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1.

図書
図書
1. Numerical methods for bifurcation problems and large-scale dynamical systems
Eusebius Doedel, Laurette S. Tuckerman, editors
出版情報: New York : Springer, c2000  x, 471 p. ; 25 cm
シリーズ名: The IMA volumes in mathematics and its applications ; v. 119
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2.

図書
図書
2. Geometric numerical integration : structure-preserving algorithms for ordinary differential equations
Ernst Hairer, Christian Lubich, Gerhard Wanner
出版情報: Berlin ; Tokyo : Springer, c2002  xiii, 515 p. ; 24 cm
シリーズ名: Springer series in computational mathematics ; v. 31
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3.

図書
図書
3. Numerical methods for ordinary differential equations
J.C. Butcher
出版情報: Chichester : Wiley, c2003  xiv, 425 p. ; 24 cm
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目次情報: 続きを見る
Preface
Differential and Difference Equations / 1：
Differential Equation Problems / 10：
Differential Equation Theory / 11：
Difference Equation Problems / 100：
Introduction to differential equations
Difference Equation Theory / 13：
The Kepler problem / 2：
Numerical Differential Equation Methods
The Euler Method / 102：
Many-body gravitational problems
Analysis of the Euler Method / 21：
A problem arising from the method of lines / 22：
Generalizations of the Euler Method
Runge-Kutta Methods / 104：
The simple pendulum
Linear Multistep Methods / 24：
A chemical kinetics problem / 25：
Taylor Series Methods
Hybrid Methods / 106：
The van der Pol equation and limit cycles
The Lotka-Volterra problem and periodic orbits / 3：
Preliminaries
Order Conditions / 31：
Low Order Explicit Methods / 32：
Existence and uniqueness of solutions / 33：
Runge-Kutta Methods with Error Estimates
Implicit Runge-Kutta Methods / 111：
Linear systems of differential equations
Stability of Implicit Runge-Kutta Methods / 35：
Stiff differential equations / 36：
Implementable Implicit Runge-Kutta Methods
Order Barriers / 113：
Laplace transforms
Algebraic Properties of Runge-Kutta Methods / 38：
Implementation Issues / 39：
Introduction to difference equations / 120：
A linear problem / 40：
The Order of Linear Multistep Methods
Errors and Error Growth / 122：
The Fibonacci difference equation
Stability Characteristics / 43：
Three quadratic problems / 44：
Order and Stability Barriers
One-leg Methods and G-stability / 124：
Iterative solutions of a polynomial equation
The arithmetic-geometric mean / 46：
General Linear Methods
Representing Methods in General Linear Form / 50：
Consistency, Stability and Convergence / 51：
Linear difference equations / 52：
The Stability of General Linear Methods
The Order of General Linear Methods / 131：
Constant coefficients
Methods with Runge-Kutta Stabiulity / 54：
References
Powers of matrices
Index
The Z-transform / 133：
Introduction to the Euler method / 200：
Some numerical experiments / 201：
Calculations with stepsize control / 202：
Calculations with mildly stiff problems / 203：
Calculations with the implicit Euler method / 204：
Formulation of the Euler method / 210：
Local truncation error / 211：
Global truncation error / 212：
Convergence of the Euler method / 213：
Order of convergence / 214：
Asymptotic error formula / 215：
Stability characteristics / 216：
Local truncation error estimation / 217：
Rounding error / 218：
Introduction / 220：
More computations in a step / 221：
Greater dependence on previous values / 222：
Use of higher derivatives / 223：
Multistep-multistage-multiderivative methods / 224：
Implicit methods / 225：
Local error estimates / 226：
Historical introduction / 230：
Second order methods / 231：
The coefficient tableau / 232：
Third order methods / 233：
Introduction to order conditions / 234：
Fourth order methods / 235：
Higher orders / 236：
Implicit Runge-Kutta methods / 237：
Numerical examples / 238：
Adams methods / 240：
General form of linear multistep methods / 242：
Consistency, stability and convergence / 243：
Predictor-corrector Adams methods / 244：
The Milne device / 245：
Starting methods / 246：
Introduction to Taylor series methods / 247：
Manipulation of power series / 251：
An example of a Taylor series solution / 252：
Other methods using higher derivatives / 253：
The use of f derivatives / 254：
Further numerical examples / 255：
Pseudo Runge-Kutta methods / 260：
Generalized linear multistep methods / 262：
General linear methods / 263：
Rooted trees / 264：
Functions on trees / 301：
Some combinatorial questions / 302：
The use of labelled trees / 303：
Differentiation / 304：
Taylor's theorem / 305：
