Iterative Methods for Linear and Nonlinear Equations

Preface How to Get the Software Part I. Linear Equations. 1. Basic Concepts and Stationary Iterative Methods 2. Conjugate Gradient Iteration 3. GMRES Iteration Part II. Nonlinear Equations. 4. Basic Concepts and Fixed Point Iteration 5. Newton's Method 6. Inexact Newton Methods 7. Broyden's Method 8. Global Convergence Bibliography Index.

[1]  P. Gill,et al.  Practical optimization , 2019 .

[2]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[3]  H. Walker.  Implementation of the GMRES method using householder transformations , 1988 .

[4]  P. Saylor,et al.  An optimum iterative method for solving any linear system with a square matrix , 1988 .

[5]  Layne T. Watson,et al.  Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms , 1987, TOMS.

[6]  C. Kelley,et al.  A Quasi-Newton Method for Elliptic Boundary Value Problems , 1987 .

[7]  P. Brown.  A local convergence theory for combined inexact-Newton/finite-difference projection methods , 1987 .

[8]  J. L. Nazareth.  Conjugate gradient methods less dependent on conjugacy , 1986 .

[9]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[10]  Y. Saad,et al.  A hybrid Chebyshev Krylov subspace algorithm for solving nonsymmetric systems of linear equations , 1986 .

[11]  Ekkehard W. Sachs,et al.  Broyden's method in Hilbert space , 1986, Math. Program..

[12]  W. Rheinboldt.  Numerical analysis of parametrized nonlinear equations , 1986 .

[13]  Y. Saad,et al.  Practical Use of Polynomial Preconditionings for the Conjugate Gradient Method , 1985 .

[14]  S. Nash.  Preconditioning of Truncated-Newton Methods , 1985 .

[15]  Richard H. Byrd,et al.  A Family of Trust Region Based Algorithms for Unconstrained Minimization with Strong Global Convergence Properties. , 1985 .

[16]  R. Schnabel,et al.  Tensor Methods for Nonlinear Equations. , 1984 .

[17]  T. Manteuffel,et al.  Necessary and Sufficient Conditions for the Existence of a Conjugate Gradient Method , 1984 .

[18]  P. Bosma,et al.  Efficient methods to calculate Chandrasekhar's H-functions , 1983 .

[19]  C. Kelley,et al.  A New Acceleration Method for Newton’s Method at Singular Points , 1983 .

[20]  H. Keller,et al.  CONVERGENCE RATES FOR NEWTON'S METHOD AT SINGULAR POINTS* , 1983 .

[21]  J. J. Moré,et al.  Estimation of sparse jacobian matrices and graph coloring problems , 1983 .

[22]  P. Toint,et al.  Local convergence analysis for partitioned quasi-Newton updates , 1982 .

[23]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[24]  D. Sorensen.  Newton's method with a model trust region modification , 1982 .

[25]  H. Walker,et al.  Convergence Theorems for Least-Change Secant Update Methods, , 1981 .

[26]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[27]  D. Rose,et al.  Global approximate Newton methods , 1981 .

[28]  Gilbert Strang,et al.  The application of quasi‐Newton methods in fluid mechanics , 1981 .

[29]  F. Luk,et al.  A Generalized Broyden''s Method for Solving Simultaneous Linear Equations , 1980 .

[30]  C. Kelley.  Solution of the Chandrasekhar H‐equation by Newton’s Method , 1980 .

[31]  C. Kelley,et al.  Newton’s Method at Singular Points. I , 1980 .

[32]  K. Atkinson.  An introduction to numerical analysis , 1979 .

[33]  Roland A. Sweet,et al.  Algorithm 541: Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial Differential Equations [D3] , 1979, TOMS.

[34]  E. Marwil,et al.  Convergence Results for Schubert’s Method for Solving Sparse Nonlinear Equations , 1979 .

[35]  J. H. Wilkinson,et al.  Inverse Iteration, Ill-Conditioned Equations and Newton’s Method , 1979 .

[36]  G. Reddien.  On Newton’s Method for Singular Problems , 1978 .

[37]  A. H. Sherman.  On Newton-Iterative Methods for the Solution of Systems of Nonlinear Equations , 1978 .

[38]  R. Schnabel,et al.  Least Change Secant Updates for Quasi-Newton Methods , 1978 .

[39]  T. Manteuffel.  Adaptive procedure for estimating parameters for the nonsymmetric Tchebychev iteration , 1978 .

[40]  P. Swarztrauber.  THE METHODS OF CYCLIC REDUCTION, FOURIER ANALYSIS AND THE FACR ALGORITHM FOR THE DISCRETE SOLUTION OF POISSON'S EQUATION ON A RECTANGLE* , 1977 .

[41]  P. Toint.  On sparse and symmetric matrix updating subject to a linear equation , 1977 .

[42]  R. Brent.  Table errata: Algorithms for minimization without derivatives (Prentice-Hall, Englewood Cliffs, N. J., 1973) , 1975 .

[43]  J. J. Moré,et al.  On the Global Convergence of Broyden''s Method , 1974 .

[44]  J. Ortega.  Numerical Analysis: A Second Course , 1974 .

[45]  M. Powell,et al.  On the Estimation of Sparse Jacobian Matrices , 1974 .

