MATLAB Built-in Function: pdepe 12.12 Summary Exercises Appendix A: Partial Fraction Expansions Case-I Partial Fraction Expansion when Q(s) has Distinct Roots Case-II Partial Fraction Expansion when Q(s) has Complex Conjugate Roots Case-III Partial Fraction Expansion when Q(s) has Repeated Roots Exercises Appendix B: Basic Engineering Mathematics B.1 Algebra B.1.1 Basic Laws B.1.2 Sums of Numbers B.1.3 Progressions B.1.4 Powers and Roots B.1.5 Binomial Theorem B.1.6 Absolute Values B.1.7 Logarithms B.2 Trigonometry B.2.1 Trigonometric Identities B.2.2 Cosine Law (Law of Cosines) B.2.3 Sine Law (Law of Sines) B.3 Differential Calculus B.3.1 List of Derivatives B.3.2 Expansion in Series B.4 Integral Calculus B.4.1 List of Most Common Integrals Appendix C: Cramer’s Rule Exercises Appendix D: Matlab Built-In M-File Functions Appendix E: Matlab Programs Appendix F: Answers to Odd Numbered Exercises Bibliography Index
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