Elementary differentials / 310：
The Taylor expansion of the exact solution / 311：
Elementary weights / 312：
The Taylor expansion of the approximate solution / 313：
Independence of the elementary differentials / 314：
Conditions for order / 315：
Order conditions for scalar problems / 316：
Independence of elementary weights / 317：
Methods of orders less than four / 318：
Simplifying assumptions / 321：
Methods of order four / 322：
New methods from old / 323：
Methods of order five / 324：
Methods of order six / 325：
Methods of orders greater than six / 326：
Richardson error estimates / 330：
Methods with built-in estimates / 332：
A class of error-estimating methods / 333：
The methods of Fehlberg / 334：
The methods of Verner / 335：
The methods of Dormand and Prince / 336：
Solvability of implicit equations / 340：
Methods based on Gaussian quadrature / 342：
Reflected methods / 343：
Methods based on Radau and Lobatto quadrature / 344：
A-stability, A([alpha])-stability and L-stability / 350：
Criteria for A-stability / 351：
Pade approximations to the exponential function / 352：
A-stability of Gauss and related methods / 353：
Order stars / 354：
Order arrows and the Ehle barrier / 355：
AN-stability / 356：
Non-linear stability / 357：
BN-stability of collocation methods / 358：
The V and W transformations / 359：
Implementation of implicit Runge-Kutta methods / 360：
Diagonally-implicit Runge-Kutta methods / 361：
The importance of high stage order / 362：
Singly implicit methods / 363：
Generalizations of singly-implicit methods / 364：
Effective order and DESIRE methods / 365：
Explicit barriers / 370：
An upper bound on the required number of stages / 371：
Motivation / 380：
Equivalence classes of Runge-Kutta methods / 381：
The group of Runge-Kutta methods / 382：
The Runge-Kutta group / 383：
A homomorphism between two groups / 384：
A generalization of G[subscript 1] / 385：
Recursive formula for the product / 386：
Some special elements of G / 387：
Some subgroups and quotient groups / 388：
An algebraic interpretation of effective order / 389：
Optimal sequences / 390：
Acceptance and rejection of steps / 392：
Error per step versus error per unit step / 393：
Control theoretic considerations / 394：
Solving the implicit equations / 395：
Fundamentals / 400：
Convergence / 401：
Stability / 403：
Consistency / 404：
Necessity of conditions for convergence / 405：
Sufficiency of conditions for convergence / 406：
Criteria for order / 410：
Derivation of methods / 411：
Backward difference methods / 412：
Further remarks on error growth / 420：
The underlying one-step method / 422：
Weakly stable methods / 423：
Variable stepsize / 424：
Stability regions / 430：
Examples of the boundary locus method / 432：
Examples of the Schur criterion / 433：
Stability of predictor-corrector methods / 434：
Survey of barrier results / 440：
Maximum order for a convergent k step method / 441：
Order stars for linear multistep methods / 442：
Order arrows for linear multistep methods / 443：
The one-leg counterpart to a linear multistep method / 450：
The concept of G-stability / 451：
Transformations relating one-leg and linear multistep methods / 452：
Effective order interpretation / 453：
Concluding remarks on G-stability / 454：
Survey of implementation considerations / 460：
Representation of data / 461：
Variable stepsize for Nordsieck methods / 462：
Local error estimation / 463：
Multivalue multistage methods / 500：
Transformations of methods / 501：
Runge-Kutta methods as general linear methods / 502：
Linear multistep methods as general linear methods / 503：
Some known unconventional methods / 504：
Some recently discovered general linear methods / 505：
Definitions of consistency and stability / 510：
Covariance of methods / 511：
Definition of convergence / 512：
The necessity of stability / 513：
The necessity of consistency / 514：
Stability and consistency imply convergence / 515：
Methods with maximal stability order / 520：
Possible definitions of order / 530：
Algebraic analysis of order / 531：
An example of the algebraic approach to order / 532：
Methods with Runge-Kutta stability / 533：
Design criteria for general linear methods / 540：
The types of DIMSIM methods / 541：
Runge-Kutta stability / 542：
Almost Runge-Kutta methods / 543：
Fourth order, four stage ARK methods / 544：
Doubly companion matrices / 545：
Inherent Runge-Kutta stability / 546：
Derivation of methods with IRK stability / 547：
Some nonstiff methods / 548：
Some stiff methods / 549：
Preface
Differential and Difference Equations / 1：
Differential Equation Problems / 10：
4.

