An Introduction to Scientific Computing with Matlab and Python Tutorials

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Author
1. An Overview of Scientific Computing
	1.1. What Is Scientific Computing?
	1.2. Errors in Scientific Computing
		1.2.1. Absolute and Relative Errors
		1.2.2. Upper Bounds
		1.2.3. Sources of Errors
	1.3. Algorithm Properties
	1.4. Exercises
2. Taylor’s Theorem
	2.1. Polynomials
		2.1.1. Polynomial Evaluation
	2.2. Taylor’s Theorem
		2.2.1. Taylor Polynomials
		2.2.2. Taylor Series
		2.2.3. Taylor’s Theorem
	2.3. Alternating Series Theorem
	2.4. Exercises
	2.5. Programming Problems
3. Roundoff Errors and Error Propagation
	3.1. Numbers
		3.1.1. Integers
	3.2. Floating-Point Numbers
		3.2.1. Scientific Notation and Rounding
		3.2.2. DP Floating-Point Representation
	3.3. Error Propagation
		3.3.1. Catastrophic Cancellation
		3.3.2. Algorithm Stability
	3.4. Exercises
	3.5. Programming Problems
4. Direct Methods for Linear Systems
	4.1. Matrices and Vectors
	4.2. Triangular Systems
	4.3. GE and A=LU
		4.3.1. Elementary Matrices
		4.3.2. A=LU
		4.3.3. Solving Ax = b by A=LU
	4.4. GEPP and PA=LU
		4.4.1. GEPP
		4.4.2. PA=LU
		4.4.3. Solving Ax = b by PA=LU
	4.5. Tridiagonal Systems
	4.6. Conditioning of Linear Systems
		4.6.1. Vector and Matrix Norms
		4.6.2. Condition Numbers
		4.6.3. Error and Residual Vectors
	4.7. Software
	4.8. Exercises
	4.9. Programming Problems
5. Root Finding for Nonlinear Equations
	5.1. Roots and Fixed Points
	5.2. The Bisection Method
	5.3. Newton’s Method
		5.3.1. Convergence Analysis of Newton’s Method
		5.3.2. Practical Issues of Newton’s Method
	5.4. Secant Method
	5.5. Fixed-Point Iteration
	5.6. Newton’s Method for Systems of Nonlinear Equations
		5.6.1. Taylor’s Theorem for Multivariate Functions
		5.6.2. Newton’s Method for Nonlinear Systems
	5.7. Unconstrained Optimization
	5.8. Software
	5.9. Exercises
	5.10. Programming Problems
6. Interpolation
	6.1. Terminology of Interpolation
	6.2. Polynomial Space
		6.2.1. Chebyshev Basis
		6.2.2. Legendre Basis
	6.3. Monomial Interpolation
	6.4. Lagrange Interpolation
	6.5. Newton’s Interpolation
	6.6. Interpolation Error
		6.6.1. Error in Polynomial Interpolation
		6.6.2. Behavior of Interpolation Error
			6.6.2.1. Equally-Spaced Nodes
			6.6.2.2. Chebyshev Nodes
	6.7. Spline Interpolation
		6.7.1. Piecewise Linear Interpolation
		6.7.2. Cubic Spline
		6.7.3. Cubic Spline Interpolation
	6.8. Discrete Fourier Transform (DFT)
	6.9. Exercises
	6.10. Programming Problems
7. Numerical Integration
	7.1. Definite Integrals
	7.2. Numerical Integration
		7.2.1. Change of Intervals
	7.3. The Midpoint Rule
		7.3.1. Degree of Precision (DOP)
		7.3.2. Error of the Midpoint Rule
	7.4. The Trapezoidal Rule
	7.5. Simpson’s Rule
	7.6. Newton-Cotes Rules
	7.7. Gaussian Quadrature Rules
	7.8. Other Numerical Integration Techniques
		7.8.1. Integration with Singularities
		7.8.2. Adaptive Integration
	7.9. Exercises
	7.10. Programming Problems
8. Numerical Differentiation
	8.1. Differentiation Using Taylor’s Theorem
		8.1.1. The Method of Undetermined Coefficients
	8.2. Differentiation Using Interpolation
		8.2.1. Differentiation Using DFT
