Computational Methods in Physics: Compendium for Students

Preface
	Updates in This Edition
Acknowledgments
Contents
1 Basics of Numerical Analysis
	1.1 Introduction
		1.1.1 Finite-Precision Arithmetic
	1.2 Approximation of Expressions
		1.2.1 Optimal (Minimax) and Almost Optimal Approximations
		1.2.2 Rational (Padé) Approximation
	1.3 Power and Asymptotic Expansion, Asymptotic Analysis
		1.3.1 Power Expansion
		1.3.2 Asymptotic Expansion
		1.3.3 Asymptotic Analysis of Integrals by Integration by Parts
		1.3.4 Asymptotic Analysis of Integrals by the Laplace Method
		1.3.5 Stationary-Phase Approximation
		1.3.6 Differential Equations with Large Parameters
	1.4 Summation of Finite and Infinite Series
		1.4.1 Tests of Convergence
		1.4.2 Summation of Series in Floating-Point Arithmetic
		1.4.3 Acceleration of Convergence
		1.4.4 Alternating Series
		1.4.5 Levin's Transformations
		1.4.6 Poisson Summation
		1.4.7 Borel Summation
		1.4.8 Abel Summation
	1.5 Series Reversion
	1.6 Problems
		1.6.1 Integral of the Normal Distribution
		1.6.2 Airy Functions
		1.6.3 Bessel Functions
		1.6.4 Alternating Series
		1.6.5 Coulomb Scattering Amplitude and Borel Resummation
	References
2 Solving Non-linear Equations
	2.1 Scalar Equations
		2.1.1 Bisection
		2.1.2 The Family of Newton's Methods and the Newton–Raphson Method
		2.1.3 The Secant Method and Its Relatives
		2.1.4 Müller's Method
	2.2 Vector Equations
		2.2.1 Newton–Raphson's Method
		2.2.2 Broyden's (Secant) Method
	2.3 Convergence Acceleration
	2.4 Polynomial Equations of a Single Variable
		2.4.1 Locating the Regions Containing Zeros
		2.4.2 Descartes' Rule and the Sturm's Method
		2.4.3 Newton's Sums and Vièto's Formulas
		2.4.4 Eliminating Multiple Zeros of the Polynomial
		2.4.5 Conditioning of the Computation of Zeros
		2.4.6 General Hints for the Computation of Zeros
		2.4.7 The Hubbard–Schleicher–Sutherland Method
	2.5 Algebraic Equations of Several Variables
	2.6 Problems
		2.6.1 Wien's Law and Lambert's Function
		2.6.2 Heisenberg's Model in the Mean-Field Approximation
		2.6.3 Energy Levels of Simple One-Dimensional Quantum Systems
		2.6.4 Propane Combustion in Air
		2.6.5 Fluid Flow Through Systems of Pipes
		2.6.6 Automated Assembly of Structures
	References
3 Numerical Integration and Differentiation
	3.1 Integration Over Finite Intervals
		3.1.1 Trapezoidal Rule
		3.1.2 Exponential Convergence of the Trapezoidal Rule
		3.1.3 Simpson's Rule
		3.1.4 Romberg Integration
		3.1.5 Gaussian Quadrature
		3.1.6 Clenshaw–Curtis Quadrature
		3.1.7 Gauss–Kronrod Quadrature
		3.1.8 Contour Integrals
	3.2 Integrals Over (Semi-)infinite Intervals
		3.2.1 Trapezoidal Rule
		3.2.2 Variable Transformation: Double-Exponential Methods
		3.2.3 Gaussian Quadrature
	3.3 Computation of Principal Values
		3.3.1 Finite Interval
		3.3.2 Semi-infinite and Infinite Domains
	3.4 Integration of Rapidly Oscillating Functions
		3.4.1 Asymptotic Method
		3.4.2 Filon's Method
		3.4.3 Double-Exponential Methods for Fourier-Type Integrals
		3.4.4 A Short Tale of Long Tail Integration
	3.5 Quadrature in Two or More Dimensions
		3.5.1 Integration Over a Square
		3.5.2 Software Resources
	3.6 Stable Numerical Differentiation
		3.6.1 Regularized Finite Differences
		3.6.2 Differentiation by Using Smoothing Kernels
		3.6.3 Variational Regularization of the Derivative
	3.7 Problems
		3.7.1 Hyperbolic Volume