図書
図書
4. An introduction to frames and Riesz bases
Ole Christensen
出版情報: Boston : Birkhäuser, c2003  xx, 440 p ; 24 cm
シリーズ名: Applied and numerical harmonic analysis / series editor, John J. Benedetto
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目次情報: 続きを見る
Preface
Frames in Finite-dimensional Inner Product Spaces / 1：
Some basic facts about frames / 1.1：
Frame bounds and frame algorithms / 1.2：
Frames in C[superscript n] / 1.3：
The discrete Fourier transform / 1.4：
Pseudo-inverses and the singular value decomposition / 1.5：
Finite-dimensional function spaces / 1.6：
Exercises / 1.7：
Infinite-dimensional Vector Spaces and Sequences / 2：
Sequences / 2.1：
Banach spaces and Hilbert spaces / 2.2：
L[superscript 2] (R) and l[superscript 2] (N) / 2.3：
The Fourier transform / 2.4：
Operators on L[superscript 2] (R) / 2.5：
Bases / 2.6：
Bases in Banach spaces / 3.1：
Bessel sequences in Hilbert spaces / 3.2：
Bases and biorthogonal systems in H / 3.3：
Orthonormal bases / 3.4：
The Gram matrix / 3.5：
Riesz bases / 3.6：
Fourier series and Gabor bases / 3.7：
Wavelet bases / 3.8：
Bases and their Limitations / 3.9：
Gabor systems and the Balian-Low Theorem / 4.1：
Bases and wavelets / 4.2：
General shortcomings / 4.3：
Frames in Hilbert Spaces / 5：
Frames and their properties / 5.1：
Frame sequences / 5.2：
Frames and operators / 5.3：
Frames and bases / 5.4：
Characterization of frames / 5.5：
The dual frames / 5.6：
Tight frames / 5.7：
Continuous frames / 5.8：
Frames and signal processing / 5.9：
Frames versus Riesz Bases / 5.10：
Conditions for a frame being a Riesz basis / 6.1：
Riesz frames and near-Riesz bases / 6.2：
Frames containing a Riesz basis / 6.3：
A frame which does not contain a basis / 6.4：
A moment problem / 6.5：
Exercise / 6.6：
Frames of Translates / 7：
Sequences in R[superscript d] / 7.1：
Frames of translates / 7.2：
Frames of integer-translates / 7.3：
Irregular frames of translates / 7.4：
The sampling problem / 7.5：
Frames of exponentials / 7.6：
Gabor Frames in L[superscript 2] (R) / 7.7：
Continuous representations / 8.1：
Gabor frames / 8.2：
Necessary conditions / 8.3：
Sufficient conditions / 8.4：
The Wiener space W / 8.5：
Special functions / 8.6：
General shift-invariant systems / 8.7：
Selected Topics on Gabor Frames / 8.8：
Popular Gabor conditions / 9.1：
Representations of the Gabor frame operator and duality / 9.2：
The duals of a Gabor frame / 9.3：
The Zak transform / 9.4：
Tight Gabor frames / 9.5：
The lattice parameters / 9.6：
Irregular Gabor systems / 9.7：
Applications of Gabor frames / 9.8：
Wilson bases / 9.9：
Gabor Frames in l[superscript 2] (Z) / 9.10：
Translation and modulation on l[superscript 2] (Z) / 10.1：
Discrete Gabor systems through sampling / 10.2：
Gabor frames in C[superscript L] / 10.3：
Shift-invariant systems / 10.4：
Frames in l[superscript 2] (Z) and filter banks / 10.5：
General Wavelet Frames / 10.6：
The continuous wavelet transform / 11.1：
Sufficient and necessary conditions / 11.2：
Irregular wavelet frames / 11.3：
Oversampling of wavelet frames / 11.4：
Dyadic Wavelet Frames / 11.5：
Wavelet frames and their duals / 12.1：
Tight wavelet frames / 12.2：
Wavelet frame sets / 12.3：
Frames and multiresolution analysis / 12.4：
Frame Multiresolution Analysis / 12.5：
Frame multiresolution analysis / 13.1：
Relaxing the conditions / 13.2：
Construction of frames / 13.4：
Frames with two generators / 13.5：
Some limitations / 13.6：
Wavelet Frames via Extension Principles / 13.7：
