Contents and Web Index

A

A Model Result: Backward Error for a Dot Product (1)

A Posteriori Eigenvalue Error Bounds (6)

A Posteriori Residual Estimation (11)

Accumulation of Rounding Errors (1)

Adams–Bashforth Methods (10)

Adams–Moulton Methods (10)

Adaptive Mesh Refinement (11)

Adaptive Quadrature (9)

Adaptive Step-Size Control (10)

ADI Methods (12)

Adjoint Methods for Parameter Optimization (14)

Advanced problems (1)

Advanced problems (2)

Advanced problems (3)

Advanced problems (4)

Advanced problems (5)

Advanced problems (6)

Advanced problems (7)

Advanced problems (8)

Advanced problems (9)

Advanced problems (10)

Advanced problems (11)

Advanced problems (12)

Advanced problems (13)

Advanced problems (14)

Advanced problems (15)

Advection: Upwind, Lax–Friedrichs, and Lax–Wendroff (12)

Aitken Acceleration and Steffensen’s Method (2)

Algebraic and Geometric Multiplicity (6)

Algorithms (1)

Algorithms Summary (2)

Algorithms Summary (3)

Aliasing (13)

Aliasing and the Dealiasing Rule (12)

Anderson Acceleration (2)

Appendix A: Formula Handbook (A)

Arnoldi Method for Large Nonsymmetric Problems (6)

Arnoldi Process (5)

Augmented Lagrangian and ADMM (14)

Automatic Differentiation versus Numerical Differentiation (8)

B

B-Splines (3)

Back Substitution (4)

Backward Differentiation Formulas (10)

Backward Euler and the Theta Method (10)

Banded and Sparse Direct Methods (4)

Barycentric Interpolation (3)

Barycentric Interpolation (13)

Basic problems (1)

Basic problems (2)

Basic problems (3)

Basic problems (4)

Basic problems (5)

Basic problems (6)

Basic problems (7)

Basic problems (8)

Basic problems (9)

Basic problems (10)

Basic problems (11)

Basic problems (12)

Basic problems (13)

Basic problems (14)

Basic problems (15)

Bauer–Fike Theorem and Pseudospectra (6)

Bernstein Polynomials and the Weierstrass Theorem (3)

Best Uniform Approximation and Chebyshev Alternation (3)

Bibliography (Refs)

Bibliography (Refs)

Bisection (2)

Boundary-Value Problems (11)

Boundary-value problems and Galerkin notation (Symbols)

Boundary-Value Problems for Ordinary Differential Equations (11)

Bracketing Methods (2)

Broyden and Quasi-Newton Methods for Systems (14)

Burgers Equation and Nonlinear Conservation Laws (12)

C

Cancellation and Loss of Significance (1)

Caputo versus Riemann–Liouville (15)

Caputo–Katugampola L1 Scheme (15)

Chapter 10: Initial-Value Problems for ODEs (A)

Chapter 11: Boundary-Value Problems for ODEs (A)

Chapter 12: Partial Differential Equations (A)

Chapter 13: Spectral and Pseudospectral Methods (A)

Chapter 14: Optimization and Nonlinear Systems (A)

Chapter 15: Fractional Differential Equations (A)

Chapter 1: Floating-Point Arithmetic (A)

Chapter 2: Nonlinear Equations (A)

Chapter 3: Interpolation and Approximation (A)

Chapter 4: Direct Linear Systems (A)

Chapter 5: Iterative Linear Systems (A)

Chapter 6: Eigenvalue Problems (A)

Chapter 7: Least Squares (A)

Chapter 8: Numerical Differentiation (A)

Chapter 9: Numerical Integration (A)

Chapter Summary (1)

Chebyshev Accuracy versus Finite-Difference Accuracy (8)

Chebyshev Collocation for Boundary-Value Problems (13)

Chebyshev Collocation for Evolution PDEs (13)

Chebyshev Differentiation for Boundary-Value Problems (8)

Chebyshev Differentiation Matrix (8)

Chebyshev Differentiation Matrix (13)

Chebyshev Expansions and Spectral Interpolation (3)

Chebyshev Nodes (8)

