Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

/pdf/iterative-methods-for-sparse-linear-systems-34ozhpwq89.pdf

Iterative Methods for Sparse Linear Systems

In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed Orthogonal polynomials on the unit circle are not discussed

Orthogonal Polynomials

I Mathematical Theory- 1 Introduction- 11 Conservation laws- 12 Applications- 13 Mathematical difficulties- 14 Numerical difficulties- 15 Some references- 2 The Derivation of Conservation Laws- 21 Integral and differential forms- 22 Scalar equations- 23 Diffusion- 3 Scalar Conservation Laws- 31 The linear advection equation- 311 Domain of dependence- 312 Nonsmooth data- 32 Burgers' equation- 33 Shock formation- 34 Weak solutions- 35 The Riemann Problem- 36 Shock speed- 37 Manipulating conservation laws- 38 Entropy conditions- 381 Entropy functions- 4 Some Scalar Examples- 41 Traffic flow- 411 Characteristics and "sound speed"- 42 Two phase flow- 5 Some Nonlinear Systems- 51 The Euler equations- 511 Ideal gas- 512 Entropy- 52 Isentropic flow- 53 Isothermal flow- 54 The shallow water equations- 6 Linear Hyperbolic Systems 58- 61 Characteristic variables- 62 Simple waves- 63 The wave equation- 64 Linearization of nonlinear systems- 641 Sound waves- 65 The Riemann Problem- 651 The phase plane- 7 Shocks and the Hugoniot Locus- 71 The Hugoniot locus- 72 Solution of the Riemann problem- 721 Riemann problems with no solution- 73 Genuine nonlinearity- 74 The Lax entropy condition- 75 Linear degeneracy- 76 The Riemann problem- 8 Rarefaction Waves and Integral Curves- 81 Integral curves- 82 Rarefaction waves- 83 General solution of the Riemann problem- 84 Shock collisions- 9 The Riemann problem for the Euler equations- 91 Contact discontinuities- 92 Solution to the Riemann problem- II Numerical Methods- 10 Numerical Methods for Linear Equations- 101 The global error and convergence- 102 Norms- 103 Local truncation error- 104 Stability- 105 The Lax Equivalence Theorem- 106 The CFL condition- 107 Upwind methods- 11 Computing Discontinuous Solutions- 111 Modified equations- 1111 First order methods and diffusion- 1112 Second order methods and dispersion- 112 Accuracy- 12 Conservative Methods for Nonlinear Problems- 121 Conservative methods- 122 Consistency- 123 Discrete conservation- 124 The Lax-Wendroff Theorem- 125 The entropy condition- 13 Godunov's Method- 131 The Courant-Isaacson-Rees method- 132 Godunov's method- 133 Linear systems- 134 The entropy condition- 135 Scalar conservation laws- 14 Approximate Riemann Solvers- 141 General theory- 1411 The entropy condition- 1412 Modified conservation laws- 142 Roe's approximate Riemann solver- 1421 The numerical flux function for Roe's solver- 1422 A sonic entropy fix- 1423 The scalar case- 1424 A Roe matrix for isothermal flow- 15 Nonlinear Stability- 151 Convergence notions- 152 Compactness- 153 Total variation stability- 154 Total variation diminishing methods- 155 Monotonicity preserving methods- 156 l1-contracting numerical methods- 157 Monotone methods- 16 High Resolution Methods- 161 Artificial Viscosity- 162 Flux-limiter methods- 1621 Linear systems- 163 Slope-limiter methods- 1631 Linear Systems- 1632 Nonlinear scalar equations- 1633 Nonlinear Systems- 17 Semi-discrete Methods- 171 Evolution equations for the cell averages- 172 Spatial accuracy- 173 Reconstruction by primitive functions- 174 ENO schemes- 18 Multidimensional Problems- 181 Semi-discrete methods- 182 Splitting methods- 183 TVD Methods- 184 Multidimensional approaches

