Numerical Methods
Edition: 1
Copyright: 2019
Pages: 336
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Numerical Methods is written for undergraduate and graduate students majoring in engineering and sciences. Mathematical theories and techniques as modified, extended and customized to tackle and solve applied problems arising in the various fields of engineering and sciences fall under the general subject area of numerical methods.
It provides a basic mathematical concepts and connects them to the numerical techniques. This book provides explanations of the numerical techniques in such a way that students should be able to develop their own algorithms and implement them with the help of software of their choice. Straightforward numerical examples have been provided to help students understand the techniques.
This book is written based on the course notes the author has developed over two decades of teaching the subject matter at the University of Saskatchewan. Depending on the course objectives of a discipline, the chapters can be taught in a different sequence than the one presented in this book. It is necessary that the students have already taken first-year university level courses on elementary calculus, linear algebra and introductory differential equations.
CHAPTER 1 POLYNOMIAL APPROXIMATIONS AND TAYLOR POLYNOMIALS
1.1 Taylor Polynomials
1.2 Taylor’s Remainder
1.3 Taylor Series
1.4 Maclaurin Series
CHAPTER 2 SOLUTIONS OF EQUATIONS IN ONE VARIABLE
2.1 The Bisection Method
2.1.1 Error bound in the bisection method
2.2 Fixed-point Iteration
2.3 Newton’s Method
2.3.1 Error term of Newton’s method
2.3.2 The Secant method
2.3.3 Method of false position
2.4 Müller’s Method
2.5 Brent’s Algorithm
2.6 Repeated Roots
2.7 Convergence
2.8 Accelerating Convergence
2.8.1 Aitken’s extrapolation for linearly convergent sequences
2.8.2 Steffensen’s acceleration
CHAPTER 3 SOLUTIONS OF LINEAR SYSTEMS OF EQUATIONS
3.1 Systems of Linear Equations
3.2 Linear Dependence and Independence
3.3 Matrix Representation of Systems of Linear Equations
3.4 Gauss Elimination
3.4.1 Pivoting
3.5 Gauss-Jordan Elimination
3.6 Solution by Matrix Inversion
3.7 Solution by Cramer’s Rule
3.8 Solution by Matrix Factorization
3.8.1 Triangular matrices
3.8.2 LU factorization
3.8.3 Elementary matrices and LU factorization
3.8.4 Choleski’s factorization
3.8.5 LDLT factorization
3.8.6 Solution by QR factorization
3.9 Ill-Conditioned Systems
CHAPTER 4 ITERATIVE TECHNIQUES FOR SOLVING LINEAR AND NON-LINEAR SYSTEMS
4.1 Jacobi Iterative Technique
4.2 Gauss-Seidel Iterative Technique
4.3 Iterative Techniques and the Convergence of Linear Systems
4.4 Iterative Techniques for Non-linear Systems
4.5 Newton’s Method
CHAPTER 5 INTERPOLATION
5.1 Polynomial Interpolations
5.2 Lagrange Interpolating Polynomials
5.2.1 Error bound of Lagrange interpolating polynomial
5.3 Newton’s Interpolating Polynomials
5.3.1 Divided difference and the coefficients
5.3.2 Error bound of Newton’s interpolating polynomial
5.4 Hermite Interpolation
5.5 Cubic Spline Interpolation
5.5.1 Constructing cubic splines from the second derivative
CHAPTER 6 NUMERICAL DIFFERENTIATION AND INTEGRATION
6.1 Numerical Differentiation
6.2 Alternate Approach and Error Estimates
6.3 Second-order Derivatives
6.4 Numerical Integration
6.4.1 Trapezoidal rule
6.4.2 Simpson’s rule
6.5 Error Analysis
6.5.1 Error of the Trapezoidal rule
6.5.2 Error of Simpson’s rule
6.5 Composite Numerical Integration
6.5.1 Composite Trapezoidal rule
6.5.2 Composite Simpson’s rule
6.6 Gaussian Integration
CHAPTER 7 INITIAL-VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS
7.1 Some Basics of Differential Equations
7.2 Forward Euler’s Method
7.2.1 Error and convergence of forward Euler’s method
7.3 Backward Euler’s Method
7.4 Trapezoidal and Modified Euler’s Method (Heun’s Method)
7.5 Higher-Order Taylor’s Methods
7.6 Runge-Kutta Method of Order Four
7.7 Runge-Kutta-Fehlberg Method
7.8 Multistep Methods
7.8.1 4th order Adams-Bashforth technique
7.8.2 4th order Adams-Moulton technique
CHAPTER 8 PERIODIC FUNCTIONS AND FOURIER SERIES
8.1 Periodic Functions
8.2 Fourier’s Theorem
8.2.1 Dirichlet’s conditions
8.3 Fourier Coefficients
8.4 Convergence of Fourier Series and Gibb’s Phenomenon
8.5 Even and Odd Functions
8.6 Functions Having Arbitrary Period
8.7 Complex Form of Fourier Series
8.8 Functions Defined Over a Finite Interval
CHAPTER 9 EIGENVALUES AND EIGENVECTORS AND THEIR APPLICATIONS
9.1 Eigenvalues and Eigenvectors
9.2 State-Space Models
9.2.1 A series RLC electrical circuit
9.2.2 An automobile suspension system
9.2.3 A two-tank liquid level system
9.2.4 State-space representations
9.2.5 Transfer function matrices and stability
9.2.6 Coordinate transformation and eigenvalues
9.3 Solving Systems of Differential Equations
CHAPTER 10 BOUNDARY-VALUE PROBLEMS AND PARTIAL DIFFERENTIAL EQUATIONS
10.1 Existence of Solutions to Boundary-value Problems
10.2 Shooting Methods
10.2.1 Linear shooting method
10.2.2 Nonlinear shooting method
10.3 Partial Differential Equations
10.4 Solution to PDEs: Method of Separation of Variables
10.5 Finite Difference Method for the Wave Equation
10.6 Finite Difference Method for the Heat Equation
CHAPTER 11 OPTIMIZATION
11.1 Convex Sets
11.2 The Mean Value Theorem
11.3 Relative Maximum and Minimum
11.4 Optimization of Functions of a Single Variable
11.5 Sufficient Conditions for an Optimum Point
11.6 Global Maximum (Minimum) of One Variable
11.7 Optima of Convex and Concave Functions
11.8 Search Methods for Functions of One Variable
11.8.1 Exhaustive search method
11.8.2 Fibonacci method
11.9 Optimization of Functions of Several Variables
11.10 Multivariate Grid Search Method
11.11 Univariate Search Method
11.12 Gradient Methods: Directional Derivatives
11.13 Direction of the Steepest Descent (Ascent)
11.13.1 Steepest descent (ascent) method: one variable at a time
11.13.2 Steepest ascent (descent) method: multiple variable at a time
11.14 Constrained Optimization