Chebyshev Polynomial Methods for IVPs (10)

Chebyshev Polynomials (13)

Chebyshev Pseudospectral Nonlinear BVPs (13)

Chebyshev Spectral Collocation (11)

Chebyshev Spectral Methods (12)

Chebyshev–Gauss–Lobatto Nodes (13)

Chebyshev–Lobatto Panel Quadrature (15)

Chebyshev’s Method (2)

Cholesky Factorization (4)

Choosing the Regularization Parameter (7)

Classical Fourth-Order Runge–Kutta Method (10)

Classical Splittings: Jacobi, Gauss–Seidel, and SOR (5)

Classification and Numerical Consequences (12)

Clenshaw–Curtis Quadrature (9)

Collocation Methods (10)

Collocation Methods (11)

Comparison of Fractional Differentiation Methods (15)

Comparison with Splines and High-Order Finite Elements (13)

Complete Pivoting, Rook Pivoting, and Scaling (4)

Complex-Step Differentiation (8)

Componentwise Backward Error (4)

Composite Quadrature (9)

Conditioning of Least Squares (7)

Conditioning of Linear Systems (1)

Conditioning of Mathematical Problems (1)

Conjugate Gradient Method (5)

Conjugate Gradient Methods for Optimization (14)

Consistency, Stability, and Convergence (10)

Constrained Least Squares (7)

Constrained Optimization and KKT Conditions (14)

Convergence Criteria for Classical Methods (5)

Cubic Chebyshev–Lobatto Formula (15)

Cubic One-Step Methods (2)

D

Damped and Globalized Newton Methods (14)

Damped, Safeguarded, and Trust-Region Newton Methods (2)

Dealiasing: The 2/3 -Rule and Zero Padding (13)

Dense Output and Event Detection (10)

Diagonalization and Schur Decomposition (6)

Differentiating Integral Objectives (14)

Differentiation Matrices on Uniform Grids (8)

Direct Methods for Linear Systems (4)

Discontinuous Galerkin Methods (12)

Discrete Fourier Transform and FFT Differentiation (13)

Dormand–Prince 5(4) (10)

E

Eigenvalue Conditioning (6)

Eigenvalue problems (Symbols)

Eigenvalue Problems (6)

Endpoint Singularities and Transformations (9)

Error, Residual, and Conditioning of a Root (2)

Euler–Maclaurin Formula and Periodic Integrands (9)

Euler’s Method (10)

Example 1: Caputo derivative of a constant (15)

Example 2: Fractional relaxation (15)

Example 3: Fractional diffusion (15)

Example 4: Residual minimization (15)

Examples (15)

Exercise solutions (1)

Exercise solutions (2)

Exercise solutions (3)

Exercise solutions (4)

Exercise solutions (5)

Exercise solutions (6)

Exercise solutions (7)

Exercise solutions (8)

Exercise solutions (9)

Exercise solutions (10)

Exercise solutions (11)

Exercise solutions (12)

Exercise solutions (13)

Exercise solutions (14)

Exercise solutions (15)

Exercises (1)

Exercises (2)

Exercises (3)

Exercises (4)

Exercises (5)

Exercises (6)

Exercises (7)

Exercises (8)

Exercises (9)

Exercises (10)

Exercises (11)

Exercises (12)

Exercises (13)

Exercises (14)

Exercises (15)

Existence and Uniqueness (7)

Existence, Uniqueness, and Flow Maps (10)

F

Fast Convolution and Short-Memory Principle (15)

Filter Factors and Truncated SVD (7)

Filtering (13)

Finite Differences for Linear BVPs (11)

Finite Differences for Nonlinear BVPs (11)

Finite Differences for the Heat Equation (12)

Finite Differences for the Wave Equation (12)

Finite Element Weak Forms for PDEs (12)

Finite Elements with Piecewise Linear Basis Functions (11)

Finite-Difference Weights from Interpolation (8)

Finite-Volume Numerical Fluxes (12)

Fixed-Point Iteration (2)

Floater–Hormann Rational Interpolation (3)

Floating-point arithmetic and error analysis (Symbols)