The general setup / 14.1：
The unitary extension principle / 14.2：
Applications to B-splines I / 14.3：
The oblique extension principle / 14.4：
Fewer generators / 14.5：
Applications to B-splines II / 14.6：
Approximation orders / 14.7：
Construction of pairs of dual wavelet frames / 14.8：
Applications to B-splines III / 14.9：
Perturbation of Frames / 14.10：
A Paley-Wiener Theorem for frames / 15.1：
Compact perturbation / 15.2：
Perturbation of frame sequences / 15.3：
Perturbation of Gabor frames / 15.4：
Perturbation of wavelet frames / 15.5：
Perturbation of the Haar wavelet / 15.6：
Approximation of the Inverse Frame Operator / 15.7：
The first approach / 16.1：
A general method / 16.2：
Applications to Gabor frames / 16.3：
Integer oversampled Gabor frames / 16.4：
The finite section method / 16.5：
Expansions in Banach Spaces / 16.6：
Representations of locally compact groups / 17.1：
Feichtinger-Grochenig theory / 17.2：
Banach frames / 17.3：
p-frames / 17.4：
Gabor systems and wavelets in L[superscript p] (R) and related spaces / 17.5：
Appendix A / 17.6：
Normed vector spaces and inner product spaces / A.1：
Linear algebra / A.2：
Integration / A.3：
Some special normed vector spaces / A.4：
Operators on Banach spaces / A.5：
Operators on Hilbert spaces / A.6：
The pseudo-inverse / A.7：
Some special functions / A.8：
B-splines / A.9：
Notes / A.10：
List of symbols
References
Index
Preface
Frames in Finite-dimensional Inner Product Spaces / 1：
Some basic facts about frames / 1.1：
5.

図書
図書
5. Perturbation methods for differential equations
Bhimsen K. Shivamoggi
出版情報: Boston : Birkhäuser, c2003  xiv, 354 p. ; 24 cm
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6.

図書
図書
6. An introduction to minimax theorems and their applications to differential equations
by Maria do Rosário Grossinho and Stepan Agop Tersian
出版情報: Dordrecht : Kluwer Academic Publishers, c2001  xii, 269 p. ; 25 cm
シリーズ名: Nonconvex optimization and its applications ; v. 52
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7.

図書
図書
7. Introduction to the h-principle
Y. Eliashberg, N. Mishachev
出版情報: Providence, R.I. : American Mathematical Society, c2002  xvii, 206 p. ; 26 cm
シリーズ名: Graduate studies in mathematics ; v. 48
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目次情報: 続きを見る
Intrigue Holonomic approximation: Jets and holonomy
Thom transversality theorem Holonomic approximation
Applications Differential relations and Gromov's $h$-principle: Differential relations
Homotopy principle Open Diff $V$-invariant differential relations
Applications to closed manifolds
The homotopy principle in symplectic geometry: Symplectic and contact basics
Symplectic and contact structures on open manifolds
Symplectic and contact structures on closed manifolds
Embeddings into symplectic and contact manifolds
Microflexibility and holonomic $\mathcal{R}$-approximation
First applications of microflexibility
Microflexible $\mathfrak{U}$-invariant differential relations
Further applications to symplectic geometry
Convex integration: One-dimensional convex integration
Homotopy principle for ample differential relations
Directed immersions and embeddings
First order linear differential operators Nash-Kuiper theorem
Bibliography
Index
Intrigue Holonomic approximation: Jets and holonomy
Thom transversality theorem Holonomic approximation
Applications Differential relations and Gromov's $h$-principle: Differential relations
8.