		3.7.2 A Parametric Integral
		3.7.3 Differentiation of a Noisy Function
	References
4 Matrix Methods
	4.1 Basic Operations
		4.1.1 Matrix Multiplication
		4.1.2 Computing the Determinant
	4.2 Systems of Linear Equations
		4.2.1 Analysis of Errors
		4.2.2 Gauss Elimination
		4.2.3 Systems with Banded Matrices
		4.2.4 Toeplitz Systems
		4.2.5 Vandermonde Systems
		4.2.6 Condition Estimates for Matrix Inversion
	4.3 Solving bold italic upper A bold italic x bold equals bold italic bAx=b with sparse matrices
		4.3.1 Direct Methods
		4.3.2 Iterative Methods Based on Krylov Subspaces
		4.3.3 Preconditioning in Projection Methods
	4.4 Solving Matrix Equations
		4.4.1 Sylvester Equations
		4.4.2 Lyapunov Equations
	4.5 Linear Least-Squares Problem and Orthogonalization
		4.5.1 The bold italic upper Q bold italic upper RQR decomposition
		4.5.2 Singular Value Decomposition (SVD)
		4.5.3 The Minimum-Norm Solution of the Least-Squares Problem
	4.6 Eigenvalue Problems
		4.6.1 Non-symmetric Problems
		4.6.2 The Power Method and Inverse Iteration
		4.6.3 Symmetric Problems
		4.6.4 Generalized Eigenvalue Problems
		4.6.5 The Quadratic Eigenvalue Problem
		4.6.6 Converting a Matrix to Its Jordan Form
	4.7 Eigenvalue Problems with Sparse Matrices
		4.7.1 Arnoldi's Method
		4.7.2 Hermitian and Non-Hermitian Lanczos Algorithm
		4.7.3 Preconditioning and Filtering
	4.8 Pseudospectra of Matrices
		4.8.1 Definition of Pseudospectrum
		4.8.2 Pseudospectra of Linear Operators
	4.9 Random Matrices
		4.9.1 General Random Matrices
		4.9.2 Gaussian Orthogonal or Unitary Ensemble
		4.9.3 Circular Ensembles
	4.10 Problems
		4.10.1 Percolation in a Random-Lattice Model
		4.10.2 Electric Circuits of Linear Elements
		4.10.3 Systems of Oscillators
		4.10.4 Image Compression by Singular Value Decomposition
		4.10.5 Eigenstates of Particles in the Anharmonic Potential
		4.10.6 Anderson Localization
		4.10.7 Spectra of Random Symmetric Matrices
	References
5 Transformations of Functions and Signals
	5.1 Fourier Transformation
	5.2 Fourier Series
		5.2.1 Continuous Fourier Expansion
		5.2.2 Discrete Fourier Expansion
		5.2.3 Aliasing
		5.2.4 Leakage
		5.2.5 Fast Discrete Fourier Transformation (FFT)
		5.2.6 Multiplication of Polynomials by Using the FFT
		5.2.7 Power Spectral Density
		5.2.8 Sparse FFT
		5.2.9 Non-uniform (Non-equispaced) FFT
	5.3 Transformations with Orthogonal Polynomials
		5.3.1 Legendre Polynomials
		5.3.2 Chebyshev Polynomials
		5.3.3 Laguerre Polynomials and Basis Functions
		5.3.4 Rational Chebyshev Functions MathID656TL
		5.3.5 Hermite Polynomials and Basis Functions
		5.3.6 Rational Chebyshev Functions MathID719TB
	5.4 Laplace Transformation
		5.4.1 Use of Laplace Transformation with Differential Equations
	5.5 Hilbert Transformation
		5.5.1 Analytic Signal
		5.5.2 Kramers–Kronig Relations
		5.5.3 Numerical Computation of the Continuous Hilbert Transform
		5.5.4 Discrete Hilbert Transformation
	5.6 Continuous Wavelet Transformation
		5.6.1 Numerical Computation of the Wavelet Transform
	5.7 Discrete Wavelet Transformation
		5.7.1 One-Dimensional DWT
		5.7.2 Two-Dimensional DWT
	5.8 Problems
		5.8.1 Fourier Spectrum of Signals
		5.8.2 Fourier Analysis of the Doppler Effect
		5.8.3 Use of Laplace Transformation and Its Inverse
		5.8.4 Use of the Wavelet Transformation
	References
6 Statistical Analysis and Modeling of Data