Floating-Point Arithmetic and Error Analysis (1)

Floating-Point Systems (1)

Forward Error, Backward Error, and Stability (1)

Forward Substitution (4)

Fourier Pseudospectral Burgers Solver (13)

Fourier Series and Periodic Problems (13)

Fourier Spectral Methods (12)

Fourier Spectral Methods for PDEs (13)

Fractional calculus and fractional differential equations (Symbols)

Fractional Integrals (15)

Fractional Residual Minimization and Fractional Clocks (15)

Frequently overloaded symbols (Symbols)

G

Gaussian Elimination (4)

Gaussian Quadrature (9)

Gauss–Kronrod Rules (9)

Gauss–Legendre Nodes and Weights (9)

Gauss–Seidel Method (5)

General mathematical notation (Symbols)

Generalized Eigenvalue Problems (6)

Gershgorin Localization (6)

Gibbs Phenomenon (13)

Givens Rotations (4)

GMRES (5)

Gradient Descent (14)

Growth Factor (4)

Grunwald–Letnikov and Shifted GL Methods (15)

H

Halley’s Method (2)

Hermite Interpolation (3)

Hermite–Obreschkoff Methods (10)

Hessenberg Reduction (6)

High-Precision Quadrature for Galerkin Integrals (11)

Higher-Order Explicit Runge–Kutta Methods (10)

Higher-Order Finite Differences (8)

Householder QR (4)

How to read the book (Guide)

I

Implicit and Crank–Nicolson Heat Schemes (12)

Incomplete-Beta Weights for the Caputo Kernel (15)

Incomplete-Beta Weights for the Caputo–Katugampola Kernel (15)

Inexact Newton and Newton–Krylov Methods (14)

Infinite Intervals (9)

Initial-Value Problems (10)

Initial-value problems for ODEs (Symbols)

Initial-Value Problems for Ordinary Differential Equations (10)

Intended audience (Guide)

Interactive tools (1)

Interactive tools (2)

Interactive tools (3)

Interactive tools (4)

Interactive tools (5)

Interactive tools (6)

Interactive tools (7)

Interactive tools (8)

Interactive tools (9)

Interactive tools (10)

Interactive tools (11)

Interactive tools (12)

Interactive tools (13)

Interactive tools (14)

Interactive tools (15)

Intermediate problems (1)

Intermediate problems (2)

Intermediate problems (3)

Intermediate problems (4)

Intermediate problems (5)

Intermediate problems (6)

Intermediate problems (7)

Intermediate problems (8)

Intermediate problems (9)

Intermediate problems (10)

Intermediate problems (11)

Intermediate problems (12)

Intermediate problems (13)

Intermediate problems (14)

Intermediate problems (15)

Interpolation and approximation (Symbols)

Interpolation and Approximation Theory (3)

Interpolation Error (3)

Interpolatory Quadrature (9)

Interval Newton and Certified Roots (2)

Introduction (Symbols)

Inverse Iteration and Shifted Inverse Iteration (6)

Iterative methods and Krylov spaces (Symbols)

Iterative Methods for Linear Systems (5)

Iterative Refinement (4)

J

Jacobi Method (5)

Jacobi Method for Symmetric Eigenvalue Problems (6)

Jarratt-Type Fourth-Order Methods (2)

K

King’s Fourth-Order Family (2)

Krylov Subspaces (5)

L

L1-2, Alikhanov, L2, and L3 Ideas (15)

Lagrange Interpolation (3)

Lanczos Method for Hermitian Problems (6)

Lanczos Process and Symmetric Krylov Methods (5)

Large-Scale Least Squares: CGLS and LSQR (7)

LDL^T Factorization (4)

Least squares and regularization (Symbols)

Least-Squares Approximation (7)

Least-Squares Approximation and Orthogonal Projection (3)

Least-Squares Problems (4)

Lebesgue Constants and Stability (3)

Legendre Spectral Methods (13)

Line Search Methods (14)

Linear Galerkin Method (11)

Linear Multistep Methods (10)

Linear systems and matrix factorizations (Symbols)

List of Symbols and Notation (Symbols)

LU Factorization and Solving (4)

