CHAPTER 1 Numerical Computation
1.1
Motivation and Objectives
1.1.1
A Simple Calculation
1.1.2
Chapter Objectives
1.1.3
Mathematical Background
1.2
Number Representation
1.2.1
Binary, Decimal, and Hexadecimal
1.2.2
Integers
1.2.3
Floats
1.3
Machine Precision
1.4
Round-Off Error
1.4.1
Chopping and Rounding
1.4.2
Error Propagation
1.5
Truncation Error
1.6
Random Number Generation
1.6.1
Uniform Distribution
1.6.2
Gaussian Distribution
1.7
Numerical Software
1.7.1
A Numerical Library: NLIB
1.7.2
NLIB Example Browser
1.7.3
Pseudo-Prototypes
1.8
Applications
1.8.1
Throwing Darts to Estimate : MATLAB
1.8.2
Monte Carlo Integration: C
1.9
Summary
Problems
1.10.1
Analysis
1.10.2
Computation
CHAPTER 2 Linear Algebraic Systems
2.1
Motivation and Objectives
2.1.1
Robotic Arm
2.1.2
Converter Circuit
2.1.3
DC Motor
2.1.4
Chapter Objectives
2.2
Gauss-Jordan Elimination
2.3
Gaussian Elimination
2.4
LU Decomposition
2.5.1
LU Factorization
2.5.2
Forward and Back Substitution
2.5.3
Tridiagonal Systems
2.5
Ill-Conditioned Systems
2.5.1
Vector and Matrix Norms
2.5.2
Condition Number
2.5.3
Approximate Condition Number
2.5.4
Iterative Improvement
2.6 Iterative Methods
2.6.1
Jacobi''s Method
2.6.2
Gauss-Seidel Method
2.6.3
Relaxation Methods
2.6.4
Convergence
2.7
Applications
2.7.1
Chemical Absorption Process: MATLAB
2.7.2
Planar Truss: C
2.7.3
DC Bridge Circuit: MATLAB
2.7.4
Mass-Spring-Damper System: C
2.8
Summary
Problems
2.9.1
Analysis
2.9.2
Computation
CHAPTER 3 Eigenvalues and Eigenvectors
3.1
Motivation and Objectives
3.1.1
Seismograph
3.1.2
Convergence of Iterative Methods
3.1.3
Chapter Objectives
3.2
The Characteristic Polynomial
3.3
Power Methods
3.3.1
Direct Power Method
3.3.2
Inverse Power Method
3.4
Jacobi''s Method
3.5
Householder Transformation
3.6
QR Method
3.6.1
Deflation
3.6.2
Shifting
3.7
Danilevsky''s Method
3.8
Polynomial Roots
3.9
Applications
3.9.1
Transient Analysis of an Absorption Process: C
3.9.2
Population Growth Model: MATLAB
3.9.3
Telescope Position Control: C
3.9.4
Rotating Masses and Torsional Springs: MATLAB
3.10
Summary
Problems
3.11.1
Analysis
3.11.2
Computation
CHAPTER 4 Curve Fitting
4.1
Motivation and Objectives
4.1.1
Gravitational Acceleration
4.1.2
Circadian Rhythms
4.1.3
Chapter Objectives
4.2
Interpolating
4.2.1
Piecewise-Linear Interpolation
4.2.2
Polynomial Interpolation
4.2.3
Lagrange Interpolation Polynomials
4.2.4
Polynomials
4.3
Newton''s Difference Formula
4.4
Cubic Splines
4.5
Least Squares
4.5.1
Straight Line Fit
4.5.2
Polynomial Fit
4.5.3
Orthogonal Polynomials
4.6
Two-Dimensional Interpolation
4.7
Applications
4.7.1
Pressure-Temperature Curves: MATLAB
4.7.2
Water Resource Management: C
4.7.3
Voltage Regulator Circuit: MATLAB
4.7.4
Nonlinear Friction Model: C
4.8
Summary
Problems
4.9.1
Analysis
4.9.2
Computation
CHAPTER 5 Root Finding
5.1
Motivation and Objectives
5.1.1
Tunnel Diode Circuit
5.1.2
Leaky Tank
5.1.3
Bacterial Chemostat
5.1.4
Chapter Objectives
5.2
Bracketing Methods