	8.3. Richardson Extrapolation
	8.4. Exercises
	8.5. Programming Problems
9. Initial Value Problems and Boundary Value Problems
	9.1. Initial Value Problems (IVPs)
		9.1.1. Euler’s Method
			9.1.1.1. Local Truncation Error and Global Error
			9.1.1.2. Consistency, Convergence and Stability
			9.1.1.3. Explicit and Implicit Methods
		9.1.2. Taylor Series Methods
		9.1.3. Runge-Kutta (RK) Methods
	9.2. Boundary Value Problems (BVPs)
		9.2.1. Finite Difference Methods
			9.2.1.1. Local Truncation Error and Global Error
			9.2.1.2. Consistency, Stability and Convergence
	9.3. Exercises
	9.4. Programming Problems
10. Basic Iterative Methods for Linear Systems
	10.1. Jacobi and Gauss-Seidel Methods
		10.1.1. Jacobi Method
		10.1.2. Gauss-Seidel (G-S) Method
	10.2. Convergence Analysis
	10.3. Other Iterative Methods
	10.4. Exercises
	10.5. Programming Problems
11. Discrete Least Squares Problems
	11.1. The Discrete LS Problems
	11.2. The Normal Equation by Calculus
	11.3. The Normal Equation by Linear Algebra
	11.4. LS Problems by A=QR
	11.5. Artificial Neural Network
	11.6. Exercises
	11.7. Programming Problems
12. Monte Carlo Methods and Parallel Computing
	12.1. Monte Carlo Methods
	12.2. Parallel Computing
	12.3. Exercises
	12.4. Programming Problems
Appendices
	A. An Introduction of MATLAB for Scientific Computing
		A.1. What is MATLAB?
			A.1.1. Starting MATLAB
			A.1.2. MATLAB as an Advanced Calculator
			A.1.3. Order of Operations
			A.1.4. MATLAB Built-in Functions and Getting Help
			A.1.5. Keeping a Record for the Command Window
			A.1.6. Making M-Scripts
		A.2. Variables, Vectors and Matrices
			A.2.1. Variables
			A.2.2. Suppressing Output
			A.2.3. Vectors and Matrices
			A.2.4. Special Matrices
			A.2.5. The Colon Notation and linspace
			A.2.6. Accessing Entries in a Vector or Matrix
		A.3. Matrix Arithmetic
			A.3.1. Scalar Multiplication
			A.3.2. Matrix Addition
			A.3.3. Matrix Multiplication
			A.3.4. Transpose
			A.3.5. Entry-wise Convenience Operations
		A.4. Outputting/Plotting Results
			A.4.1. disp
			A.4.2. fprintf
			A.4.3. plot
		A.5. Loops and Decisions
			A.5.1. for Loops
			A.5.2. Logicals and Decisions
			A.5.3. while Loops
		A.6. Functions
			A.6.1. M-Functions
			A.6.2. Anonymous Functions
			A.6.3. Passing Functions to Functions
		A.7. Creating Live Scripts in the Live Editor
		A.8. Concluding Remarks
		A.9. Programming Problems
	B. An Introduction of Python for Scientific Computing
		B.1. What is Python?
			B.1.1. Starting Python
			B.1.2. Python as an Advanced Calculator
			B.1.3. Python Programs
		B.2. Variables, Lists and Dictionaries
			B.2.1. Variables
			B.2.2. Lists
			B.2.3. Dictionaries
		B.3. Looping and Making Decisions
			B.3.1. for Loops
			B.3.2. if Statements
			B.3.3. while Loops
			B.3.4. break and continue in Loops
		B.4. Functions
			B.4.1. Passing Arguments
			B.4.2. Passing Lists
		B.5. Classes
			B.5.1. Attributes
			B.5.2. Methods
			B.5.3. Inheritance
			B.5.4. Objects as Attributes
		B.6. Modules
		B.7. numpy, scipy, matplotlib
			B.7.1. numpy
			B.7.2. scipy
			B.7.3. matplotlib
		B.8. Jupyter Notebook
		B.9. Concluding Remarks
		B.10. Programming Problems
Index