PARAMETRIC SENSITIVITIES NORMAL EQUATIONS

Gauss (1809) used a Newton-like iteration scheme to minimize S 6) for nonlinear models a single iteration suffices for linear models. He approximated the departures of the data from a model as linear expansions F (0) around the starting point of the current iteration  

Here Xyr denotes dFy/dOr evaluated at 6. This set of equations can be expressed concisely as 

the correction vector — X 8 — 8 ) is orthogonal (normal) to each column vector of X. The rows of Eq. (6.3-5) are accordingly known as the normal equations of the given problem. 

If the model functions E (0) all are linear in 8. then the sensitivity matrix X is constant and a single application of Eq. (6.3-5) will give the least-squares solution. In practice, X is often far from constant, making iteration necessary as described in the following section. 

One can extend Eq. (6.3-1) to second order to get a fuller quadratic function S 8). This gives the full Newton equations of the problem, whereas Eqs. (6.3-3)-(6.3-5) are called the Gauss-Newton equations. Either form can be expressed as a symmetric matrix expansion 

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To facilitate the discussion on the influence of the above-defined parameters (Equations 5.37-5.39) on the reactor behavior and the parametric sensitivity, Equations 5.34 and 5.36 are given in a dimensionless form. According to the studies of Barkelew [25] the mean residence time is referred to the characteristic reaction time and the temperature is given in the form of a relative temperature difference normalized with the Arrhenius number (Equation 5.41). 

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