this is for holding javascript data
Kim H. Parker added The_value_of_B_for__.tex
over 8 years ago
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The value of $B$ for fitted curve is found by calculation $R$ for the given data and using this curve to find $\beta T_d$. The negative values of $\beta T_d$ are unrealistic, but occasionally occur in the iterative solution of the inversion of this function and are included to show that the function is well-behaved over the whole range of $\beta T_d$. Note the relatively small range of $R$; $R(0) = \frac{1}{(3-e^1)} \approx 3.550$ and $R(\infty) =\frac{e^2-3}{2(e^1-2)} \approx 3.055$. This indicates that it is important to determine $R$ for the curve to be fitted as accurately as possible.
In the algorithm, the inversion of $R=f(\beta t_d)$ is done using the Matlab function \textit{fzero}. This function is used to find the solution of the function $f(\beta T_d) - R_{exp} = 0$, where $R_{exp}$ is the ratio of exponential moments determined for the measured data. Simpson's rule integration is used to evaluate the exponential integrals rather than simple summing or trapezoidal rule integration because of its higher accuracy. For idealised data, the algorithm yields values of the parameters with fractional errors of the order $10^{-6}$. The results for noisy, real data will obviously be less good.
