Applied Numerical Methods Using MATLAB

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