Quadratic stationary points

The simplest smooth fiuictioii which has a local miiiimum is a quadratic. Such a function has only one, easily detemiinable stationary point. It is thus not surprising that most optimization methods try to model the unknown fiuictioii with a local quadratic approximation, in the fomi of equation (B3.5.1). 

The advantage of the NR method is that the convergence is second-order near a stationary point. If the function only contains tenns up to second-order, the NR step will go to the stationary point in only one iteration. In general the function contains higher-order terms, but the second-order approximation becomes better and better as the stationary point is approached. Sufficiently close to tire stationary point, the gradient is reduced quadratically. This means tlrat if the gradient norm is reduced by a factor of 10 between two iterations, it will go down by a factor of 100 in the next iteration, and a factor of 10 000 in the next ... 

Find df(x)/dx = 0, a stationary point of the quadratic model of the function. The result obtained by differentiating Equation (5.6) with respect to x is... 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

A sequence of Newton-Raphson iterations is obtained by solving equation (4 4) redefining the zero point, p0, as the new set of parameters recalculating g and H and returning to equation (4 4). Such a procedure converges quadratically, that is, the error vector in iteration n is a quadratic function of the error vector in iteration n-1. This does not necessarily mean that the NR procedure will converge fast, or even at all. However, close to the stationary point we can expect a quadratic behaviour. We shall return later to a more precise definition of what close means in this respect. 

To determine the stationary points of the quadratic model we differentiate the model and set the result equal to zero. We obtain a linear set of equations... 

The quadratic model is an improvement on the linear model since it gives information about the curvature of the function and contains a stationary point. However, the model is still unbounded and it is a good approximation to fix) only in some region around xc. The region where we can trust the model to represent fix) adequately is called the trust region. Usually it is impossible to specify this region in detail and for convenience we assume that it has the shape of a hypersphere s <, h where h is the trust... 

To summarize, in the RF approach we make the quadratic model bounded by adding higher-order terms. This introduces n+1 stationary points, which are obtained by diagonalizing the augmented Hessian Eq. (3.22). The figure below shows three RF models with S equal to unity, using the same function and expansion points as for the linear and quadratic models above. Each RF model has one maximum and one minimum in contrast to the SO models that have one stationary point only. The minima lie in the direction of the true minimizer. 

Moreover, the number and type of stationary points is even larger in the case of a polyatomic molecule, as can be shown from the analysis of the structure of the hessian matrix of the force constants (Fy = d2 V/dRt 8Rj). Take R° as one such stationary point, and expand K(R) near R° in a Taylor series (note that grad V = 0 for s = 0). Up to quadratic terms, one gets... 

To see this, consider the Taylor expansion of a quadratic function q about a stationary point x ... 

As we noted above, the kinetic energy is positive definite. Furthermore, it is quadratic in the momenta. As a consequence, we can reduce the search for points of stationary flow in phase space to one of finding the stationary points of the potential energy surface. To see how this comes about, consider the Hamilton s equations for the three velocities... 

If A is a square matrix, the quadratic form x Ax is a scalar product. In Chapter 12 on response surface models was discussed how the stationary point on the response surface was determined as the roots of the systems of equations defined by setting all partial derivatives of the response surface model to zero. In matrix language this corresponds to determining for which values of the x variables, the vector d Mdx is the null vector. This derivative is computed as... 

When 0stationary state (1, 1) is a stable node. When a = otj equation (6.71) has two equal real roots, kr = k2 < 0. The value kt is determined from the requirement of vanishing of the discriminant A of the quadratic equation (6.71). This is a sensitive state when a increasing exceeds the value ax, a change in the phase portrait of the system (6.67) takes place the stationary state is, for < a < a0 = (y — 1) a stable focus (the real part of kx >2 is negative). In accordance with what we established in Chapter 5, the catastrophe does not alter a phase portrait nearby a stationary point but is of a global character. 

Vibrational frequencies are calculated to obtain IR spectra, to characterize stationary points, and to obtain zero point energies (below). The calcnlation of meaningful frequencies is valid only at a stationary point and only using the same method that was used to optimize to that stationary point (e.g. an ab initio method with a particular correlation level and basis set - see chapter 5). This is because (1) the use of second derivatives as force constants presupposes that the PES is quadratically curved along each geometric coordinate q (Eig. 2.2) but it is only near a stationary point that this is true, and (2) use of a method other than that nsed to obtain the stationary point presupposes that the PES s of the two methods are parallel (that they have the same... 

Given wavefunclions belonging to one or more states that are obtained from an MCSCF, HF, CI RSPT, or CC calculation, one is often interested in subsequently using these wavefunctions to compute physical properties of the system other than the total electronic energy. Below we discuss how the three distinct classes of properties—expectation values, transition properties, and response properties—may be evaluated, and we show also how stationary points on the potential energy surface may be determined using a quadratic cally convergent procedure. 

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