# 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…

## 201 Citations

### Constructive characterization ofA-stable approximations to exp(z) and its connection with algebraically stable Runge-Kutta methods

- Mathematics
- 1982

SummaryAll rational approximations to exp(z) of order ≧2m−β (m denotes the maximal degree of nominator and denominator) are given by a closed formula involving β real parameters. Using the theory of…

### Generalized Padé approximations to the exponential function

- Mathematics
- 1992

The stability properties of the Padé rational approximations to the exponential function are of importance in determining the linear stability properties of several classes of Runge-Kutta methods. It…

### Order stars, contractivity and a pick-type theorem

- Mathematics
- 1984

Given a function f that is analytic in the complex domain V and such that |f|≡1 along ∂V (with the possible exception of essential singularities) we examine analytic approximations R to f that are…

### Characterization of allA-stable methods of order 2m-4

- Computer Science
- 1980

This paper characterize theA-acceptable approximations of orderp ≧2m-4 and apply the result to 12-parameter families of implicit Runge-Kutta methods.

### Order stars and the structure of Padé tableaux

- Mathematics
- 1984

We demonstrate by using the theory of order stars that analytic properties of complex functions impose bounds on the maximal block size in their Pade tableau. After a short survey of the relevant…

### The A-Stability of Methods with Padé and Generalized Padé Stability Functions

- MathematicsNumerical Algorithms
- 2004

This paper surveys some stability results and suggests the use of ‘order arrows’ as an alternative to order stars in studying questions about the possible A-stability of a numerical method. A…

### A(α)-acceptability of pade approximations to function exp (Q)

- Mathematics
- 1999

AbstractIn this paper, a necessary and sufficient condition of A(a)-acceptability for rational approximations to the functionexp(q) and some sufficient conditions to guarantee A(α)-acceptability of…

### Order stars and stability for delay differential equations

- MathematicsNumerische Mathematik
- 1999

It is proved that all Gauss methods are $\tau (0)$-stable, and the theory of order stars is used to characterize high-order symmetric methods with this property.

### On Energy Laws and Stability of Runge-Kutta Methods for Linear Seminegative Problems

- Computer ScienceSIAM J. Numer. Anal.
- 2022

A unified energy identity for all the diagonal Padé approximations is discovered, based on an analytical Cholesky type decomposition of a class of symmetric matrices, which provides a precise characterization on whether and how the energy dissipates in the RK discretization, thereby leading to weak and strong stability criteria of RK methods.

## References

SHOWING 1-10 OF 13 REFERENCES

### On rational approximations to the exponential

- Mathematics, Philosophy
- 1977

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

- Mathematics
- 1976

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

- Mathematics
- 1978

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

- Mathematics
- 1974

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

- Mathematics
- 1977

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…

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

- Mathematics
- 1977

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…

### C-Polynomials for rational approximation to the exponential function

- Business
- 1975

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

- Computer Science, Mathematics
- 1973

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

- Mathematics
- 1974

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…