## Order Stars: Theory and ApplicationsAccording to Hilbert's dictum, the scaffolding should be invisible in a math ematical edifice. Less kind interpretation of this common principle of writing and presenting mathematics is that we should always strive to do it baek to-front, forever wise after the event. Nobody should be allowed to see the seams in the supposedly seamless robe or eompare authors' intentions with the outeome of their endeavour. In particular, the short pieee of prose oeea sionally labelIed 'Prefaee' or 'Forward' ought to be written after the main body of the book. And so it is, and we, the authors, can refleet (with much trepidation) on an enterprise that for us is finally over. Order stars have been originally introduced in the context of numerical solution of ordinary differential equations and, as far as many numerical an alysts are concerned, they still belong there. It is our case in this book that the seope of order stars ranges considerably wider and that the cornerstone of the order star theory is a function-theoretic interpretation of complex approximation theory. An application to numerical analysis is a matter of serendipity, not of essen ce. |

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### Contents

General order stars | 9 |

Rational approximants to the exponential | 28 |

Aacceptability barriers | 50 |

Multistep methods | 92 |

The advection equation | 120 |

The diffusion equation | 155 |

Fade approximants | 177 |

Contractive approximation | 193 |

Open problems | 207 |

Advection equation with two space | 224 |

239 | |

### Common terms and phrases

4_-loop A-acceptable adjoin the origin analysis analytic function approach the origin approximants to exp argument principle asymptotic behaviour Blaschke product boundary bounded 4+-regions branch points Chapter characteristic function coefficients collocation complex fitting complex plane Consequently convergence Dahlquist defined degree denote error constant essential singularities essentially analytic Example explicit exponential expz Fade approximants Figure finite follows at once full discretization Hairer hence imaginary axis implies inequality integer interpolation points interval Iserles Jeltsch Laguerre polynomials left half-plane Lemma linear mathematical maximal order modulus Moreover multiderivative multiplicity multistep methods nonanalytic Norsett number of sectors number of zeros obeys optimal order stars ordinary differential equations p-restricted approximants Pade Pm/n polynomial problem Proposition 2.3 prove Rainville rational approximants rational function restricted Riemann surfaces Rm/n root condition Runge-Kutta methods scheme sectors of A+ semi-discretization singly p-restricted Sm/n solution theory underlying unit circle unit disc V-contractivity zeros and poles

### Popular passages

Page 235 - One-step methods of Hermite type for numerical integration of stiff systems, BIT 14 (1974) 63-77.

Page 235 - Attainable Order of Rational Approximations to the Exponential Function with only real Poles, BIT 17 (1977, 200-208.

Page 231 - Rational approximations to the exponential function with two complex conjugate interpolation points. SIAM J.

Page 228 - Akhiezer, NI (1965), The classical moment problem, Oliver and Boyd, Edinburgh and London. [2] Ashurst, WT and Meiron, D. (1987), "Numerical Study of Vortex Reconnection,

### References to this book

Dynamical Systems and Numerical Analysis, Volume 8 Andrew Stuart,A. R. Humphries Limited preview - 1998 |