Ebook: Nonlinear Numerical Methods and Rational Approximation

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gu!ik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; use Stein spaces. And in addition to this there are and prediction and electrical engineering can such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

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Content:
Front Matter....Pages i-xviii
Front Matter....Pages 1-1
Integral Approximants for Functions of Higher Monodromic Dimension....Pages 3-22
Asymptotics of Hermite-Pad? Polynomials and Related Convergence Results — A Summary of Results....Pages 23-53
Front Matter....Pages 55-55
On the Behavior of Zeros and Poles of Best Uniform Polynomial and Rational Approximants....Pages 57-75
Once Again: The Adamjan-Arov-Krein Approximation Theory....Pages 77-91
Diagonal Pad? Approximants, Rational Chebyshev Approximants and Poles of Functions....Pages 93-103
On the Use of the Carath?odory-Fej?r Method for Investigating ‘1/9’ and Similar Constants....Pages 105-132
Front Matter....Pages 133-133
Simultaneous Rational Approximation to Some q-Hypergeometric Functions....Pages 135-142
Minimal Pade-Sense Matrix Approximations Around s = 0 and s = ?....Pages 143-154
(Pade)y of (Pade)x Approximants of F(x,y)....Pages 155-166
Different Techniques for the Construction of Multivariate Rational Interpolants....Pages 167-190
Front Matter....Pages 191-210
Some Orthogonal Systems of p+1Fp-Type Laurent Polynomials....Pages 211-211
The Moment Problem on Equipotential Curves....Pages 213-227
Difference Equations, Continued Fractions, Jacobi Matrices and Orthogonal Polynomials....Pages 229-238
Multipoint Pad? Approximation and Orthogonal Rational Functions....Pages 239-257
L-Polynomials Orthogonal on the Unit Circle....Pages 259-270
Front Matter....Pages 271-278
Schur’s Algorithm Extended and Schur Continued Fractions....Pages 279-279
Some Recent Results in the Analytic Theory of Continued Fractions....Pages 281-298
Best a Posteriori Truncation Error Estimates for Continued Fractions K(an/1) with Twin Element Regions....Pages 299-333
Convergence Acceleration for Miller’s Algorithm....Pages 335-347
Front Matter....Pages 349-370
A New Approach to Convergence Acceleration Methods....Pages 371-371
Front Matter....Pages 373-405
General T—Fraction Solutions to Riccati Differential Equations....Pages 407-407
A Simple Alternative Principle for Rational ? — Method Approximation....Pages 409-425
Evaluation of Fermi—Dirac Integral....Pages 427-434
An Application of Operator Pade Approximants to Multireggeon Processes....Pages 435-444
Back Matter....Pages 445-452
....Pages 453-458