5.2.1
Bisection Method
5.2.2
False Position Method
5.3
Contraction Mapping Method
5.3.1
Root Finding
5.3.2
Aitken Extrapolation
5.4
Secant Method
5.5
Muller''s Method
5.6
Newton''s Method
5.7
Polynomial Roots
5.7.1
Quadratic Formula
5.7.2
Synthetic Division
5.7.3
Laguerre''s Method
5.8
Nonlinear Systems of Equations
5.9
Applications
5.9.1
Propane Cylinder: C
5.9.2
Bacterial Chemostat: MATLAB
5.9.3
Industrial High-Temperature Oven: C
5.9.4
Suspension Cable: MATLAB
5.10
Summary
Problems
5.11.1
Analysis
5.11.2
Computation
CHAPTER 6 Optimization
6.1
Motivation and Objectives
6.1.1
Nonlinear Regression
6.1.2
Electrical Load Design
6.1.3
Container Design
6.1.4
Chapter Objectives
6.2
Local and Global Minima
6.3
Line Searches
6.3.1
Golden Section
6.3.2
Derivative Bisection
6.3.3
Inverse Parabolic Interpolation
6.4
Steepest Descent Method
6.5
Conjugate-Gradient Method
6.6
Quasi-Newton Methods
6.7
Penalty Functions
6.8
Simulated Annealing
6.8.1
Annealing Schedules
6.8.2
Constrained Optimization
6.9
Applications
6.9.1
Heat Exchanger: MATLAB
6.9.2
Transportation Planning: C
6.9.3
Maximum Power Extraction: MATLAB
6.9.4
Container Design: C
6.10
Summary
Problems
6.11.1
Analysis
6.11.2
Computation
CHAPTER 7 Differentiation and Integration
7.1
Motivation and Objectives
7.1.1
Magnetic Levitation
7.1.2
Mechanical Work
7.1.3
Water Management
7.1.4
Chapter Objectives
7.2
Numerical Differentiation
7.2.1
First Derivative
7.2.2
Second Derivative
7.2.3
Richardson Extrapolation
7.3
Noise-Corrupted Data
7.4
Newton-Cotes Integration Formulas
7.4.1
Trapezoid Rule
7.4.2
Simpson''s Rules
7.4.3
Midpoint Rule
7.5
Romberg Integration
7.6
Gauss Quadrature
7.6.1
Legendre Polynomials
7.6.2
Chebyshev Polynomials
7.6.3
Laguerre Polynomials
7.6.4
Hermite Polynomials
7.7
Improper Integrals
7.8
Multiple Integrals
7.8.1
Parameterization Method
7.8.2
Monte Carlo Integration
7.9
Applications
7.9.1
Change in Enthalpy: C
7.9.2
Dam Design: MATLAB
7.9.3
RC Network: C
7.9.4
Link of Robotic Arm: MATLAB
7.10
Summary
Problems
7.11.1
Analysis
7.11.2
Computation
CHAPTER 8 Ordinary Differential Equations
8.1
Motivation and Objectives
8.1.1
Satellite Attitude Control
8.1.2
Pendulum
8.1.3
Predator-Prey Ecological System
8.1.4
Chapter Objectives
8.2
Euler''s Method
8.3
Runge-Kutta Methods
8.4
Step Size Control
8.4.1
Interval Halving
8.4.2
Runge-Kutta-Fehlberg Method
8.4.3
Step Size Adjustment
8.5
MultiStep Methods
8.5.1
Adams-Bashforth Predictor
8.5.2
Adams-Moulton Corrector
8.6
Bulirsch-Stoer Extrapolation Methods
8.6.1
Modified Midpoint Method
8.6.2
Richardson Extrapolation
8.7
Stiff Differential Equations
8.7.1
Implicit Methods
8.7.2
Semi-Implicit Extrapolation Method
8.7.3
Differential-Algebraic Systems
8.8
Boundary Value Problems
8.8.1
Shooting Method
8.8.2
Finite Difference Method
8.9
Applications
8.9.1
Chemical Reactor: MATLAB
8.9.2
Cantilever Beam: C
8.9.3
Phase-Locked Loop: MATLAB
8.9.4
Turbulent Flow and Chaos: C
8.10
Summary
Problems
