Numerical Methods for Engineers and Scientists Using MATLAB

ISBN-10: 1466585692

ISBN-13: 9781466585690

Edition: 2013

Authors: Ramin S. Esfandiari

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Description:

This book provides the reader with a working knowledge of the fundamentals of numerical methods utilized in various disciplines in engineering and science. MATLAB® is introduced at the outset and is used throughout the book to perform symbolic, graphical, and numerical tasks. The textbook, written at the junior level, methodically covers a wide array of methods ranging from curve fitting a set of data to numerically solving ordinary and partial differential equations. Each method is accompanied by either a user-defined function or a MATLAB script file. MATLAB built-in functions are also presented for each main topic.
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Book details

List price: $141.00
Copyright year: 2013
Publisher: CRC Press LLC
Publication date: 6/4/2013
Binding: Paperback
Pages: 550
Size: 6.00" wide x 9.00" long x 1.00" tall
Weight: 1.694
Language: English

Preface
Acknowledgments
Author
Background and Introduction
Background
Differential Equations
Linear First-Order ODEs
Second-Order ODEs with Constant Coefficients
Matrix Analysis
Matrix Operations
Special Matrices
Determinant of a Matrix
Inverse of a Matrix
Matrix Eigenvalue Problem
Solving the Eigenvalue Problem
Similarity Transformation
Introduction to Numerical Methods
Errors and Approximations
Computational Errors
Binary and Hexadecimal Numbers
Floating Point and Rounding Errors
Absolute and Relative Errors
Iterative Methods
A Fundamental Iterative Method
Rate of Convergence of an Iterative Method
Problem Set
Introduction to MATLAB�
MATLAB� Built-in Functions
Rounding Commands
Relational Operators
Format Options
Vectors and Matrices
Linspace
Matrices
Determinant, Transpose, and Inverse
Slash Operators
Element-by-Element Operations
User-Defined Functions and Script Files
Setting Default Values for Input Variables
Creating Script Files
Anonymous Functions
Inline
Program Flow Control
for Loop
if Command
while Loop
Displaying Formatted Data
Symbolic Toolbox
Differentiation
Integration
Differential Equations
Plotting
Subplot
Plotting Analytical Expressions
Multiple Plots
Problem Set
Solution of Equations of a Single Variable
Numerical Solution of Equations
Bisection Method
MATLAB� Built-in Function fzero
Regula Falsi Method (Method of False Position)
Modified Regula Falsi Method
Fixed-Point Method
Selection of a Suitable Iteration Function
A Note on Convergence
Rate of Convergence of the Fixed-Point Iteration
Newton's Method (Newton-Raphson Method)
Rate of Convergence of Newton's Method
A Few Notes on Newton's Method
Modified Newton's Method for Roots with Multiplicity 2 or Higher
Secant Method
Rate of Convergence of Secant Method
A Few Notes on Secant Method
Equations with Several Roots
Finding Zeros to the Right of a Specified Point
Finding Zeros on Two Sides of a Specified Point
Using f zero to Find Several Roots
Points of Discontinuity Mistaken for Roots
Problem Set
Solution of Systems of Equations
Linear Systems of Equations
Numerical Solution of Linear Systems
Gauss Elimination Method
Choosing the Pivot Row: Partial Pivoting with Row Scaling
Permutation Matrices
Counting the Number of Operations
Elimination
Back Substitution
Tridiagonal Systems
Thomas Method
MATLAB� Built-in Function "\"
LU Factorization Methods
Doolittle Factorization
Doolittle's Method to Solve a Linear System
Operations Count
Cholesky Factorization
Cholesky's Method to Solve a Linear System
Operations Count
MATLAB� Built-in Functions lu and chol
Iterative Solution of Linear Systems
Vector Norms
Matrix Norms
Compatibility of Vector and Matrix Norms
General Iterative Method
Convergence of the General Iterative Method
Jacobi Iteration Method
Convergence of the Jacobi Iteration Method
Gauss-Seidel Iteration Method
Convergence of the Gauss-Seidel Iteration Method
Indirect Methods versus Direct Methods for Large Systems
Ill-Conditioning and Error Analysis
Condition Number
Ill-Conditioning
Indicators of Ill-Conditioning
Computational Error
Consequences of Ill-Conditioning
Effects of Parameter Changes on the Solution
Systems of NonLinear Equations
Newton's Method for a System of Nonlinear Equations
Method for Solving a System of Two Nonlinear Equations
Method for Solving a System of n Nonlinear Equations
Convergence of Newton's Method
Fixed-Point Iteration Method for a System of Nonlinear Equations
Convergence of the Fixed-Point Iteration Method
Problem Set
Curve Fitting (Approximation) and Interpolation
Least-Squares Regression
Linear Regression
Deciding a "Best" Fit Criterion
Linear Least-Squares Regression
Linearization of Nonlinear Data
Exponential Function
Power Function
Saturation Function
Polynomial Regression
Quadratic Least-Squares Regression
Cubic Least-Squares Regression
MATLAB� Built-in Functions polyfit and polyval
Polynomial Interpolation
Lagrange Interpolating Polynomials
Drawbacks of Lagrange Interpolation
Newton Divided-Difference Interpolating Polynomials
Special Case: Equally Spaced Data
Newton Forward-Difference Interpolating Polynomials