	6.1 Basic Data Analysis
		6.1.1 Probability Distributions
		6.1.2 Moments of Distributions
		6.1.3 Uncertainties of Moments of Distributions
	6.2 Robust Statistics
		6.2.1 Hunting for Outliers
		6.2.2 bold italic upper MM-Estimates of Location
		6.2.3 bold italic upper MM-Estimates of Scale
	6.3 Statistical Tests
		6.3.1 Computing the Confidence Interval for the Population Mean
		6.3.2 Comparing the Means of Two Samples with Equal Variances
		6.3.3 Comparing the Means of Two Samples with Different Variances
		6.3.4 Determining the Confidence Interval for the Population Variance
		6.3.5 Comparing Two Sample Variances
		6.3.6 Comparing Histogrammed Data to a Known Distribution
		6.3.7 Comparing Two Sets of Histogrammed Data
		6.3.8 Kolmogorov–Smirnov Test
	6.4 Correlation
		6.4.1 Linear Correlation
		6.4.2 Non-parametric Correlation
	6.5 Linear Regression
		6.5.1 Fitting by a Polynomial, Straight Line, or Constant
		6.5.2 Generalized Linear Regression by Using SVD
		6.5.3 Robust Methods for One-Dimensional Regression
	6.6 Non-linear Regression
	6.7 Multiple Linear Regression
		6.7.1 The Basic Method
		6.7.2 Principal Component Multiple Regression
	6.8 Principal Component Analysis
		6.8.1 Principal Components by Diagonalizing the Covariance Matrix
		6.8.2 Standardization of Data for PCA
		6.8.3 Principal Components from the SVD of the Data Matrix
		6.8.4 Improvements of PCA: Non-linearity, Robustness
	6.9 Cluster Analysis
		6.9.1 Hierarchical Clustering
		6.9.2 Partitioning Methods: kk-Means
		6.9.3 Gaussian Mixture Clustering and the EM Algorithm
		6.9.4 Spectral Methods
	6.10 Linear Discriminant Analysis
		6.10.1 Binary Classification
		6.10.2 Logistic Discriminant Analysis
		6.10.3 Assignment to Multiple Classes
	6.11 Canonical Correlation Analysis
	6.12 Factor Analysis
		6.12.1 Determining the Factors and Weights from the Covariance Matrix
		6.12.2 Standardization of Data and Robust Factor Analysis
	6.13 Problems
		6.13.1 Multiple Regression
		6.13.2 Nutritional Value of Food
		6.13.3 Discrimination of Radar Signals from Ionospheric Reflections
		6.13.4 Canonical Correlation Analysis of Objects in the CDFS Area
	References
7 Modeling and Analysis of Time Series
	7.1 Random Variables
		7.1.1 Basic Definitions
		7.1.2 Generation of Random Numbers
	7.2 Random Processes
		7.2.1 Basic Definitions
	7.3 Stable Distributions and Random Walks
		7.3.1 Central Limit Theorem
		7.3.2 Stable Distributions
		7.3.3 Generalized Central Limit Theorem
		7.3.4 Discrete-Time Random Walks
		7.3.5 Continuous-Time Random Walks
	7.4 Markov Chains
		7.4.1 Discrete-Time or Classical Markov Chains
		7.4.2 Continuous-Time Markov Chains
	7.5 Noise
		7.5.1 Types of Noise
		7.5.2 Generation of Noise
	7.6 Time Correlation and Auto-Correlation
		7.6.1 Sample Correlations of Signals
		7.6.2 Representation of Time Correlations
		7.6.3 Fast Computation of Discrete Sample Correlations
	7.7 Auto-Regressive Analysis of Discrete-Time Signals
		7.7.1 Auto-Regressive (AR) Model
		7.7.2 Application of AR Models
		7.7.3 Estimate of the Fourier Spectrum
	7.8 Optimal Filtering
		7.8.1 Linear Kalman Filter
		7.8.2 Kalman Predictor
		7.8.3 (Fixed-Interval) Kalman Smoother
		7.8.4 Extended Kalman Filter
		7.8.5 Unscented Kalman Filter
		7.8.6 Adaptive Non-linear Estimation
	7.9 Independent Component Analysis
		7.9.1 Estimate of the Separation Matrix and the FastICA Algorithm
