Algorithm Stability

Compliance with 11CFR part 35 should provide adequate proof of the ruggedness and robustness of the system software. Method developers must consider tests to evaluate algorithm stability during the development and validation processes. 

Conditioning status of the mathematical formulation. Algorithm stability. 

Algorithm stability decreases as algorithm order increases. 

In many practical applications, the integration of well-conditioned systems carried out using traditional methods requires a very small integration step to ensure algorithm stability. Such a step is generally much smaller than required to ensure reasonable method accuracy. 

This fact perplexed the pioneers who tackled the first ODE problems numerically because they were unaware of two phenomena the well conditioning and ill-conditioning of systems and numerical algorithm stability. 

Although qualitative, this discussion shows that controlling only the local error is usually a good way to keep the integration step within the region of algorithm stability since is overestimated as it exits such a region. 

The choice must be made, as usual, by giving the due priority to algorithm stability rather than efficiency. 

A novel optimization approach based on the Newton-Kantorovich iterative scheme applied to the Riccati equation describing the reflection from the inhomogeneous half-space was proposed recently [7]. The method works well with complicated highly contrasted dielectric profiles and retains stability with respect to the noise in the input data. However, this algorithm like others needs the measurement data to be given in a broad frequency band. In this work, the method is improved to be valid for the input data obtained in an essentially restricted frequency band, i.e. when both low and high frequency data are not available. This... 

O. Gonzales and J. C. Simo. On the stability of symplectic and energy-momentum conserving algorithms for nonlinear Hamiltonian systems with symmetry. Comp. Meth. App. Mech. Engin., 134 197, 1994. 

U. Ascher. Stabilization of invariants of discretized diflferential systems. Numerical Algorithms, 14 1-23, 1997. 

Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations other aspects involve the integration of ordinaiy differential equations apphcations include chaos and fractals as well as unusual operation of some chemical engineering eqmpment. Ref. 176 gives an excellent introduction to the subject and the details needed to apply the methods. Ref. 66 gives more details of the algorithms. A concise survey with some chemical engineering examples is given in Ref. 91. Bifurcation results are closely connected with stabihty of the steady states, which is essentially a transient phenomenon. 

MD runs for polymers typically exceed the stability Umits of a micro-canonical simulation, so using the fluctuation-dissipation theorem one can define a canonical ensemble and stabilize the runs. For the noise term one can use equally distributed random numbers which have the mean value and the second moment required by Eq. (13). In most cases the equations of motion are then solved using a third- or fifth-order predictor-corrector or Verlet s algorithms. 

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