図書
図書
8. Computational physics : an introduction. 2nd ed
Franz J. Vesely
出版情報: New York : Kluwer Academic/Plenum Publishers, c2001  xvi, 259 p. ; 26 cm
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目次情報: 続きを見る
The Three Pillars of Computational Physics / I：
Finite Differences / 1：
Interpolation Formulae / 1.1：
NGF Interpolation / 1.1.1：
NGB Interpolation / 1.1.2：
ST Interpolation / 1.1.3：
Difference Quotients / 1.2：
DNGF Formulae / 1.2.1：
DNGB Formulae / 1.2.2：
DST Formulae / 1.2.3：
Finite Differences in Two Dimensions / 1.3：
Sample Applications / 1.4：
Classical Point Mechanics / 1.4.1：
Diffusion and Thermal Conduction / 1.4.2：
Linear Algebra / 2：
Exact Methods / 2.1：
Gauss Elimination and Back Substitution / 2.1.1：
Simplifying Matrices: The Householder Transformation / 2.1.2：
LU Decomposition / 2.1.3：
Tridiagonal Matrices: Recursion Method / 2.1.4：
Iterative Methods / 2.2：
Jacobi Relaxation / 2.2.1：
Gauss-Seidel Relaxation (GSR) / 2.2.2：
Successive Over-Relaxation (SOR) / 2.2.3：
Alternating Direction Implicit Method (ADI) / 2.2.4：
Conjugate Gradient Method (CG) / 2.2.5：
Eigenvalues and Eigenvectors / 2.3：
Largest Eigenvalue and Related Eigenvector / 2.3.1：
Arbitrary Eigenvalue/-vector: Inverse Iteration / 2.3.2：
Potential Equation / 2.4：
Electronic Orbitals / 2.4.3：
Stochastics / 3：
Equidistributed Random Variates / 3.1：
Linear Congruential Generators / 3.1.1：
Shift Register Generators / 3.1.2：
Other Distributions / 3.2：
Fundamentals / 3.2.1：
Transformation Method / 3.2.2：
Generalized Transformation Method / 3.2.3：
Rejection Method / 3.2.4：
Multivariate Gaussian Distribution / 3.2.5：
Equidistribution in Orientation Space / 3.2.6：
Random Sequences / 3.3：
Markov Processes / 3.3.1：
Autoregressive Processes / 3.3.3：
Wiener-Levy Process / 3.3.4：
Markov Chains and the Monte Carlo method / 3.3.5：
Stochastic Optimization / 3.4：
Simulated Annealing / 3.4.1：
Genetic Algorithms / 3.4.2：
Everything Flows / II：
Ordinary Differential Equations / 4：
Initial Value Problems of First Order / 4.1：
Euler-Cauchy Algorithm / 4.1.1：
Stability and Accuracy of Difference Schemes / 4.1.2：
Explicit Methods / 4.1.3：
Implicit Methods / 4.1.4：
Predictor-Corrector Method / 4.1.5：
Runge-Kutta Method / 4.1.6：
Extrapolation Method / 4.1.7：
Initial Value Problems of Second Order / 4.2：
Verlet Method / 4.2.1：
Nordsieck Formulation of the PC Method / 4.2.2：
Symplectic Algorithms / 4.2.4：
Numerov's Method / 4.2.6：
Boundary Value Problems / 4.3：
Shooting Method / 4.3.1：
Relaxation Method / 4.3.2：
Partial Differential Equations / 5：
Initial Value Problems I (Hyperbolic) / 5.1：
FTCS Scheme; Stability Analysis / 5.1.1：
Lax Scheme / 5.1.2：
Leapfrog Scheme (LF) / 5.1.3：
Lax-Wendroff Scheme (LW) / 5.1.4：
Lax and Lax-Wendroff in Two Dimensions / 5.1.5：
Initial Value Problems II (Parabolic) / 5.2：
FTCS Scheme / 5.2.1：
Implicit Scheme of First Order / 5.2.2：
Crank-Nicholson Scheme (CN) / 5.2.3：