	7.10 State-Space Reconstruction
		7.10.1 Establishing the Optimal Time Delay
		7.10.2 Determining the Embedding Dimension
	7.11 Problems
		7.11.1 Logistic Map
		7.11.2 Diffusion and Chaos in the Standard Map
		7.11.3 Phase Transitions in the Two-Dimensional Ising Model
		7.11.4 Reconstruction of Vehicle Position and Velocity
		7.11.5 Independent Component Analysis
	References
8 Initial-Value Problems for ODE
	8.1 Evolution Equations
	8.2 Explicit Euler's Methods
	8.3 Explicit Methods of the Runge–Kutta Type
	8.4 Errors of Explicit Methods
		8.4.1 Discretization and Round-Off Errors
		8.4.2 Consistency, Convergence, Stability
		8.4.3 Richardson Extrapolation
		8.4.4 Embedded Methods
		8.4.5 Automatic Step Size Control
	8.5 Stability of Explicit Single-Step Methods
	8.6 Extrapolation Methods
	8.7 Conservative Second-Order Equations
		8.7.1 Runge–Kutta–Nyström Methods
	8.8 Implicit Single-Step Methods
		8.8.1 Solution by Newton's Iteration
		8.8.2 Rosenbrock Linearization
	8.9 Multi-step Methods
		8.9.1 Predictor-Corrector Methods
		8.9.2 Stability of Multi-step Methods
		8.9.3 Backward Differentiation Methods
		8.9.4 Multi-step Methods for Second-Order Conservative Equations
		8.9.5 Integration of Gravitational Many-Body Problems
	8.10 Stiff Problems
		8.10.1 Eigenvalue Characterization of Stiffness
		8.10.2 Stiffness and Pseudospectra
		8.10.3 Automatic Stiffness Detection
		8.10.4 Implicit Multi-step Methods for Stiff Problems
	8.11 Geometric Integration
		8.11.1 Preservation of Invariants
		8.11.2 Preservation of the Symplectic Structure
		8.11.3 Reversibility and Symmetry
		8.11.4 Modified Hamiltonians and Equations of Motion
	8.12 Lie-Series Integration
		8.12.1 Taylor Expansion of the Trajectory
	8.13 Problems
		8.13.1 Time Dependence of Filament Temperature
		8.13.2 Oblique Projectile Motion with Drag Force and Wind
		8.13.3 Influence of Fossil Fuels on Atmospheric bold upper C upper O bold 2CO2 Content
		8.13.4 Synchronization of Globally Coupled Oscillators
		8.13.5 Excitation of Muscle Fibers
		8.13.6 Restricted Three-Body Problem (Arenstorf Orbits)
		8.13.7 Lorenz System
		8.13.8 Sine Pendulum
		8.13.9 Charged Particles in Electric and Magnetic Fields
		8.13.10 Chaotic Scattering
		8.13.11 Hydrogen Burning in the pp I Chain
		8.13.12 Oregonator
		8.13.13 Kepler's Problem
		8.13.14 Northern Lights
		8.13.15 Galactic Dynamics
	References
9 Boundary-Value Problems for ODE
	9.1 Difference Methods for Scalar Boundary-Value Problems
	9.2 Difference Methods for Systems of Boundary-Value Problems
		9.2.1 Linear Systems
		9.2.2 Schemes of Higher Orders
	9.3 Shooting Methods
		9.3.1 Second-Order Linear Equations
		9.3.2 Systems of Linear Second-Order Equations
		9.3.3 Non-linear Second-Order Equations
		9.3.4 Systems of Non-linear Equations
		9.3.5 Multiple (Parallel) Shooting
	9.4 Asymptotic Discretization Schemes
		9.4.1 Discretization
	9.5 Collocation Methods
		9.5.1 Scalar Linear Second-Order Boundary-Value Problems
		9.5.2 Scalar Linear Boundary-Value Problems of Higher Orders
		9.5.3 Scalar Non-linear Boundary-Value Problems of Higher Orders
		9.5.4 Systems of Boundary-Value Problems
	9.6 Weighted-Residual Methods
	9.7 Regular Boundary-Value Problems with Eigenvalues
		9.7.1 Transformation to Liouville Normal Form
		9.7.2 Properties of Eigenvalues
		9.7.3 Properties of Eigenfunctions