Numerical methods for conservation laws

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

Padé-type approximation and general orthogonal polynomials

Introduction to the Theory. First Steps. What is an Extrapolation Method? What is an Extrapolation Algorithm? Quasi-linear Sequence Transformations. Sequence Transformations as Ratios of Determinants. Triangular Recursive Schemes. Normal Forms of the Algorithms. Progressive Forms of the Algorithms. Particular Rules of the Algorithms. Accelerability and Non-accelerability. Optimality. Asymptotic Behaviour of Sequences. Scalar Extrapolation Algorithms. The E-algorithm. Richardson Extrapolation Process. The -algorithm. The G-transformation. Rational Extrapolation. Generalizations of the -algorithm. Levin's Transformations. Overholt's Process. -type Algorithms. The Iterated 2 Process. Miscellaneous Algorithms. Special Devices. Error Estimates and Acceleration. Convergence Tests and Acceleration. Construction of Asymptotic Expansions. Construction of Extrapolation Processes. Extraction Procedures. Automatic Selection. Composite Sequence Transformations. Error Control. Contractive Sequence Transformations. Least Squares Extrapolation. Vector Extrapolation Algorithms. The Vector -algorithm. The Topological -algorithm. The Vector E-algorithm. The Recursive Projection Algorithm. The H-algorithm. The Ford-Sidi Algorithms. Miscellaneous Algorithms. Continuous Prediction Algorithms. The Taylor Expansion. Confluent Overholt's process. Confluent -algorithms. Confluent -algorithm. Confluent G-transform. Confluent E-algorithm. -type Confluent Algorithms. Applications. Sequences and Series: Simple Sequences, Double Sequences, Chebyshev and Fourier Series, Continued Fractions, Vector Sequences. Systems of Equations: Linear Systems, Projection Methods, Regularization and Penalty Techniques, Nonlinear Equations, Continuation Methods. Eigenelements: Eigenvalues and eigenvectors, Derivatives of Eigensystems. Integral and Differential Equations: Implicit Runge-Kutta Methods, Boundary Value Problems, Nonlinear Methods, Laplace Transform Inversion, Partial Differential Equations. Interpolation and Approximation. Statistics: The Jackknife, ARMA Models, Monte-Carlo Methods. Integration and Differentiation: Acceleration of Quadrature Formulae, Nonlinear Quadrature Formulae, Cauchy's Principal Values, Infinite Integrals, Multiple Integrals, Numerical Differentiation. Prediction. Software. Programming the Algorithms. Computer Arithmetic. Programs. Bibliography. Index.

Extrapolation Methods: Theory and Practice

In this paper a general formalism for linear and rational extrapolation processes is developped. This formalism includes most of the sequence transformations actually used for convergence acceleration. A general recursive algorithm for implementing the method is given. Convergence results and convergence acceleration results are proved. The vector case and some other extensions are also studied.

A general extrapolation algorithm

1 The Early Ages.- 1.1 Euclid's algorithm.- 1.2 The square root.- 1.3 Indeterminate equations.- 1.4 History of notations.- 2 The First Steps.- 2.1 Ascending continued fractions.- 2.2 The birth of continued fractions.- 2.3 Miscellaneous contributions.- 2.4 Pell's equation.- 3 The Beginning of the Theory.- 3.1 Brouncker and Wallis.- 3.2 Huygens.- 3.3 Number theory.- 4 Golden Age.- 4.1 Euler.- 4.2 Lambert.- 4.3 Lagrange.- 4.4 Miscellaneous contributions.- 4.5 The birth of Pade approximants.- 5 Maturity.- 5.1 Arithmetical continued fractions.- 5.1.1 Algebraic properties.- 5.1.2 Arithmetic.- 5.1.3 Applications.- 5.1.4 Number theory.- 5.1.5 Convergence.- 5.2 Algebraic continued fractions.- 5.2.1 Expansion methods and properties.- 5.2.2 Examples and applications.- 5.2.3 Orthogonal polynomials.- 5.2.4 Convergence and analytic theory.- 5.2.5 Pade approximants.- 5.3 Varia.- 6 The Modern Times.- 6.1 Number theory.- 6.2 Set and probability theories.- 6.3 Convergence and analytic theory.- 6.4 Pade approximants.- 6.5 Extensions and applications.- Documents.- Document 1: L'algebre des geometres grecs.- Document 2: Histoire de l'Academie Royale des Sciences.- Document 3: Encyclopedie (Supplement).- Document 4: Elementary Mathematics from an advanced standpoint.- Document 5: Sur quelques applications des fractions continues.- Document 6: Rapport sur un Memoire de M. Stieltjes.- Document 7: Correspondance d'Hermite et de Stieltjes.- Document 8: Notice sur les travaux et titres.- Document 9: Note annexe sur les fractions continues.- Scientific Bibliography.- Works.- Historical Bibliography.- Name Index.

Claude Brezinski

Papers

Padé-type approximation and general orthogonal polynomials

Extrapolation Methods: Theory and Practice

A general extrapolation algorithm

History of Continued Fractions and Padé Approximants

Extrapolation methods