8.11.1
Analysis
8.11.2
Computation
CHAPTER 9 Partial Differential Equations
9.1
Motivation and Objectives
9.1.1
Laplace''s Equation
9.1.2
Heat Equation
9.1.3
Wave Equation
9.1.4
Equation Classification
9.1.5
Chapter Objectives
9.2
Elliptic Equations
9.2.1
Central Difference Method
9.2.2
Boundary Conditions
9.2.3
Iterative Solution Methods
9.3
One-Dimensional Parabolic Equations
9.3.1
Explicit Forward Euler Method
9.3.2
Implicit Backward Euler Method
9.3.3
Crank-Nicolson Method
9.4
Two-Dimensional Parabolic Equations
9.5
One-Dimensional Hyperbolic Equations
9.5.1
d''Alembert''s Solution
9.5.2
Explicit Central Difference Method
9.6
Two-Dimensional Hyperbolic Equations
9.7
Applications
9.7.1
Heated Rod: C
9.7.2
Plate Deflection: MATLAB
9.7.3
Electrostatic Field: C
9.7.4
Twisted Bar: MATLAB
9.8
Summary
Problems
9.9.1
Analysis
9.9.2
Computation
CHAPTER 10 Digital Signal Processing
10.1
Motivation and Objectives
10.1.1
Harmonic Distortion
10.1.2
Radar
10.1.3
Chapter Objectives
10.2
Fourier Transform
10.3
Fast Fourier Transform FFT
10.4
Correlation
10.5
Convolution
10.5.1
Pulse Response
10.5.2
Stability
10.6
Digital Filters
10.6.1
Frequency Response
10.6.2
FIR Filter Design
10.7
Two-Dimensional FFT
10.8
System Identification
10.8.1
Least-Squares Method
10.8.2
Adaptive LMS Method
10.9
Applications
10.9.1
Heat Exchanger Frequency Response: MATLAB
10.9.2
Flagpole Motion: C
10.9.3
Band Pass Filter: MATLAB
10.9.4
Helicopter Noise: C
10.10
Summary
Problems
10.11.1
Analysis
10.11.2
Computation
References and Further Reading
APPENDIX 1 NUB Using MATLAB
1.1
A Numerical Toolbox: NLIB
1.1.1
Toolbox Installation
1.1.2
NLIB Example Browser
1.2
Main-Program Support
1.2.1
Tabular Display
1.2.2
Graphical Display
1.2.3
Utility Functions
1.3
Linear Algebraic Systems
1.4
Eigenvalues and Eigenvectors
1.5
Curve Fitting
1.6
Root Finding
1.7
Optimization
1.8
Differentiation and Integration
1.9
Ordinary Differential Equations
1.10
Partial Differential Equations
1.11
Digital Signal Processing
APPENDIX 2 NUB Using C
2.1
A Numerical Library: NLIB
2.1.1
NLIB Installation
2.1.2
Library Usage
2.1.3
NLIB Example Browser
2.2
NLIB Data Types
2.2.1
Scalars
2.2.2
Vectors
2.2.3
Matrices
2.2.4
Precision
2.3
NLIB Core
2.3.1
Vector and Matrix Allocation
2.3.2
Vector and Matrix Input/Output
2.3.3
Matrix Algebra
2.3.4
Complex Arithmetic
2.3.5
Random Number Generation
2.3.6
Utility Functions
2.4
Tabular Display
2.4.1
Screen
2.4.2
Keyboard
2.4.3
Printer
2.5
Graphical Display
2.5.1
Curves
2.5.2
Surfaces
2.6
Linear Algebraic Systems
2.7
Eigenvalues and Eigenvectors
2.8
Curve Fitting
2.9
Root Finding
2.10
Optimization
2.11
Differentiation and Integration
2.12
Ordinary Differential Equations
2.13
Partial Differential Equations
2.14
Digital Signal Processing
APPENDIX 3 Vectors and Matrices
3.1
Vector and Matrix Notation
3.2
Basic Operations
3.3
Matrix Inverse
3.4
Eigenvalues and Eigenvectors
3.5
Vector Norms
APPENDIX 4 Answers to Selected Problems
Index