Dufort-Frankel Scheme (DF) / 5.2.4：
Boundary Value Problems: Elliptic DE / 5.3：
Relaxation and Multigrid Techniques / 5.3.1：
ADI Method for the Potential Equation / 5.3.2：
Fourier Transform Method (FT) / 5.3.3：
Cyclic Reduction (CR) / 5.3.4：
Anchors Aweigh / III：
Simulation and Statistical Mechanics / 6：
Model Systems of Statistical Mechanics / 6.1：
A Nutshellfull of Fluids and Solids / 6.1.1：
Tricks of the Trade / 6.1.2：
Monte Carlo Method / 6.2：
Molecular Dynamics Simulation / 6.3：
Hard Spheres / 6.3.1：
Continuous Potentials / 6.3.2：
Beyond Basic Molecular Dynamics / 6.3.3：
Evaluation of Simulation Experiments / 6.4：
Pair Correlation Function / 6.4.1：
Autocorrelation Functions / 6.4.2：
Particles and Fields / 6.5：
Ewald summation / 6.5.1：
Particle-Mesh Methods (PM and P3M) / 6.5.2：
Stochastic Dynamics / 6.6：
Quantum Mechanical Simulation / 7：
Diffusion Monte Carlo (DMC) / 7.1：
Path Integral Monte Carlo (PIMC) / 7.2：
Wave Packet Dynamics (WPD) / 7.3：
Density Functional Molecular Dynamics (DFMD) / 7.4：
Hydrodynamics / 8：
Compressible Flow without Viscosity / 8.1：
Explicit Eulerian Methods / 8.1.1：
Particle-in-Cell Method (PIC) / 8.1.2：
Smoothed Particle Hydrodynamics (SPH) / 8.1.3：
Incompressible Flow with Viscosity / 8.2：
Vorticity Method / 8.2.1：
Pressure Method / 8.2.2：
Free Surfaces: Marker-and-Cell Method (MAC) / 8.2.3：
Lattice Gas Models for Hydrodynamics / 8.3：
Lattice Gas Cellular Automata / 8.3.1：
The Lattice Boltzmann Method / 8.3.2：
Direct Simulation Monte Carlo / Bird method / 8.4：
Appendixes
Machine Errors / A：
Discrete Fourier Transformation / B：
Fast Fourier Transform (FFT) / B.1：
Bibliography
Index
The Three Pillars of Computational Physics / I：
Finite Differences / 1：
Interpolation Formulae / 1.1：
9.

図書
図書
9. Solution of ordinary differential equations by continuous groups
George Emanuel
出版情報: Boca Raton, FL : Chapman & Hall/CRC, 2001  220 p.
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Preface
Background / I：
Introduction / 1：
Continuous One-Parameter Groups-I / 2：
Group Concept / 2.1：
Infinitesimal Transformation / 2.2：
Global Group Equations / 2.3：
Problems / 2.4：
Method of Characteristics / 3：
Theory / 3.1：
Examples / 3.2：
Continuous One-Parameter Groups-II / 3.3：
Invariance / 4.1：
The Once-Extended Group / 4.2：
Higher-Order Extended Groups / 4.3：
Ordinary Differential Equations / 4.4：
First-Order ODEs / 5：
Invariance Under a One-Parameter Group / 5.1：
Canonical Coordinates / 5.2：
Special Procedures / 5.3：
Compendium / 5.4：
Higher-Order ODEs / 5.5：
Invariant Equations / 6.1：
Finding the Groups / 6.2：
System of First-Order ODEs / 6.3：
Second-Order ODEs / 6.4：
Classification of Two-Parameter Groups / 7.1：
Invariance and Canonical Coordinates / 7.2：
Appendices / 7.3：
Bibliography and References / A：
The Rotation Group / B：
Basic Relations / C：
Tables / D：
Answers to Selected Problems / E：
Preface
Background / I：
Introduction / 1：
10.

