Quadratic step

Truncating the path expansion at second-order (after the third term) yields a simple quadratic approximation to the path. Determining the coefficient of the second order term requires second energy derivatives. So if force constants are available, then a quadratic step can be taken along the path. This quadratic step accounts for the curvature of the reaction path at the point of expansion and consequently allows a larger step size than the Euler method does for a given accuracy. 

When comparing the data of these figures one can clearly see the symbasis between the accumulation of phenoxyl radicals and the dynamics of quadratic steps with their participation, steps (22) and (24). Here, the growth of negative influence of steps (27), (29), (10) may be tracked at the detectable consumption of the inhibitor, when the hydroperoxide of ethylbenzene and the quinolide peroxides are accumulated in the system. 

Note 2.5.- It should be noted in this mechanism that a quadratic step [2.R9c] intervenes. This is a step where two intermediates react with one another. 

A step during which two intermediates react with each other is sometimes called a quadratic step. 

Simulations of the adaptive reconstruction have been performed for a single slice of a porosity in ferritic weld as shown in Fig. 2a [11]. The image matrix has the dimensions 230x120 pixels. The number of beams in each projection is M=131. The total number of projections K was chosen to be 50. For the projections the usual CT setup was used restricted to angels between 0° and 180° with the uniform step size of about 3.7°. The diagonal form of the quadratic criteria F(a,a) and f(a,a) were used for the reconstruction algorithms (5) and (6). 

In simple relaxation (the fixed approximate Hessian method), the step does not depend on the iteration history. More sophisticated optimization teclmiques use infonnation gathered during previous steps to improve the estimate of the minunizer, usually by invoking a quadratic model of the energy surface. These methods can be divided into two classes variable metric methods and interpolation methods. 

In these methods, also known as quasi-Newton methods, the approximate Hessian is improved (updated) based on the results in previous steps. For the exact Hessian and a quadratic surface, the quasi-Newton equation and its analogue = Aq must hold (where - g " and... 

Xk) is the inverse Hessian matrix of second derivatives, which, in the Newton-Raphson method, must therefore be inverted. This cem be computationally demanding for systems u ith many atoms and can also require a significant amount of storage. The Newton-Uaphson method is thus more suited to small molecules (usually less than 100 atoms or so). For a purely quadratic function the Newton-Raphson method finds the rniriimum in one step from any point on the surface, as we will now show for our function f x,y) =x + 2/. 

This formula is exact for a quadratic function, but for real problems a line search may be desirable. This line search is performed along the vector — x. . It may not be necessary to locate the minimum in the direction of the line search very accurately, at the expense of a few more steps of the quasi-Newton algorithm. For quantum mechanics calculations the additional energy evaluations required by the line search may prove more expensive than using the more approximate approach. An effective compromise is to fit a function to the energy and gradient at the current point x/t and at the point X/ +i and determine the minimum in the fitted function. 

This approximates the root x = 4.93488 from Program QROOT in only two steps. Solution by the quadratic equation yields x = 4.93487. 

There are several reasons that Newton-Raphson minimization is rarely used in mac-romolecular studies. First, the highly nonquadratic macromolecular energy surface, which is characterized by a multitude of local minima, is unsuitable for the Newton-Raphson method. In such cases it is inefficient, at times even pathological, in behavior. It is, however, sometimes used to complete the minimization of a structure that was already minimized by another method. In such cases it is assumed that the starting point is close enough to the real minimum to justify the quadratic approximation. Second, the need to recalculate the Hessian matrix at every iteration makes this algorithm computationally expensive. Third, it is necessary to invert the second derivative matrix at every step, a difficult task for large systems. 

On page 235-241 is the explicit solution used in Excel format to make studies, or mathematical experiments, of any desired and possible nature. The same organization is used here as in previous Excel applications. Column A is the name of the variable, the same as in the FORTRAN program. Column B is the corresponding notation and Column C is the calculation scheme. This holds until line 24. From line 27 the intermediate calculation steps are in coded form. This agrees with the notation used toward the end of the FORTRAN listing. An exception is at the A, B, and C constants for the final quadratic equation. The expression for B was too long that we had to cut it in two. Therefore, after the expression for A, another forD is included that is then included in B. 

These options to the IRC keyword increase the maximum number of points on each side of the path to 15 and the step size between points to 0.3 amu bohr (30 units of 0.1 amu bohr), where the defaults are 6 steps and 0.1 amu bohr, respectively. The SCF=QC keyword requests the quadratic convergence SCF procedure, a somewhat slower but significantly more reliable SCF procedure. 

Many gradient methods approximate the energy surface at step by a quadratic expression in terms of the coordinate vector the total energy the gradient and the Hessian... 

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 ... 

Anotlrer way of choosing A is to require that the step length be equal to the trust radius R, this is in essence the best step on a hypersphere with radius R. This is known as the Quadratic Approximation (QA) method. ... 

The A for the minimization modes is determined as for the RFO method, eq. (14.8). The equation for Ays is quadratic, and by choosing the solution which is larger than 8ts it is guaranteed that the step component in this direction is along the gradient, i.e. a maximization. As for the RFO step, there is no guarantee that the total step length will be within the trust radius. 

Finally, we mention an approach based on using a different energy (or difference) function. The quadratic form used above (equation 10.58) is simple to use but it is not the only form possible. In fact, we could go through the same derivation steps as above by using any function /(Of, Sf) of the net s output Of and actual output... 

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