Matlab script

The Matlab script Main PCR.m first reads in the complete corn data. Then it execute stepwise all the tasks that are described in the following. 

Typically two of the lower-order moments m, m2, and m3 are selected. However, in order to accommodate more moments of the original set, sometimes the final moments are calculated as the arithmetic average of the moments of two log-normal distributions. A typical choice could be to use for the first log-normal distribution mo, mi, and m3, and for the second mo, m2, and m3. In this way, after the correction only mo and m3 will be identical to the original ones, but a certain degree of control is achieved on both mi and m2. A Matlab script implementing this correction algorithm can be found in Section A.2.2 of Appendix A. 

To conclude this section on tensor-product QMOM, it is important to highlight that, although it is not necessary, the formulation of the problem in terms of translated (i.e. centered on the mean) and rotated (i.e. with diagonal covariance matrix) internal coordinates can be advantageous. In fact, if a change of variables is implemented so that the distribution is rewritten with respect to its principal coordinates, the calculations for the derivation of the quadrature approximation are simplified. These concepts will be illustrated in Exercise 3.8. A Matlab script implementing a tensor-product QMOM can be found in Section A.3.2 of Appendix A. 

In this appendix Matlab scripts for the moment-inversion algorithms discussed in Chapter 3 are reported. 

Below a Matlab script for the calculation of a quadrature approximation of order N from a known set of moments iti using the Wheeler algorithm is reported. The script computes the intermediate coefficients sigma and the jacobi matrix, and, as for the PD algorithm, determines the nodes and weights of the quadrature approximation from the eigenvalues and eigenvectors of the matrix. 

Below a Matlab script with the implementation of the algorithm for the correction of an unrealizable moment set is reported. The script first analyzes the moment set and then, if it is unrealizable, the script identifies the moment that has to be changed the least to make the set realizable. The procedure is iterated until the moment set becomes realizable or the maximum iteration number is reached. Realizability is verified by ensuring positiveness of the second-order differences however, other more stringent conditions could more effectively be used. 

Below a simple Matlab script implementing the Wright version of the correction algorithm is reported. 

Below a Matlab script implementing the tensor-product QMOM for a simple bivariate case described in this section is reported. The required inputs are the number of nodes for the first (Nl) and for the second (N2) internal coordinates. Since in the formulation described above the moments used for the calculation of the quadrature approximation are defined by the method itself, no exponent matrix is needed. The moments used are passed though a matrix variable m, whose elements are defined by two indices. The first one indicates the order of the moments with respect to the first internal coordinates (index 1 for moment 0, index 2 for moment order 1, etc.), whereas the second one is for the order of the moments with respect to the second internal coordinate. The final matrix is very similar to that reported in Table 3.8. The script returns the quadrature approximation in the usual form the weights are stored in the weight vector w of size N = Mi M2, whereas the nodes are stored in a matrix with two rows (corresponding to the first and second internal coordinate) and M = M1M2 columns (corresponding to the different nodes). 

To complete the book, four appendices are included. Appendix A contains the Matlab scripts for the most common moment-inversion algorithms presented in Chapter 3. Appendix B discusses in more detail the kinetics-based finite-volume methods introduced in Chapter 8. Einally, the key issues of PTC in phase space, which occurs in systems far from collisional equilibrium, and moment conservation with some QBMM are discussed in Appendix C and Appendix D, respectively. 

For the system given in Example 21.3, the MATLAB script that generates the dynamic RGA in Figure 21.5 is... 

In order to obtain the parameter estimates and analyse the results, the following MATLAB script will be used ... 

Matlab script to compute dipole angle theta from trajectory of positions x ... 

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