The PD algorithm

This system of 2 M nonlinear equations is ill-conditioned for large M, but can be efficiently solved using the product-difference (PD) algorithm introduced by McGraw (1997). Thus, given the set of 2 M moments on the left-hand side of Eq. (107), the PD algorithm returns wm and lm for m — 1., M. The closed microscopic transport equation for the moments can then be written for k — 0,..., 2 M— 1 as... 

Note that each environment in the micromixing model will have its own set of concentrations can and moments mkn, reflecting the fact that the PSD is coupled to the chemistry and will thus be different at every SGS point in the flow. The PD algorithm is applied separately in each environment to compute the weights (wmn) and abscissa (lmn) from the quadrature formula as follows ... 

By using the PD algorithm, this set of A x M delta functions is reduced to a set of only M delta functions, but with the same values for the first 2 M moments (mp) as the original set. It should be obvious that this cannot be accomplished by simply averaging the weights and abscissas. 

Thus, it would be natural to attempt to extend the QMOM approach to handle a bivariate NDF. Unfortunately, the PD algorithm needed to solve the weights and abscissas given the moments cannot be extended to more than one variable. Other methods for inverting Eq. (125) such as nonlinear equation solvers can be used (Wright et al., 2001 Rosner and Pykkonen, 2002), but in practice are computationally expensive and can suffer from problems due to ill-conditioning. 

In the PDS algorithm, a small set of calibration-transfer samples are measured on a primary instrument and a secondary instrument, producing spectral response matrices X, and X2. A permutation matrix F (Procrustes transfer matrix) is used to map spectra measured on the secondary instrument so that they match the spectra measured on the primary instrument. 

After applying the PD algorithm the following Jacobi matrix is obtained ... 

The PD algorithm is quite efficient in a number of practical cases however, it generally becomes less stable as N increases. It is difficult to predict a priori when this will occur since it depends on the absolute values of the moments, but typically problems can be expected when N > 10. Another important issue is related to the fact that for distributions with zero mean (or in other words with mi = 0) the algorithm blows up, due to a division by zero during the calculation of the coefficients of the continued fraction fa. The Wheeler algorithm, which will be reported in the next section, does not suffer from these problems. 

But mi is usually not zero when the internal coordinate represents particle mass, surface area, size, etc. In these cases the PD algorithm can be safely used. The case of null mi occurs more often when the internal coordinate is a particle velocity that, ranging from negative to positive real values, can result in distributions with zero mean velocity. Another frequent case in which the mean is null is when central moments (moments translated with respect to the mean of the distribution) are used to build the quadrature approximation. These cases will be discussed later on, when describing the algorithms for building multivariate quadratures. 

It is now sufficient to modify the second-order moment to 25 (instead of 26, corresponding to a difference of only 4%) to make the moment set unrealizable. The Hankel-Hadamard determinants are now equal to -179352 and -12 362 344, respectively, whereas the difference table (see Table 3.2) presents negative elements in the column containing the second-order differences (namely A2). If this moment set is fed to the PD algorithm, the resulting quadrature is unable to reproduce the moment set. 

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. 

If a two-unequal-weight quadrature is employed, the first four low-order sectional moments are used to find two secondary particle positions and weights. The PD algorithm was simplified resulting in the following set of algebraic equations ... 

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