		9.7.4 Solution by Difference Methods
		9.7.5 Shooting Methods with Prüfer Transformation
		9.7.6 Pruess Method
		9.7.7 Eigenvalue-Dependent Boundary Conditions
	9.8 Singular Sturm–Liouville Problems
		9.8.1 The ``Shapes'' of Spectra
		9.8.2 Classification of Singular Endpoints
		9.8.3 Titchmarsh–Weyl mm-Function and Spectral Density
		9.8.4 Resonances
		9.8.5 Available Software
	9.9 Problems
		9.9.1 Gelfand–Bratu Equation
		9.9.2 Measles Epidemic
		9.9.3 Diffusion-Reaction Kinetics in a Catalytic Pellet
		9.9.4 Deflection of a Beam with Inhomogeneous Elastic Modulus
		9.9.5 A Boundary-Layer Problem
		9.9.6 Small Oscillations of an Inhomogeneous String
		9.9.7 One-Dimensional Schrödinger Equation
		9.9.8 A Fourth-Order Eigenvalue Problem
		9.9.9 Harmonic Oscillator with a Complex Potential
	References
10 Difference Methods for One-Dimensional PDE
	10.1 Discretization of the Differential Equation
	10.2 Discretization of Initial and Boundary Conditions
	10.3 Consistency
	10.4 Implicit Schemes
	10.5 Stability and Convergence
		10.5.1 Initial-Value Problems
		10.5.2 Initial-Boundary-Value Problems
	10.6 Higher Order Schemes
	10.7 Hyperbolic Equations
		10.7.1 Explicit Schemes
		10.7.2 Implicit Schemes
		10.7.3 Wave Equation
	10.8 Non-linear Equations and Equations of Mixed Type
	10.9 Dispersion and Dissipation
	10.10 Systems of Linear Hyperbolic and Parabolic PDE
	10.11 Conservation Laws and High-Resolution Schemes
		10.11.1 High-Resolution Schemes: TVD Versus ENO
		10.11.2 High-Resolution Flux-Limiter Schemes
		10.11.3 High-Resolution Slope-Limiter Schemes
		10.11.4 High-Order Essentially Non-oscillatory (ENO) Schemes
	10.12 Problems
		10.12.1 Diffusion Equation
		10.12.2 Initial-Boundary Value Problem for v Subscript t Baseline plus c v Subscript x Baseline equals 0vt+cvx=0
		10.12.3 Dirichlet Problem for a System of Non-linear Hyperbolic PDE
		10.12.4 Second-Order and Fourth-Order Wave Equations
		10.12.5 Burgers Equation
		10.12.6 Buckley–Leverett Equation
		10.12.7 The Shock-Tube Problem
		10.12.8 Korteweg–de Vries Equation
		10.12.9 Non-stationary Schrödinger Equation
		10.12.10 Non-stationary Cubic Schrödinger Equation
	References
11 Difference Methods for PDE in Two or More Dimensions
	11.1 Parabolic and Hyperbolic PDE
		11.1.1 Parabolic Equations
		11.1.2 Explicit Scheme
		11.1.3 Crank–Nicolson Scheme
		11.1.4 Alternating Direction Implicit Schemes
		11.1.5 Three Space Dimensions
		11.1.6 Hyperbolic Equations
		11.1.7 Explicit Schemes
		11.1.8 Schemes for Equations in the Form of Conservation Laws
		11.1.9 Implicit and ADI Schemes
	11.2 Elliptic PDE
		11.2.1 Dirichlet Boundary Conditions
		11.2.2 Neumann Boundary Conditions
		11.2.3 Mixed Boundary Conditions
		11.2.4 Iterative Solution Methods: Relaxation
		11.2.5 Iterative Solution Methods: Conjugate Gradients
	11.3 High-Resolution Schemes
		11.3.1 Two Basic Schemes
		11.3.2 Zalesak–Smolarkiewicz Scheme
		11.3.3 Kurganov–Tadmor Scheme
	11.4 Physically Motivated Discretizations
		11.4.1 Two-Dimensional Diffusion Equation in Polar Coordinates
		11.4.2 Two-Dimensional Poisson Equation in Polar Coordinates
	11.5 Boundary Element Method
	11.6 Finite Element Method
		11.6.1 One Space Dimension
		11.6.2 Two Space Dimensions
	11.7 Mimetic Discretizations
	11.8 Multi-grid and Mesh-Free Methods
		11.8.1 A Mesh-Free Method Based on Radial Basis Functions
	11.9 Problems
