On the Continuous Realization of Iterative Processes

Many important mathematical problems are solved by iterative methods. Many of these iterative schemes may be regarded as the discrete realization of certain continuous dynamical systems. This paper summarizes some of the recent developments in the continuous realization of several popular basic iterative methods.

[1]  L. Collatz Functional analysis and numerical mathematics , 1968 .

[2]  J. G. F. Francis,et al.  The QR Transformation A Unitary Analogue to the LR Transformation - Part 1 , 1961, Comput. J..

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

[4]  J. H. Wilkinson The algebraic eigenvalue problem , 1966 .

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

[6]  F. H. Branin Widely convergent method for finding multiple solutions of simultaneous nonlinear equations , 1972 .

[7]  G. Stewart,et al.  An Algorithm for Generalized Matrix Eigenvalue Problems. , 1973 .

[8]  B. Parlett The Rayleigh Quotient Iteration and Some Generalizations for Nonnormal Matrices , 1974 .

[9]  J. J. Moré,et al.  Quasi-Newton Methods, Motivation and Theory , 1974 .

[10]  J. Yorke,et al.  Finding zeroes of maps: homotopy methods that are constructive with probability one , 1978 .

[11]  H. B. Keller Global Homotopies and Newton Methods , 1978 .

[12]  K. Tanabe Continuous Newton-Raphson method for solving and underdetermined system of nonlinear equations☆ , 1979 .

[13]  M. Hirsch,et al.  On Algorithms for Solving f(x)=0 , 1979 .

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

[15]  Werner C. Rheinboldt,et al.  Solution Fields of Nonlinear Equations and Continuation Methods , 1980 .

[16]  R. B. Kearfott A derivative-free arc continuation method and a bifurcation technique , 1981 .

[17]  K. Georg Numerical integration of the Davidenko equation , 1981 .

[18]  Philip E. Gill,et al.  Practical optimization , 1981 .

[19]  D. Faddeev,et al.  Computational methods of linear algebra , 1981 .

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

[21]  W. Symes The QR algorithm and scattering for the finite nonperiodic Toda Lattice , 1982 .

[22]  David S. Watkins,et al.  Understanding the $QR$ Algorithm , 1982 .

[23]  R. Kellogg,et al.  Pathways to solutions, fixed points, and equilibria , 1983 .

[24]  M. T. Chu,et al.  On a numerical treatment for the curve-tracing of the homotopy method , 1983 .

[25]  Werner C. Rheinboldt,et al.  Algorithm 596: a program for a locally parameterized , 1983, TOMS.

[26]  Gene H. Golub,et al.  Matrix computations , 1983 .

[27]  Moody T. Chu,et al.  On the Global Convergence of the Toda Lattice for Real Normal Matrices and Its Applications to the Eigenvalue Problem , 1984 .

[28]  M. Chu The Generalized Toda Flow, the QR Algorithm and the Center Manifold Theory , 1984 .

[29]  F. Brauer,et al.  The Qualitative Theory of Ordinary Differential Equations: An Introduction , 1989 .