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__Nordsieck methods with inherent quadratic stability__

M. Bras

**Abstract**

We will discuss general linear methods (GLMs) for the numerical solution of systems of ordinary differential equations.
They are characterized by four integers $(p,q,r,s)$, where $p$ is the order of the method, $q$ is the stage order of the method, $s$ is the number of internal stages and $r$ is the number of external approximations.
In order to simplify the analysis of this class, we introduce the inherent quadratic stability property (IQS), i.e., the conditions on method's coefficients that guarantee
that the stability function assumes quadratic form
$p(w,z) = w^{r-1} (w^2 -p_1(z)w+p_0(z)),$
where $p_1(z)$ and $p_0(z)$ are rational function of $z$. Then the stability properties are determined by the quadratic part of the stability polynomial.
We assume, that the vector of external stages approximates the Nordsieck vector. After applying order, stage order and IQS conditions we search for methods with desired stability properties.

**Bibliography**

[1]
M. Bras,
Nordsieck Methods with Inherent Quadratic Stability, Math. Model. Anal., 16 (2011), pp. 82-96.

[2]
M. Bras and A. Cardone,
Construction of efficient general linear methods for non-stiff differential systems, Math. Model. Anal., 17 (2012), pp. 171-189.

[3]
Z. Jackiewicz,
General linear methods for ordinary differential equation, John Wiley & Sons Inc., Hoboken, NJ, 2009.