Topics in Numerical Analysis

The paper shows that by means of a simple variable transformation many formulas given in the literature for bounds including the positive root of certain polynomials can already be derived out of those for special simple cases of these polynomials. In the same way some more complicated cases can now be treated with the help of known results. These polynomials are occurring in mathematics of finance and in the theory of the determination of the order of convergence of iterative numerical processes. AMS Subject Classifications: 26C05, 65H05, 91B28.

[1]  F. Ana,et al.  A new method for large-scale box constrained convex quadratic minimization problems , 1995 .

[2]  José Mario Martínez,et al.  On the Resolution of Linearly Constrained Convex Minimization Problems , 1994, SIAM J. Optim..

[3]  P. Toint,et al.  Global convergence of a class of trust region algorithms for optimization with simple bounds , 1988 .

[4]  L. B. Rall A Comparison of the Existence Theorems of Kantorovich and Moore , 1980 .

[5]  Roberto Andreani,et al.  The reformulation of nonlinear complementarity problems using the Fischer-burmeister function , 1999 .

[6]  E. Birgin,et al.  Determination of thickness and optical constants of amorphous silicon films from transmittance data , 2000 .

[7]  Zdenek Dostál,et al.  Augmented Lagrangians with Adaptive Precision Control for Quadratic Programming with Equality Constraints , 1999, Comput. Optim. Appl..

[8]  M. Raydan On the Barzilai and Borwein choice of steplength for the gradient method , 1993 .

[9]  P. Deuflhard,et al.  Affine Invariant Convergence Theorems for Newton’s Method and Extensions to Related Methods , 1979 .

[10]  José Mario Martínez,et al.  Reformulation of Variational Inequalities on a Simplex and Compactification of Complementarity Problems , 1999, SIAM J. Optim..

[11]  B. Friedlander,et al.  An Adaptive Algorithm for Bound Constrained Quadratic Minimization , 1997 .

[12]  Nicholas I. M. Gould,et al.  CUTE: constrained and unconstrained testing environment , 1995, TOMS.

[13]  Ana Friedlander,et al.  On the Maximization of a Concave Quadratic Function with Box Constraints , 1994, SIAM J. Optim..

[14]  Roberto Andreani,et al.  On the solution of the extended linear complementarity problem , 1998 .

[15]  E. Birgin,et al.  Estimation of the optical constants and the thickness of thin films using unconstrained optimization , 2006 .

[16]  J. M. Martínez BOX-QUACAN and the implementation of Augmented Lagrangian algorithms for minimization with inequalit , 1998 .

[17]  J. M. Martínez,et al.  Gradient Method with Retards and Generalizations , 1998 .

[18]  José Mario Martínez,et al.  Nonmonotone Spectral Projected Gradient Methods on Convex Sets , 1999, SIAM J. Optim..

[19]  J. Júdice,et al.  Solution of a General Linear Complementarity Problem Using Smooth Optimization and Its Application to Bilinear Programming and LCP , 2001 .

[20]  José Mario Martínez,et al.  A Curvilinear Search Using Tridiagonal Secant Updates for Unconstrained Optimization , 1991, SIAM J. Optim..

[21]  Marcos Raydan,et al.  The Barzilai and Borwein Gradient Method for the Large Scale Unconstrained Minimization Problem , 1997, SIAM J. Optim..

[22]  Shen Zuhe,et al.  A note on the comparison of the Kantorovich and Moore theorems , 1990 .

[23]  L. T. Santos,et al.  Restricted optimization: a clue to a fast and accurate implementation of the Common Reflection Surface Stack method , 1999 .

[24]  V. Torczon,et al.  A GLOBALLY CONVERGENT AUGMENTED LAGRANGIAN ALGORITHM FOR OPTIMIZATION WITH GENERAL CONSTRAINTS AND SIMPLE BOUNDS , 2002 .

[25]  Ramon E. Moore A Test for Existence of Solutions to Nonlinear Systems , 1977 .

[26]  J. M. Martínez,et al.  A new trust region algorithm for bound constrained minimization , 1994 .

[27]  Zdeněk Dostál,et al.  Solution of Coercive and Semicoercive Contact Problems by FETI Domain Decomposition , 1998 .

[28]  J. M. Martínez,et al.  Solution of Finite-Dimensional Variational Inequalities Using Smooth Optimization with Simple Bounds , 1997 .

[29]  José Mario Martínez,et al.  Nonlinear programming algorithms using trust regions and augmented Lagrangians with nonmonotone penalty parameters , 1999, Math. Program..

[30]  J. M. Martínez,et al.  A new strategy for solving variational inequalities in bounded polytopes , 1995 .

[31]  José Mario Martínez,et al.  Solution of linear complementarity problems using minimization with simple bounds , 1995, J. Glob. Optim..

[32]  Arnold Neumaier,et al.  The Krawczyk operator and Kantorovich's theorem , 1990 .

[33]  Roberto Andreani,et al.  Solving Complementarity Problems by Means of a New Smooth Constrained Nonlinear Solver , 1998 .

[34]  Zdenek Dostál,et al.  Box Constrained Quadratic Programming with Proportioning and Projections , 1997, SIAM J. Optim..

[35]  Ramon E. Moore,et al.  A Simple Test for Accuracy of Approximate Solutions to Nonlinear (or Linear) Systems , 1980 .

[36]  José Mario Martínez,et al.  Validation of an Augmented Lagrangian Algorithm with a Gauss-Newton Hessian Approximation Using a Set of Hard-Spheres Problems , 2000, Comput. Optim. Appl..