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Preface
Preface to the Julia edition
Preface to the original edition
Main text
1. Introduction
1.1. Floating-point numbers
1.2. Problems and conditioning
1.3. Algorithms
1.4. Stability
1.5. Next steps
2. Linear systems of equations
2.1. Polynomial interpolation
2.2. Computing with matrices
2.3. Linear systems
2.4. LU factorization
2.5. Efficiency of matrix computations
2.6. Row pivoting
2.7. Vector and matrix norms
2.8. Conditioning of linear systems
2.9. Exploiting matrix structure
2.10. Next steps
3. Overdetermined linear systems
3.1. Fitting functions to data
3.2. The normal equations
3.3. The QR factorization
3.4. Computing QR factorizations
3.5. Next steps
4. Roots of nonlinear equations
4.1. The rootfinding problem
4.2. Fixed-point iteration
4.3. Newton’s method
4.4. Interpolation-based methods
4.5. Newton for nonlinear systems
4.6. Quasi-Newton methods
4.7. Nonlinear least squares
4.8. Next steps
5. Piecewise interpolation
5.1. The interpolation problem
5.2. Piecewise linear interpolation
5.3. Cubic splines
5.4. Finite differences
5.5. Convergence of finite differences
5.6. Numerical integration
5.7. Adaptive integration
5.8. Next steps
6. Initial-value problems for ODEs
6.1. Basics of IVPs
6.2. Euler’s method
6.3. IVP systems
6.4. Runge–Kutta methods
6.5. Adaptive Runge–Kutta
6.6. Multistep methods
6.7. Implementation of multistep methods
6.8. Zero-stability of multistep methods
6.9. Next steps
7. Matrix analysis
7.1. From matrix to insight
7.2. Eigenvalue decomposition
7.3. Singular value decomposition
7.4. Symmetry and definiteness
7.5. Dimension reduction
7.6. Next steps
8. Krylov methods in linear algebra
8.1. Sparsity and structure
8.2. Power iteration
8.3. Inverse iteration
8.4. Krylov subspaces
8.5. GMRES
8.6. MINRES and conjugate gradients
8.7. Matrix-free iterations
8.8. Preconditioning
8.9. Next steps
9. Global function approximation
9.1. Polynomial interpolation
9.2. The barycentric formula
9.3. Stability of polynomial interpolation
9.4. Orthogonal polynomials
9.5. Trigonometric interpolation
9.6. Spectrally accurate integration
9.7. Improper integrals
9.8. Next steps
10. Boundary-value problems
10.1. Two-point BVP
10.2. Shooting
10.3. Differentiation matrices
10.4. Collocation for linear problems
10.5. Nonlinearity and boundary conditions
10.6. The Galerkin method
10.7. Next steps
11. Diffusion equations
11.1. Black–Scholes equation
11.2. The method of lines
11.3. Absolute stability
11.4. Stiffness
11.5. Boundaries
11.6. Next steps
12. Advection equations
12.1. Traffic flow
12.2. Upwinding and stability
12.3. Absolute stability
12.4. The wave equation
12.5. Next steps
13. Two-dimensional problems
13.1. Tensor-product discretizations
13.2. Two-dimensional diffusion and advection
13.3. Laplace and Poisson equations
13.4. Nonlinear elliptic PDEs
13.5. Next steps
Appendices
Review of linear algebra
Glossary
Julia cheatsheet
References
Index
Elementi Di Analisi Matematica 1 Direct