Đối với trường hợp trong đó h # k, một biểu hiện cho F (x, y) có thể dễ dàng thu được từ phương trình 12-14. Lưu ý rằng bốn điểm mạng có liên quan đến việc tính toán của F (x, y) bởi phương trình 12 - 16, đại diện trongtừ đó chúng ta có được phương trình 12-16 | 314 EXCEL NUMERICAL METHODS Nonlinear Least-Squares Curve Fitting Unlike for linear regression there are no analytical expressions to obtain the set of regression coefficients for a fitting function that is nonlinear in its coefficients. To perform nonlinear regression we must essentially use trial-and-error to find the set of coefficients that minimize the sum of squares of differences between yCaic and Jobsd- For data such as in Figure 14-1 we could proceed in the following manner using reasonable guesses for Á 1 and k2 calculate B at each time data point then calculate the sum of squares of residuals SSresiduais S B caic - B expt 2. Our goal is to minimize this errorsquare sum. We could do this in a true trial-and-error fashion attempting to guess at a better set of kỵ and k2 values then repeating the calculation process to get a new and hopefully smaller value for the SSresiduais- Or we could attempt to be more systematic. Starting with our initial guesses for k and k2 we could create a twodimensional array of starting values that bracket our guesses as in Figure 14-2. The initial guesses for k and k2 were and respectively and the array of starting values are 70 80 90 100 110 120 and 130 of the respective initial estimates. Then for each set of k and k2 values we calculate the SSresiduais- The k and k values with the smallest error-square sum k Figure 14-1. A typical plot of the concentration of species B for a system of two consecutive first-order reactions the reaction scheme A- B- C CHAPTER 14 NONLINEAR REGRESSION USING THE SOLVER 315 k in Figure 14-2 become the new initial estimates and the process is repeated using smaller bracketing values. Years ago this procedure called pitmapping was performed on early digital computers. In essence we are mapping out the error surface in a sort of topographic way searching for the minimum. A typical error surface is shown in Figure 14-3 the logarithm of the residuals has been plotted to make the .
