Computation of the Residual

The central part of the Newton-Gauss algorithm is the computation of the residuals, which are now collected in the matrix R. R is a function of the measurement Y, the model, and the parameters. For the example, the parameters include the two rate constants k and fe, which we collect in the vector p and all molar absorptivities, all elements of the matrix A. For a given model we can write... 

The dimensions of these subspace representations are small enough so that Eqs (274) and (275) may be solved using direct linear equation an matrix eigenvalue equation methods. The computation of the residual vector uses the same matrix-vector products as is required for the construction of the subspace representations of the Hessian blocks. If either of the components of the residual vector are too large, a new expansion vector corresponding to the large residual component may be computed from one of the equations... 

When the scores, weights, and loadings have been determined for a latent variable (at convergence), X- and Y-block matrices are adjusted to exclude the variation explained by that latent variable. Equations 4.20 and 4.21 illustrate the computation of the residuals after the first latent variable and weights have been determined ... 

Here (,) denotes the inner product and (Av.,v, are known from the previous application of the CG method. Then computation of ° requires only computation of the residual and... 

In the following we will discuss parallel implementation of the two dominant computational steps in the LMP2 procedure, namely the two-electron integral transformation and the computation of the residual. Parallelization of other computationally less demanding, but nonnegligible, steps is straightforward. ... 

The computation of the residual is the dominant step in the iterative procedure. From Eq. 10.6, we see that a given residual matrix R,y, with elements Rfj , contains contributions from the integrals and double-substitution amplitudes with the same occupied indices, K,j and T,-y, respectively, as well as from the double-substitution amplitudes Tik and Tkj- The contributions from Tik and Tkj complicate the efficient parallelization of the computation of the residual and make communication necessary in the iterative procedure. The double-substitution amplitudes can either be replicated, in which case a collective communication (all-to-all broadcast) step is required in each iteration to copy the new amplitudes to all processes or the amplitudes can be distributed, and each process must then request amplitudes from other processes as needed throughout the computation of the residual. Achieving high parallel efficiency in the latter case requires the use of one-sided messagepassing. 

Solution The system of two equations (Eqs. (4.214) and (4.215)) and two unknowns (that is and P ) for the Baker and Luks formulation can be solved via the secant method. This is just a Newton-Raphson method on numerical derivatives. Computation of the residual in Eq. (4.214) is fairly straight forward, because it only requires expressions for the second derivatives of the Helmholtz free energy/I in terms of V and These derivatives arc provided in Example 4.8. Equation (4.215) is somewhat more complicated its determinant requires derivatives of Eq. (4.214) with respect to V and N. The procedure presented in Problem 4.14 can be used to evaluate the determinant derivatives. After both residuals are computed, the system of two equations and two unknowns are solved by the Newton-Raphson method to convergence. 

Equation 15.78 gives the radial stress at the outer surface of the monobloc core due to the windings (prior to the application of internal pressure) if rj is substituted for r. This radial stress is numerically equal to the external pressure on the core and, together with the dimensions of the monobloc, permits the computation of the residual-stress distribution. 

Only a few quantitative data are available on copolymerization of methacrylates. Direct determination of the cross-propagation constants is readily achieved in living polymer systems whenever the absorption spectra of the two propagating species are different. Unfortunately, this is not the case in the methacrylate series. A new approach to this problem was developed by Muller 43). A mixture of two monomers is copolymerized, the reaction is interrupted at various times, and the concentrations of the residual monomers are determined as functions of time. The pertinent differential equations include 4 constants ku, k12, k21, and k22. Since kn and k22 were independently determined, the remaining cross-propagation constants are obtained by computer fitting the experimental conversion curves to the calculated ones. 

Fig. 37.2. Principal components loading plot of 7 physicochemical substituent parameters, as obtained from the correlations in Table 37.5 [39,40]. The horizontal and vertical axes account for 46 and 31%, respectively, of the correlations. Most of the residual correlation is along the perpendicular to the plane of the diagram. The line segments define clusters of parameters that have been computed by means of cluster analysis.

