Ill-conditioned systems

P. Deuflhard. A Modified Method for the Solution of Ill-Conditioned Systems of Nonlinear Equations with Application to Multiple Shooting. Numer. Math., 32 289-315,1974. 

This ensures that even in the case of ill-conditioned systems, for example, in the case of very similar spectra, good results for the... 

In the case of full rank, all singular values will be obviously different from zero and the SVD solution equals that of OLS. However, one often comes up with several small singular values because of ill-conditioned systems. Therefore, the main goal of PCR is not to keep all singular values for an exact representation of the Moore-Penrose matrix, but to select a subset of singular values that best guarantee predictions of unknown cases. 

A bit of confusion may arise between stiff and nonstiff problems and well- and ill-conditioned systems. For example, the equation... 

Experimental results for comparisons of all the direct methods for the solution of systems of linear equations applied to systems with a dense coefficient matrix (Klem 1979) has shown the methods based on triangular decomposition to be better than the methods based on orthogonal decomposition, according to all the various criteria for comparison (accuracy, computer time, computer storage) this applies in cases of both well-conditioned and ill-conditioned systems. 

The assessment of whether the system is well- or ill-conditioned maybe wrong. It means, for instance, that we may think we have a good solution to the system because it looks well-conditioned, whereas it is actually very ill-conditioned and therefore the solution is wrong. The opposite may also be true we can think we have a very ill-conditioned system, but we do not. 

Among the implicit methods are the Gaussian elimination and methods such as the modified strongly implicit (MSI) procedure, the LU-SSOR, and the implicit Runge-Kutta. The parallelization of implicit methods is more elaborate than for explicit methods. Implicit methods are frequently employed for solving ill-conditioned problems, such as those that arise in reactive flows. Thus, in physical terms, implicit methods are best suited to address ill-conditioned systems, while in computational terms these methods are preferred to resolve small matrix systems. 

Considering the fact that a higher-order polynomial may result in an ill-conditional system of equations for unknown coefficients and exhibit... 

R. Neelamani, H.Choi, and R.Baraniuk, "Eorward Fourier-wavelet regularized deconvolution for ill-conditioned systems IEEE Transactions on signal processing 52 pp. 418-433, 2004... 

If however, matrix A is reasonably well-conditioned at the optimum, A could easily be ill-conditioned when the parameters are away from their optimal values. This is quite often the case in parameter estimation and it is particularly true for highly nonlinear systems. In such cases, we would like to have the means to move the parameters estimates from the initial guess to the optimum even if the condition number of matrix A is excessively high for these initial iterations. 

This system of 2 M nonlinear equations is ill-conditioned for large M, but can be efficiently solved using the product-difference (PD) algorithm introduced by McGraw (1997). Thus, given the set of 2 M moments on the left-hand side of Eq. (107), the PD algorithm returns wm and lm for m — 1., M. The closed microscopic transport equation for the moments can then be written for k — 0,..., 2 M— 1 as... 

These schemes have been frequently suggested [105-107] as possible mechanisms to achieve the chirally pure starting point for prebiotic molecular evolution toward our present homochiral biopolymers. Demonstrably successftd amplification mechanisms are the spontaneous resolution of enantiomeric mixtures under race-mizing conditions, [509 lattice-controlled solid-state asymmetric reactions, [108] and other autocatalytic processes. [103, 104] Other experimentally successful mechanisms that have been proposed for chirality amplification are those involving kinetic resolutions [109] enantioselective occlusions of enantiomers on opposite crystal faces, [110] and lyotropic liquid crystals. [Ill] These systems are interesting in themselves but are not of direct prebiotic relevance because of their limited scope and the specialized experimental conditions needed for their implementation. 

In the present case, the system has more equations (11) than unknowns (7) and may be conveniently solved for x = Al3...Ah2. Png, and Pn by the least-square solution alluded to above. The system is, in general, ill-conditioned and extended precision should be used for the inversion. 

While the point - area method is very convenient in terms of computational efforts, it has a serious drawback. The matrix of the linear system (5.6B) is inherently ill - conditioned (ref. 25), and the result is very sensitive to the errors in the observations. 

Jacobian becomes increasingly ill-conditioned. With sufficiently great scale disparity, the Jacobian can become effectively singular and the linear system can not be solved numerically. 

It should be noted that there is a growing list of conditionally essential amino acids. These are amino acids that must be augmented through the diet under a variety of special circumstances. Often these circumstances include infancy (where the various enzyme systems for metabolism have not been fully expressed) or periods of metabolic stress due to injury or illness. Conditionally essential amino acids include taurine (infants) and glutamine (metabolic stress). 

Another specific VFF feature is that it relies on the transferability of the force-constants from one molecule to chemically and structurally related systems. Thus a set / , optimized for simpler and well studied substances is used as a trial force field for the system under consideration. Due to the ill-conditioned nature of IVP special measures have to be taken in order to keep the adjustable force-constants as close as possible to the initial trial set. One possible approach is to restrict them in a physically meaningful interval of say 10% around the starting values. Alternatively a penalty function can be added to the minimized functional (4) [4] ... 

For two-time-scale systems, it is well established that inversion-based controllers designed without explicitly accounting for the time-scale multiplicity are ill-conditioned and can lead to closed-loop instability. In order to avoid such issues, controller design must be addressed on the basis of the reduced-order representations of the slow and fast dynamics, an approach referred to as composite control (see, e.g., Chow and Kokotovic 1976, 1978, Saberi and Khalil 1985, Kokotovic et al. 1986, Christofides and Daoutidis 1996a, 1996b). 

However, the stiffness/ill conditioning of the model (3.31) will strongly impact on the implementation of optimization controllers (e.g., a model predictive controller) (Baldea et al. 2010). On the other hand, for any choice of four flow rates as manipulated inputs (keeping the remaining one constant at its nominal value), the system is non-minimum phase (Kumar and Daoutidis 2002) and thus potentially closed-loop unstable with an inversion-based controller.3 As discussed in the previous section, a more systematic controller-design approach would... 

An expanded formulation of the steady-state permeation model has been presented. Two numerical problems - stiffness and an ill-conditioned boundary value problem - are encountered in solving the system equations. These problems can be circumvented by matching forward and reverse integrations at a point near the inlet (n = 0) but outside the combustion zone. The model predicts a... 

From a computational point of view it is important to ensure that the problem is not ill-conditioned so as to maintain numerical stability. Therefore, the use of rational functions as proposed by Bergmann et a/.81 is combined with the established practice of fitting polynomials to given data using orthogonal polynomial bases.83 A system of linear equations is solved at each iteration.81 Hence, the condition number83 of this system may be used to monitor numerical stability. 

Neumaier, A., Solving ill-conditioned and singular linear systems a tutorial on regularization, SIAM Rev, 40, 636-666, 1998. 

Riley, J.D., Solving systems of linear equations with a positive definite symmetric but possibly ill-conditioned matrix, Math. Table Aids Comput., 9, 96-101, 1955. 

Solving this system for unknowns a - can be achieved by the methods of Sect. 4.2 but, in most cases, the equations are very ill conditioned and, thus, the use of such polynomials is limited to low orders (less than about 10). 

Such systems are potentially stiff or ill-conditioned. The width of the domain, in which the local time constants may be found, provides a measure of the stiffness or ill-conditioning of the system. The local stiffness ratio, S, is thus given by the expression... 

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