Lubich Convolution Quadrature (15)

M

Main Fractional Derivatives (15)

Matrix Norms and Conditioning (4)

Matrix Splittings (5)

Method of Lines (12)

Methods with Memory (2)

Midpoint, Trapezoidal, and Simpson Rules (9)

Model Partial Differential Equations (12)

Monte Carlo and Quasi-Monte Carlo Integration (9)

Muller, Inverse Quadratic Interpolation, and Brent-Type Methods (2)

Multidimensional Integration (9)

Multigrid Methods (5)

Multiple Roots and Multiplicity Correction (2)

Multiple Shooting (11)

N

Nelder–Mead Simplex Method (14)

Newton Interpolation and Divided Differences (3)

Newton–Cotes Formulas (9)

Newton–Galerkin Linearization (11)

Newton–Krylov and Inexact Newton Methods (2)

Newton’s Method (2)

Newton’s Method for Optimization (14)

Noise Amplification (8)

Nonlinear equations (Symbols)

Nonlinear Equations (2)

Nonlinear Galerkin Method (11)

Nonlinear Least Squares (7)

Nonlinear Least Squares (14)

Nonlinear PDEs and Newton Linearization (12)

Nonlinear Systems (14)

Normal and Hermitian Matrices (6)

Normal Equations (7)

Numerical differentiation (Symbols)

Numerical Differentiation (8)

Numerical Integration (9)

Numerical integration and quadrature (Symbols)

Numerical Methods for Fractional Differential Equations (15)

Numerical Methods for Partial Differential Equations (12)

O

Objectives (Guide)

Objectives, Audience, and How to Use This Book (Guide)

One-Sided Boundary Formulas (8)

Operator Splitting and IMEX Methods (12)

Optimization and nonlinear systems (Symbols)

Optimization and Nonlinear Systems (14)

Optimization of Parameter-Dependent Integrals (14)

Optimization with Noisy or Expensive Objectives (14)

Order of Convergence (2)

Orthogonal Iteration and Simultaneous Iteration (6)

Orthogonal Polynomials (3)

Oscillatory Integrals (9)

Ostrowski’s Fourth-Order Method (2)

Other Derivative-Free Methods (14)

Other Explicit Fourth-Order RK Methods (10)

P

Partial differential equations (Symbols)

Partial Pivoting (4)

PECE Predictor–Corrector Methods (15)

Penalty, Barrier, and Interior-Point Methods (14)

Perturbation and Conditioning (7)

Petrov–Galerkin and Least-Squares Galerkin Methods (11)

Picard, Newton–Krylov, and Continuation for Nonlinear PDEs (12)

Pivoting (4)

Poisson and Laplace Equations (12)

Polynomial and Taylor Methods (10)

Polynomial Interpolation (3)

Polynomial Least Squares (7)

Power Functions and Fractional Clocks (15)

Power Method (6)

Practical Comparison of BVP Methods (11)

Practical Comparison of Direct Methods (4)

Practical Comparison of Eigenvalue Methods (6)

Practical Comparison of Iterative Methods (5)

Practical Comparison of IVP Methods (10)

Practical Comparison of Least-Squares Methods (7)

Practical Comparison of Methods (14)

Practical Comparison of PDE Methods (12)

Practical Comparison of Quadrature Methods (9)

Practical Comparison of Spectral Tools (13)

Practical Guidelines (15)

Preconditioned Conjugate Gradient (5)

Preconditioning (5)

Preface (Preface)

Preface (Preface)

Prerequisites (Guide)

Projection Geometry (7)

Projection Principles (5)

Pseudospectral Treatment of Nonlinear Terms (13)

Q

QR Factorization (4)

Quadrature for Nonlinear Galerkin Terms (11)

Quadrature for PDE Galerkin Integrals (12)

Quadrature in Spectral Methods (13)

Quadrature Rules for Integral Objectives (14)

Quartic and Multipoint Methods (2)

Quasi-Newton Methods: DFP, BFGS, and L-BFGS (14)

Quasi-Newton, Broyden, and Rank-One Updates (2)

Quintic Chebyshev–Lobatto Formula (15)

