Trapezoidal algorithm

To obtain averaged root-mean-square radii of gyration, partition coefficients (K ) were transformed point by point to values as appropriate integrals were summed. Integrations were by a trapezoidal algorithm. Transformations were obtained from the following calibration curves ... 

Because Xk+i appears on both sides of this equation, additional steps are required to solve for x +i before the approximation can be used to calculate it. (This can be done via iteration, such as through the Newton-Raphson method.) Hence, the backward Euler method is also referred to as an implicit method. The trapezoidal algorithm averages the information from the forward and backward Euler algorithms such that the iteration equation to be used is... 

Method Derived from the Extended Trapezoid Algorithm... 

This assumes that the panel width is changed by a factor of 2 in the two calculations. This is easily implemented as previously discussed with the trapezoidal algorithm as all the previously summed function values can be reused as the panel size is reduced at each iterative step by a factor of 2. The use of Richardson s extrapolation when applied to integration is known in the literature as Romberg integration. 

A comparison of this equation with Eqs. (10.10) and (10.14) shows that flie trapezoidal algorithm has features of both the forward difference and backwards difference equations. For stability after an infinite number of steps this expression requires that... 

V-halTf hpITf This will be satisfied by any value in the negative half plane, i.e. by any value of ha<0. Thus for an exponentially decaying solution, the trapezoidal algorithm will be stable for any desired step size, similar to the backwards difference algorithm. 

Several related rorles or algorithms for numerical integration (rectangular mle, trapezoidal rule, etc.) are described in applied mathematics books, but we shall rely on Simpson s mle. This method can be shown to be superior to the simpler rules for well-behaved functions that occur commonly in chemistry, both functions for which the analytical form is not known and those that exist in analytical form but are not integrable. 

Equations (A), (B) and (C) are used in the algorithm to obtain the information required. Step (3) is used to calculate kA from equation (B), and step (4) is not required. Results are summarized in Table 12.2, for the arbitrary step-size in fA indicated G = tl[kA 1 -/A)j, and G represents the average of two consecutive values of G. The last column lists the time required to achieve the corresponding conversion in the second column. These times were obtained as approximations for the value of the integral in equation (A) by means of the trapezoidal rule ... 

These equations, (A) to (F ), may be solved using the E-Z Solve software or the trapezoidal rule for evaluation of the integral in (A). In the latter case, the following algorithm... 

Note PSpice does not support the Gear integration option. It instead relies on a modified trapezoidal-Gear integration algorithm for transient timestep operation. 

This formula, termed the RL-algorithm, based as it is on a connect-the-dots approximation, is the semi-integration equivalent of the trapezoidal formula of integration. 

Yeh KC, Kwan KC. A comparison of numerical integrating algorithms by trapezoidal, Lagrange, and spline approximation. J Pharmacokinet Biopharm 1978 6 79-98. 

When we have too few points to justify linearizing the function between adjacent points (as the trapezoidal integration does) we can use an algorithm based on a higher-order polynomial, which thereby can more faithfully represent the curvature of the function between adjacent measurement points. The Newton-Cotes method does just that for equidistant points, and is a moving polynomial method with fixed coefficients, just as the Savitzky-Golay method used for smoothing and differentation discussed in sections 8.5 and 8.8. For example, the formula for the area under the curve between x, and xn, is... 

Algorithms Derived from the Trapezoid Method 9 Table 13 Values of x,- and w, for the Gauss-Lobatto formula with five points. 

It is therefore possible to use 13 points to calculate a series of she integrals with the extended trapezoid method (with 2, 3, 4, 5, 7, and 13 points) and a series of four integrals with the extended central point method (with 1, 2, 3, and 6 points). It is possible to extrapolate both the series to zero with the Buhrsch-Stoer algorithm and also to compare the two results to have an estimate of the error. 

The initial membership functions (MF) are defined by 1 triangular and 2 trapezoidal functions for each variable involved. In future work the shape of the membership functions will be selected by the algorithm as part of the optimization. 

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