		11.9.1 Two-Dimensional Diffusion Equation
		11.9.2 Non-linear Diffusion Equation
		11.9.3 Two-Dimensional Poisson Equation
		11.9.4 High-Resolution Schemes for the Advection Equation
		11.9.5 Two-Dimensional Buckley–Leverett Equation
		11.9.6 Two-Dimensional Diffusion Equation in Polar Coordinates
		11.9.7 Two-Dimensional Poisson Equation in Polar Coordinates
		11.9.8 Finite Element Method
		11.9.9 Boundary Element Method for the Two-Dimensional Laplace Equation
		11.9.10 Boundary Element Method for Potential Flow
	References
12 Spectral Methods for ODE and PDE
	12.1 Spectral Representation of Spatial Derivatives
		12.1.1 Fourier Spectral Derivatives
		12.1.2 Legendre Spectral Derivatives
		12.1.3 Chebyshev Spectral Derivatives
		12.1.4 Computing the Chebyshev Spectral Derivative by Fourier Transformation
		12.1.5 Laguerre Spectral Derivatives
		12.1.6 Hermite Spectral Derivatives
	12.2 Galerkin Methods
		12.2.1 Fourier–Galerkin
		12.2.2 Legendre–Galerkin
		12.2.3 Chebyshev–Galerkin
		12.2.4 Two Space Dimensions
		12.2.5 Non-stationary Problems
	12.3 Tau Methods
		12.3.1 Stationary Problems
		12.3.2 Non-stationary Problems
	12.4 Collocation Methods
		12.4.1 Stationary Problems
		12.4.2 Non-stationary Problems
		12.4.3 Spectral Elements: Collocation with upper BB-Splines
	12.5 Non-linear Equations
	12.6 Time Integration
		12.6.1 Strong Stability Preserving Methods
	12.7 Semi-infinite and Infinite Domains
		12.7.1 Domain Truncation
		12.7.2 Expansions in Terms of Laguerre Functions on left bracket 0 comma normal infinity right bracket[0,infty]
		12.7.3 Coordinate Mappings for left bracket a comma b right bracket left right arrow left bracket 0 comma normal infinity right bracket[a,b][0,infty]
		12.7.4 Eigenvalue Problems on left bracket 0 comma normal infinity right bracket[0,infty]
		12.7.5 Expansions in Terms of Hermite Functions on left bracket negative normal infinity comma normal infinity right bracket[-infty,infty]
		12.7.6 Coordinate Mappings for left bracket a comma b right bracket left right arrow left bracket negative normal infinity comma normal infinity right bracket[a,b][-infty,infty]
		12.7.7 Eigenvalue Problems on left bracket negative normal infinity comma normal infinity right bracket[-infty,infty]
	12.8 Complex Geometries
	12.9 Problems
		12.9.1 Galerkin Methods for the Helmholtz Equation
		12.9.2 Galerkin Methods for the Advection Equation
		12.9.3 Galerkin Method for the Diffusion Equation
		12.9.4 Galerkin Method for the Poisson Equation: Poiseuille Law
		12.9.5 Legendre Tau Method for the Poisson Equation
		12.9.6 Collocation Methods for the Diffusion Equation I
		12.9.7 Collocation Methods for the Diffusion Equation II
		12.9.8 Burgers Equation
	References
13 Inverse and Ill-Posed Problems
	13.1 Classification of Inverse Problems
	13.2 Generalized Solutions of upper A u equals fAu=f
		13.2.1 Compact Operators and Their Singular Value Decomposition
		13.2.2 The Pseudoinverse of Compact Operators
	13.3 Regularization of Linear Inverse Problems
		13.3.1 Truncated Singular Value Decomposition
		13.3.2 Tikhonov Regularization
		13.3.3 Landweber Iteration
		13.3.4 Conjugate Gradients Method Applied to the Normal Equation
		13.3.5 Variational Regularization
		13.3.6 Regularization of Fourier-Transform-Based Deconvolution
		13.3.7 Determination of the Optimal Regularization Parameter