R

Radial Basis Function Interpolation (3)

Randomized Least Squares (7)

Rank-Deficient Least Squares and the SVD (7)

Rank-One Updates: Sherman–Morrison and Woodbury (4)

Rank-Revealing Factorizations and the SVD (4)

Rational Approximation and Padé Approximants (3)

Rayleigh Quotient (6)

Rayleigh Quotient Iteration (6)

Regula Falsi, Illinois, Pegasus, and Ridder Methods (2)

Research-level problems (1)

Research-level problems (2)

Research-level problems (3)

Research-level problems (4)

Research-level problems (5)

Research-level problems (6)

Research-level problems (7)

Research-level problems (8)

Research-level problems (9)

Research-level problems (10)

Research-level problems (11)

Research-level problems (12)

Research-level problems (13)

Research-level problems (14)

Research-level problems (15)

Residual Minimization for Differential Equations (14)

Residual Norm and Orthogonal Projectors (7)

Restarted GMRES and Nonsymmetric Methods (5)

Richardson and Weighted Jacobi Iterations (5)

Richardson Extrapolation (8)

Richardson Extrapolation and Romberg Integration (9)

Rounding as a Projection onto a Finite Set (1)

Roundoff Error and Optimal Step Size (8)

Runge Phenomenon and Chebyshev Nodes (3)

Runge–Kutta Methods (10)

Runge–Kutta–Fehlberg 4(5) (10)

S

Savitzky–Golay Differentiation (8)

Scaling to General Intervals (13)

Schur Complements and Block Elimination (4)

Second Derivatives (8)

Second-Order Runge–Kutta Methods (10)

Semilocal Newton Theory (2)

Sequential Quadratic Programming (14)

Shifted QR Algorithm (6)

Shooting Method (11)

Singular Values as Eigenvalues (6)

Smale’s Alpha Theory (2)

Solvers and Preconditioners for PDE Systems (12)

Solving Least Squares by QR Factorization (7)

Spectral Accuracy and Loss of Spectral Accuracy (13)

Spectral and pseudospectral methods (Symbols)

Spectral and Pseudospectral Methods (13)

Spectral Approximation (13)

Spectral Deferred Correction and Polynomial Correction (10)

Spectral Element Methods (13)

Spectral Galerkin Methods (13)

Spectral, Pseudospectral, Collocation, Tau, and Galerkin Methods (13)

Spline and B-Spline Methods (12)

Spline Collocation (12)

Spline Interpolation (3)

Stability Regions (10)

Stable and Unstable Algebraic Forms (1)

Statistical Interpretation (7)

Sterbenz Lemma (1)

Stiffness (10)

Stopping Criteria (2)

Stopping Criteria and Finite Precision (5)

Successive Over-Relaxation (5)

Super-Halley and Schröder-Type Cubic Methods (2)

Symbols table (Symbols)

Systems of Nonlinear Equations (2)

T

Tau Methods (13)

Taylor Expansions and Difference Formulas (8)

The Approximation Problem (3)

The Eigenvalue Problem (6)

The Krawczyk Operator (2)

The L1 Scheme (15)

The Least-Squares Problem (7)

The Linear System Problem (4)

The Numerical Differentiation Problem (8)

The Numerical Integration Problem (9)

The Numerical Representation Problem (1)

The QR Algorithm (6)

The Root-Finding Problem (2)

The Secant Method (2)

The Spectral-Radius Convergence Theorem (5)

The Standard Model for Floating-Point Arithmetic (1)

The Vandermonde System (3)

Third-Order Runge–Kutta Methods (10)

Tikhonov Regularization (7)

Traub-Type Multipoint Methods (2)

Triangular Systems (4)

Trust-Region Methods (14)

U

Ultraspherical Spectral Methods (13)

Unconstrained Optimization (14)

V

Volterra Integral Form (15)

W

Weak Formulation for Linear BVPs (11)

Weighted Least Squares (7)

Well-Posedness and Maximum Principles (11)

Why Fractional Derivatives? (15)

Why Iterative Methods? (5)

Z

Zero-Stability and Dahlquist Equivalence (10)