Order stars and stability theorems

@article{Wanner1978OrderSA,
  title={Order stars and stability theorems},
  author={Gerhard Wanner and Ernst Hairer and Syvert P. N{\o}rsett},
  journal={BIT Numerical Mathematics},
  year={1978},
  volume={18},
  pages={475-489}
}
AbstractThis paper clears up to the following three conjectures:1.The conjecture of Ehle [1] on theA-acceptability of Padé approximations toez, which is true;2.The conjecture of Nørsett [5] on the zeros of the “E-polynomial”, which is false;3.The conjecture of Daniel and Moore [2] on the highest attainable order of certainA-stable multistep methods, which is true, generalizing the well-known Theorem of Dahlquist. We further give necessary as well as sufficient conditions forA-stable (acceptable… 

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References

SHOWING 1-10 OF 13 REFERENCES

On rational approximations to the exponential

TLDR
A simple characterization of the A-acceptability proper ty for a family of rational approximations to e~* is given, which is implicitely contained in Norsett [3].

Note onA-stability of multistep multiderivative methods

Daniel and Moore [4] conjectured that anA-stable multistep method using higher derivatives cannot have an error order exceeding 2l. We confirm partly this conjecture by showing that for a large class

Restricted Pad Approximations to the Exponential Function

In the rational Pade approximation to exp $\exp ( - q),q \in \mathbb{C}$, the parameters in the numerator and denominator are chosen to give maximum order. The zeros of the denominator of these

An algebraic approach toA-stable linear multistep-multiderivative integration formulas

A general algebraic approach and some new results are given pertaining to the synthesis of linearA-stable multistep-multiderivative formulas used for integrating stiff differential equations. This

Attainable order of rational approximations to the exponential function with only real poles

Rational approximations of the form Σi=0maiqi/Πi=1n (1+γiq) to exp(−q),qεC, are studied with respect to order and error constant. It is shown that the maximum obtainable order ism+1 and that the

C-Polynomials for rational approximation to the exponential function

SummaryA unique correspondence between (m, n) rational approximations to exp (q) of order at leastm and a polynomial of degreen, theC-polynomial, is obtained. This polynomial is then used to find an

Analysis of Discretization Methods for Ordinary Differential Equations

TLDR
The Discretization Methodology helps clarify the meaning of Consistency, Convergence, and Stability with Forward Step Methods and provides a guide to applications of Asymptotic Expansions in Even Powers of n.

One-step methods of hermite type for numerical integration of stiff systems

One-step methods of Hermite type with coefficients equal to the derivatives of Laguerre polynomials at certain points are considered. The methods areA-stable of order 1, 2, 3, 5 and for order higher

A note on a recent result of rational approximations to the exponential function

In a recent paper by Nørsett and Wolfbrandt [1] it is shown that the maximum attainable order ofN-approximationsRm,n(u) to exp (u) ism + 1. The purpose of this note is to present an alternative proof