		13.3.8 Regularization of Non-linear Inverse Problems
	13.4 Solution of Integral Equations
		13.4.1 Fredholm Equations of the Second Kind
		13.4.2 Volterra Equations of the Second Kind
		13.4.3 Fredholm Equations of the First Kind
		13.4.4 Volterra Equations of the First Kind, Smooth Kernels
		13.4.5 Volterra Equations of the First Kind, Unbounded Kernels
		13.4.6 The Radon Transformation and Its Inverse
	13.5 Inverse Sturm–Liouville Problems
		13.5.1 Information Needed to Recover q left parenthesis x right parenthesisq(x)
		13.5.2 The Gelfand–Levitan Method
		13.5.3 Successive Approximations
		13.5.4 Quasi-Newton Method
		13.5.5 Shooting Method
	13.6 Inverse Problems for Partial Differential Equations
		13.6.1 Retrospective Inverse Problem for the Heat Equation
		13.6.2 Reconstructing the Source Term in the Heat Equation
		13.6.3 Inverse Source Problems for the Wave Equation
		13.6.4 Recovering the Potential in the String Equation
		13.6.5 Inverse Scattering Problem
	13.7 Phase Retrieval
		13.7.1 Alternating-Projection Algorithms
		13.7.2 Semidefinite Programming Algorithms
	13.8 Problems
		13.8.1 Image Deblurring
		13.8.2 Gravitational Pull of Circularly Distributed Mass
		13.8.3 Polymer Sedimentation in a Centrifuge
		13.8.4 Discrete Radon Transformation and Its Inverse
		13.8.5 Reconstruction of the Potential from Two Sturm–Liouville Spectra
		13.8.6 Identifying the Source Term in the Heat Equation
		13.8.7 Reconstructing the Scatterer from the Far-Field Pattern
	References
Appendix A Mathematical Tools
A.1 Asymptotic Notation
A.2 The Norms in Spaces upper L Superscript p Baseline left parenthesis normal upper Omega right parenthesisLp(Ω) and upper L Subscript w Superscript p Baseline left parenthesis normal upper Omega right parenthesisLwp(Ω), 1 less than or equals p less than or equals normal infinity1lepleinfty
A.3 Discrete Vector Norms
A.4 Matrix and Operator Norms
A.5 Eigenvalues of Tridiagonal Matrices
A.6 Singular Values of upper XX and Eigenvalues of upper X Superscript normal upper T Baseline upper XXTX and upper X upper X Superscript normal upper TXXT
A.7 The Generalized (Moore–Penrose) Inverse of a Matrix
A.8 The ``Square Root'' of a Matrix
A.9 The Matrix Exponential
A.10 Computation of Arbitrary Matrix Functions
Appendix B Standard Numerical Data Types
B.1 Real Numbers in Floating-Point Arithmetic
B.2 Integer Numbers
B.3 (Almost) Arbitrary Precision
Appendix C Generation of Pseudorandom Numbers
C.1 Uniform Generators: From Integers to Reals
C.2 Transformations Between Distributions
C.2.1  Discrete Distribution
C.2.2  Continuous Distribution
C.3 Using and Testing Random Number Generators
C.4 Random Permutations
C.4.1 Recursive Generation of Permutations
C.4.2 Stochastic Generation of Permutations
Appendix D Dual Unscented Kalman Filter
Appendix E Computation of Poincaré Maps
Appendix F Fixed Points and Stability
F.1 Linear Stability
F.2 Spurious Fixed Points
F.3 Non-linear Stability
Appendix G Regularization in Orbital Mechanics
G.1 Burdet–Heggie Regularization
G.2 Kustaanheimo–Stiefel Regularization
G.3 Algorithmic Regularization
Appendix H Construction of Symplectic Integrators
Appendix I Transforming PDEs to Systems of ODEs
I.1 Diffusion Equation
I.2 Advection Equation
Appendix J Numerical Libraries, Auxiliary Tools, and Languages
J.1 Important Numerical Libraries
J.2 Choosing the Programming Language
Index