[46]  K. Atkinson.  Iterative variants of the Nyström method for the numerical solution of integral equations , 1974 .

[47]  John E. Dennis,et al.  On the Local and Superlinear Convergence of Quasi-Newton Methods , 1973 .

[48]  G. Stewart.  Introduction to matrix computations , 1973 .

[49]  W. E. Hart,et al.  Quasi-Newton Methods for Discretized Non-linear Boundary Problems , 1973 .

[50]  P. Gill,et al.  Quasi-Newton Methods for Unconstrained Optimization , 1972 .

[51]  C. W. Gear,et al.  Numerical initial value problem~ in ordinary differential eqttations , 1971 .

[52]  R. C. Cavanagh.  Difference equations and iterative processes , 1970 .

[53]  Herbert B. Keller,et al.  Newton's Method under Mild Differential Conditions , 1970, J. Comput. Syst. Sci..

[54]  T. Mullikin.  Some probability distributions for neutron transport in a half-space , 1968, Journal of Applied Probability.

[55]  O. Widlund,et al.  Iterative solution of elliptic systems : and applications to the neutron diffusion equations of reactor physics , 1967 .

[56]  A. Ostrowski.  Solution of equations and systems of equations , 1967 .

[57]  H. Keller,et al.  Analysis of Numerical Methods , 1967 .

[58]  J. Daniel.  The Conjugate Gradient Method for Linear and Nonlinear Operator Equations , 1967 .

[59]  L. B. Rall.  Convergence of the newton process to multiple solutions , 1966 .

[60]  J. Rice.  Experiments on Gram-Schmidt orthogonalization , 1966 .

[61]  C. G. Broyden.  A Class of Methods for Solving Nonlinear Simultaneous Equations , 1965 .

[62]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[63]  H. Brakhage.  Über die numerische Behandlung von Integralgleichungen nach der Quadraturformelmethode , 1960 .

[64]  I. W. Busbridge.  The Mathematics of Radiative Transfer. , 1960 .

[65]  Thomas E. Mott,et al.  Newton's Method and Multiple Roots , 1957 .

[66]  E. J. Craig.  The N‐Step Iteration Procedures , 1955 .

[67]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[68]  C. Lanczos.  Solution of Systems of Linear Equations by Minimized Iterations1 , 1952 .

[69]  W. Arnoldi.  The principle of minimized iterations in the solution of the matrix eigenvalue problem , 1951 .

[70]  M. Bartlett.  An Inverse Matrix Adjustment Arising in Discriminant Analysis , 1951 .

[71]  W. J. Duncan.  LXXVIII. Some devices for the solution of large sets of simultaneous linear equations: With an appendix on the reciprocation of partitioned matrices , 1944 .

[72]  Ernst Schröder,et al.  Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen , 1870 .

[73]  Stephen F. McCormick,et al.  Multilevel adaptive methods for partial differential equations , 1989, Frontiers in applied mathematics.

[74]  Thomas F. Coleman,et al.  Handbook for matrix computations , 1988 .

[75]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[76]  Zhishun A. Liu,et al.  A Look Ahead Lanczos Algorithm for Unsymmetric Matrices , 1985 .

[77]  G. Golub,et al.  Block Preconditioning for the Conjugate Gradient Method , 1985 .

[78]  G. Golub.  Matrix computations , 1983 .

[79]  T. Ypma.  The Effect of Rounding Errors on Newton-like Methods , 1983 .

[80]  J. Traub.  Iterative Methods for the Solution of Equations , 1982 .

[81]  Jorge J. Moré,et al.  Recent Developments in Algorithms and Software for Trust Region Methods , 1982, ISMP.

[82]  H. Elman.  Iterative methods for large, sparse, nonsymmetric systems of linear equations , 1982 .

[83]  Ekkehard W. Sachs,et al.  Algorithmic methods in optimal control , 1981 .

[84]  S. R. Searle,et al.  On Deriving the Inverse of a Sum of Matrices , 1981 .

[85]  E. Allgower,et al.  Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations , 1980 .

[86]  A. Griewank.  Analysis and modification of Newton's method at singularities , 1980 .

[87]  N. Bićanić,et al.  Who was ‘–Raphson’? , 1979 .

[88]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

[89]  R. Fletcher.  Conjugate gradient methods for indefinite systems , 1976 .

[90]  T. J. Rivlin.  The Chebyshev polynomials , 1974 .

[91]  D. Young,et al.  Iterative Solution of Large Linear Systems. , 1973 .

[92]  J. J. Moré,et al.  A Characterization of Superlinear Convergence and its Application to Quasi-Newton Methods , 1973 .

[93]  Lenhart K. Schubert.  Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian , 1970 .

[94]  B. Noble.  Applied Linear Algebra , 1969 .

[95]  V. Shamanskii.  A modification of Newton's method , 1967 .

[96]  Allen A. Goldstein,et al.  Constructive Real Analysis , 1967 .

[97]  L. Armijo.  Minimization of functions having Lipschitz continuous first partial derivatives. , 1966 .

[98]  L. Kantorovich,et al.  Functional analysis and applied mathematics , 1963 .

[99]  S. Banach.  Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales , 1922 .

[100]  É. Picard.  Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives , 1890 .