Application of the test substance to the test system is without doubt the most critical step of the residue field trial. Under-application may be corrected, if possible and if approved by the Study Director, by making a follow-up application if the error becomes known shortly after the application has been made. Over-application errors can usually only be corrected by starting the trial again. The Study Director must be contacted as soon as an error of this nature is detected. Immediate communication allows for the most feasible options to be considered in resolving the error. If application errors are not detected at the time of the application, the samples from such a trial can easily become the source of undesirable variability when the final analysis results are known. Because the application is critical, the PI must calculate and verify the data that will constitute the application information for the trial. If the test substance weight, the spray volume, the delivery rate, the size of the plot, and the travel speed for the application are carefully determined and then validated prior to the application, problems will seldom arise. With the advent of new tools such as computers and hand-held calculators, the errors traditionally associated with applications to small plot trials should be minimized in the future. The following paragraphs outline some of the important considerations for each of the phases of the application. 

Iversen et al, in their study of crystalline beryllium [32], were the first to make use of NUP distributions calculated by superposition of thermally-smeared spherical atoms. More recently, a superposition of thermally-smeared spherical atoms was used as NUP in model studies on noise-free structure factor amplitudes for crystalline silicon and beryllium by de Vries et al. [38]. The artefacts present in the densities computed with a uniform prior-prejudice distributions have been shown to disappear upon introduction of the NUP. No quantitative measure of the residual errors were given. 

This means that the computation of the optimum estimate of x is effectively decoupled from the estimate of the bias. Moreover, it can be computed in terms of the residuals of the bias-free estimate. In terms of x, this can be expressed as follows ... 

Non-linear regression calculations are extensively used in most sciences. The goals are very similar to the ones discussed in the previous chapter on Linear Regression. Now, however, the function describing the measured data is non-linear and as a consequence, instead of an explicit equation for the computation of the best parameters, we have to develop iterative procedures. Starting from initial guesses for the parameters, these are iteratively improved or fitted, i.e. those parameters are determined that result in the optimal fit, or, in other words, that result in the minimal sum of squares of the residuals. 

The crucial part of this algorithm is the computation of J, the derivatives of the residuals with respect to the parameters. It might be best to demonstrate this by an example. 

The derivatives of the vector r of residuals with respect to the parameter vector p are given by the following equations. Note that the first column of the Jacobian matrix contains the derivative of the residuals with respect to the first parameter, the second with respect to the second, etc. In this example the derivatives can be computed explicitly., later we will introduce the computation of numerical derivatives. 

And now we introduce numerical derivatives. In the example above, we used explicit formulas for the derivatives of the residuals with respect to the parameters. Often it is not easy, or even impossible, to work out the correct equations. Numerical computation of the derivatives is always possible. Usually it is slower and also numerically less accurate. The general formula is ... 

The central function Rcalc EqAH2. m computes the residuals and is again very similar to the ones we developed earlier. First, the total concentrations are recalculated this needs to be part of the calculation of the residuals, as we want to be able to fit initial concentrations (s. c 0) as well. Subsequently these total concentrations are passed to the Newton-Raphson function NewtonRaphson, m in order to calculate all species concentrations, see The Newton-Raphson Algorithm (p.48). The differences between measured and calculated pH define the residuals. Note that any variable used in this function to calculate the residuals can theoretically be a parameter to be fitted to the data. 

The spreadsheet in Figure 4-62 is heavily matrix based (see Chapter 2, for an introduction to basic matrix functions in Excel). It is the only way to keep the structure reasonably simple. The matrix C in cells A21 C31 is computed in the usual way, see equation (4.63) the parameters required to compute the concentration matrix are in cells Q4 S4, they include the initial concentration for species A and the two rate constants that are to be fitted. In cells E 16 018 the computation of the best absorptivity matrix A for any given concentration matrix C, is done as a matrix equation, as demonstrated in The Pseudo-Inverse in Excel (p.146). Similarly the matrix Ycaic in cells E21 031 is written as the matrix product CA. Even the calculation of the square sum of the residuals in cell R7 is written in a compact way, using the Excel function SUMXMY2, especially designed for this purpose. We refer to... 

The projection of a vector into the subspace defined by eigenvectors, and the subsequent calculation of the residual vector between the original and its projection, is a very common task. Refer back to equations (5.15) and (5.16). It is worthwhile investigating the computations in some detail. 

Next, negative elements of A are set to zero and the residuals and the sum of squares are computed as indicated in Figure 5-51. The derivatives of the residuals with respect to the parameters are computed numerically by the... 

The computation of the best b, the one for which the residual vector r is minimal, is a linear least-squares calculation. Due to the orthonormality of U it is particularly easy ... 

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