Spline Interpolation
Linear Splines
Quadratic Splines
Function Values at the Endpoints
Function Values at the Interior Knots
First Derivatives at the Interior Knots
Second Derivative at the Left Endpoint is Zero
Cubic Splines
Construction of Cubic Splines: Clamped Boundary Conditions
Construction of Cubic Splines: Free Boundary Conditions
MATLAB� Built-in Functions interp1 and spline
Boundary Conditions
Interactive Curve Fitting and Interpolation in MATLAB�
Fourier Approximation and Interpolation
Sinusoidal Curve Fitting
Linear Transformation of Data
Discrete Fourier Transform
Fast Fourier Transform
Sande-Tukey Algorithm (N = 2<sup>p</sup>, p = integer)
Cooley-Tukey Algorithm (N=2<sup>p</sup>, p = integer)
MATLAB� Built-in Function fft
Interpolation Using fft
Problem Set
Numerical Differentiation and Integration
Numerical Differentiation
Finite-Difference Formulas for Numerical Differentiation
Finite-Difference Formulas for the First Derivative
Two-Point Backward Difference Formula
Two-Point Forward Difference Formula
Two-Point Central Difference Formula
Three-Point Backward Difference Formula
Three-Point Forward Difference Formula
Finite-Difference Formulas for the Second Derivative
Three-Point Backward Difference Formula
Three-Point Forward Difference Formula
Three-Point Central Difference Formula
Summary of Finite-Difference Formulas for First to Fourth Derivatives
Estimate Improvement: Richardson's Extrapolation
Derivative Estimates for Nonevenly Spaced Data
MATLAB� Built-in Functions dif f and polyder
Numerical Integration: Newton-Cotes Formulas
Newton-Cotes Formulas
Rectangular Rule
Composite Rectangular Rule
Error Estimate for Composite Rectangular Rule
Trapezoidal Rule
Composite Trapezoidal Rule
Error Estimate for Composite Trapezoidal Rule
Simpson's Rules
Simpson's 1/3 Rule
Composite Simpson's 1/3 Rule
Error Estimate for Composite Simpson's 1/3 Rule
Simpson's 3/8 Rule
Composite Simpson's 3/8 Rule
Error Estimate for Composite Simpson's 3/8 Rule
MATLAB� Built-in Functions quad and trapz
Numerical Integration of Analytical Functions: Romberg Integration, Gaussian Quadrature
Richardson's Extrapolation, Romberg Integration
Richardson's Extrapolation
Romberg Integration
Gaussian Quadrature
Improper Integrals
Problem Set
Numerical Solution of Initial-Value Problems
One-Step Methods
Euler's Method
Error Analysis for Euler's Method
Calculation of Local and Global Truncation Errors
Higher-Order Taylor Methods
Runge-Kutta Methods
Second-Order Runge-Kutta Methods
Improved Euler's Method
Heun's Method
Ralston's Method
Graphical Representation of Heun's Method
Third-Order Runge-Kutta Methods
Classical RK3 Method
Heun's RK3 Method
Fourth-Order Runge-Kutta Methods
Classical RK4 Method
Higher-Order Runge-Kutta Methods
Runge-Kutta-Fehlberg Method
Multistep Methods
Adams-Bashforth Method
Second-Order Adams-Bashforth Formula
Third-Order Adams-Bashforth Formula
Fourth-Order Adams-Bashforth Formula
Adams-Moulton Method
Second-Order Adams-Moulton Formula
Third-Order Adams-Moulton Formula
Fourth-Order Adams-Moulton Formula
Predictor-Corrector Methods
Heun's Predictor-Corrector Method
Adams-Bashforth-Moulton Predictor-Corrector Method
Systems of Ordinary Differential Equations
Transformation into a System of First-Order ODEs
State Variables
Notation
State-Variable Equations
Numerical Solution of a System of First-Order ODEs
Euler's Method for Systems
Heun's Method for Systems
Classical RK4 Method for Systems
Stability
Euler's Method
Euler's Implicit Method
Stiff Differential Equations
MATLAB� Built-in Functions for Initial-Value Problems
Nonstiff Equations
Single First-Order IVP
Setting ODE Solver Options
System of First-Order IVPs
Stiff Equations
Problem Set
Numerical Solution of Boundary-Value Problems
Shooting Method
Finite-Difference Method
BVPs with Mixed Boundary Conditions
MATLAB� Built-in Function bvp4c for BVPs
Second-Order BVP
Problem Set
Matrix Eigenvalue Problem
Power Method: Estimation of the Dominant Eigenvalue
Inverse Power Method: Estimation of the Smallest Eigenvalue
Shifted Inverse Power Method: Estimation of the Eigenvalue Nearest a Specified Value
Shifted Power Method
MATLAB� Built-in Function eig
Deflation Methods
Wielandt's Deflation Method
Deflation Process
Householder Tridiagonalization and QR Factorization Methods
Householder's Tridiagonalization Method (Symmetric Matrices)
Determination of Symmetric Orthogonal P<sub>k</sub> (k = 1,2,&#8230;, n-2)
QR Factorization Method
Determination of Q<sub>k</sub> and R<sub>k</sub> Matrices
Structure of L<sub>k</sub> (k = 2,3,&#8230;, n)
MATLAB� Built-in Function qr
Transformation to Hessenberg Form (Nonsymmetric Matrices)
Problem Set
Numerical Solution of Partial Differential Equations
Introduction
Elliptic PDEs
Dirichlet Problem
Alternating Direction-Implicit Methods
Peaceman-Rachford Alternating Direction-Implicit Method
Neumann Problem
Existence of Solution for Neumann Problem
Mixed Problem
More Complex Regions
Parabolic PDEs
Finite-Difference (FD) Method
Stability and Convergence of the FD Method
Crank-Nicolson (CN) Method
CN Method versus FD Method
Hyperbolic PDEs
Starting the Procedure
Problem Set
Bibliography
Index
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