図書
図書
10. A first course in the numerical analysis of differential equations. 2nd ed (: pbk)
Arieh Iserles
出版情報: Cambridge : Cambridge University Press, 2009  xviii, 459 p. ; 25 cm
シリーズ名: Cambridge texts in applied mathematics
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Preface to the second edition
Preface to the first edition
Flowchart of contents
Ordinary differential equations / I：
Euler's method and beyond / 1：
Ordinary differential equations and the Lipschitz condition / 1.1：
Euler's method / 1.2：
The trapezoidal rule / 1.3：
The theta method / 1.4：
Comments and bibliography
Exercises
Multistep methods / 2：
The Adams method / 2.1：
Order and convergence of multistep methods / 2.2：
Backward differentiation formulae / 2.3：
Runge-Kutta methods / 3：
Gaussian quadrature / 3.1：
Explicit Runge-Kutta schemes / 3.2：
Implicit Runge-Kutta schemes / 3.3：
Collocation and IRK methods / 3.4：
Stiff equations / 4：
What are stiff ODEs? / 4.1：
The linear stability domain and A-stability / 4.2：
A-stability of Runge-Kutta methods / 4.3：
A-stability of multistep methods / 4.4：
Geometric numerical integration / 5：
Between quality and quantity / 5.1：
Monotone equations and algebraic stability / 5.2：
From quadratic invariants to orthogonal flows / 5.3：
Hamiltonian systems / 5.4：
Error control / 6：
Numerical software vs. numerical mathematics / 6.1：
The Milne device / 6.2：
Embedded Runge-Kutta methods / 6.3：
Nonlinear algebraic systems / 7：
Functional iteration / 7.1：
The Newton-Raphson algorithm and its modification / 7.2：
Starting and stopping the iteration / 7.3：
The Poisson equation / II：
Finite difference schemes / 8：
Finite differences / 8.1：
The finite element method / 8.2：
Two-point boundary value problems / 9.1：
A synopsis of FEM theory / 9.2：
Spectral methods / 9.3：
Sparse matrices vs. small matrices / 10.1：
The algebra of Fourier expansions / 10.2：
The fast Fourier transform / 10.3：
Second-order elliptic PDEs / 10.4：
Chebyshev methods / 10.5：
Gaussian elimination for sparse linear equations / 11：
Banded systems / 11.1：
Graphs of matrices and perfect Cholesky factorization / 11.2：
Classical iterative methods for sparse linear equations / 12：
Linear one-step stationary schemes / 12.1：
Classical iterative methods / 12.2：
Convergence of successive over-relaxation / 12.3：
Multigrid techniques / 12.4：
In lieu of a justification / 13.1：
The basic multigrid technique / 13.2：
The full multigrid technique / 13.3：
Poisson by multigrid / 13.4：
Conjugate gradients / 14：
Steepest, but slow, descent / 14.1：
The method of conjugate gradients / 14.2：
Krylov subspaces and preconditioners / 14.3：
Poisson by conjugate gradients / 14.4：
Fast Poisson solvers / 15：
TST matrices and the Hockney method / 15.1：
Fast Poisson solver in a disc / 15.2：
Partial differential equations of evolution / III：
The diffusion equation / 16：
A simple numerical method / 16.1：
Order, stability and convergence / 16.2：
Numerical schemes for the diffusion equation / 16.3：
Stability analysis I: Eigenvalue techniques / 16.4：
Stability analysis II: Fourier techniques / 16.5：
Splitting / 16.6：
Hyperbolic equations / 17：
Why the advection equation? / 17.1：
Finite differences for the advection equation / 17.2：
The energy method / 17.3：
The wave equation / 17.4：
The Burgers equation / 17.5：
Appendix Bluffer's guide to useful mathematics
Linear algebra / A.1：
Vector spaces / A.1.1：
Matrices / A.1.2：
Inner products and norms / A.1.3：
Linear systems / A.1.4：
Eigenvalues and eigenvectors / A.1.5：
Bibliography
Analysis / A.2：
Introduction to functional analysis / A.2.1：
Approximation theory / A.2.2：
Index / A.2.3：
Preface to the second edition
Preface to the first edition
Flowchart of contents
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