Curve fitting numerical

Numerical methods include those based on finite difference calculus. They are ideally suited for tabulated experimental data such as one finds in thermodynamic tables. They also include methods of solving simultaneous linear equations, curve fitting, numerical solution of ordinary and partial differential equations and matrix operations. In this appendix, numerical interpolation, integration, and differentiation are considered. Information about the other topics is available in monographs by Hornbeck [2] and Lanczos [3]. 

In this work, we advance from reducing variables to numerical curve fitting. Numerical fitting methods afford strong advantages over reducing variables and superposition plots. Numerical fits reveal weak dependences not readily apparent to the naked eye. Furthermore, reducing variables can only lead to superposition... 

Along with the curve fitting process, TableCurve also calculates the area under the curve. According to the previous discussion, this is the entropy of the test substance, lead. To find the integral, click on the numeric at the left of the desktop and find 65.06 as the area under the curve over the range of x. The literature value depends slightly on the source one value (CRC Handbook of Chemistry and Physics) is 64.8 J K mol. ... 

Experimental data that are most easily obtained are of (C, t), (p, t), (/ t), or (C, T, t). Values of the rate are obtainable directly from measurements on a continuous stirred tank reactor (CSTR), or they may be obtained from (C, t) data by numerical means, usually by first curve fitting and then differentiating. When other properties are measured to follow the course of reaction—say, conductivity—those measurements are best converted to concentrations before kinetic analysis is started. 

Numerical and some analytical solutions of the diffusion/reaction equations are represented closely by an empirical curve/fit,... 

Solution of Sets of Simultaneous Linear Equations 71. Least Squares Curve Fitting 76. Numerical Integration 78. Numerical Solution of Differential Equations 83. 

Range 1 of the mud pump performance characteristic is defined by the performance of the smallest liner, and range 2 is defined by the remaining liners. The pressure loss in a circulating system, except for bit (p ), can be estimated from numerous theoretical formulas or from a flowrate test. Data obtained from a flowrate test can be approximated using a curve-fitting technique by the following function ... 

For large values of 6, Manohar [42], and Banks [4] solved the boundary layer Eqs. (2)-(5) numerically with a finite difference method. Manohar s results for the meridional and azimuthal velocity gradients on the spherical surface have been curve-fitted by Newman [45] and Chin [18] to follow the following equations in the regime of 0 < 9 < 7t/2 ... 

The most widely-accepted dose response model at the present time is the multi-stage model, which has great flexibility in curve-fitting, and also has a strong physiological justification. Although it is difficult to implement, there are already computer codes in existence that estimate the model parameters (13). The two most widely-used models, until recently, were the one-hit model and the log-probit model. They are both easy to implement, and represent opposite extremes in terms of shape - the former represents the linear non-threshold assumption, whereas the latter has a steep threshold-like curvature. In numerous applications with different substances it has been found that these three... 

Statistics available in the system include a large set of commonly used analysis techniques, as well as advanced nonlinear curve fitting techniques. Statistical results can be displayed numerically or graphically. 

Swain (7) has discussed the general problem of determining rate constants from experimental data of this type and some of the limitations of numerical curve-fitting procedures. He suggests that a reaction progress variable for two consecutive reactions like 5.3.2 be defined as... 

Solutions are presented in the form of equations, tables, and graphs—most often the last. Serious numerical results generally have to be obtained with computers or powerful calculators. The introductory chapter describes the numerical procedures that are required. Inexpensive software has been used here for integration, differentiation, nonlinear equations, simultaneous equations, systems of differential equations, data regression, curve fitting, and graphing. 

Find k by trial by numerical integration of the E(tr) data. The solution is easier with a calculator after a curve fit of (tr, E(tr). Such a curve fit is given with the attached plot of the data. Another possible curve fit is with the linearized form of the Gamma distribution described with problem P5.O2.04 which does not require availability of the software TABLECURVE. 

The intensity function A(tr) was evaluated numerically is curve fitted, as shown on the graph, by... 

There are numerous equations in the literature describing the concentration dependence of the viscosity of dispersions. Some are from curve fitting whilst others are based on a model of the flow. A common theme is to start with a dilute dispersion, for which we may define the viscosity from the hydrodynamic analysis, and then to consider what occurs when more particles are added to replace some of the continuous phase. The best analysis of this situation is due to Dougherty and Krieger18 and the analysis presented here, due to Ball and Richmond,19 is particularly transparent and emphasises the problem of excluded volume. The starting point is the differentiation of Equation (3.42) to give the initial rate of change of viscosity with concentration ... 

The Pearson VII model contains four adjustable parameters and is particularly well suited for the curve fitting of large spectral windows containing numerous spectral features. The adjustable parameters a, p, q and v° correspond to the amplitude, line width, shape factor and band center respectively. As q —the band reduces to a Lorenzian distribution and as q approaches ca. 50, a more-or-less Gaussian distribution is obtained. If there are b bands in a data set and... 

Numerical analysis is important in digital-computer work from another viewpoint. Sometimes it is necessary to express complex functional relationships in a simpler form. Occasionally relationships may be given in a graphical or tabular form not directly suitable for processing on digital equipment. In these situations numerical methods for curve fitting and interpolation are techniques which will necessarily be employed. 

The emperical parameters S Q, S2Q, K-Q, K, A E-, and AE were determined by a numerical search routine. Figure 3 compares the theoretical curves with the experimental data and represents a satisfactory curve fit. 

A numerical search routine was applied to determine the value of K and A E. Figure 4 compares the theoretical curve with the experimental data and represents a satisfactory curve fit. The total sulfur content and SRC sulfur content for hydrotreated product were plotted (Figure 5), and a linear relationship was shown to exist between them. 

In this step Dukler made numerous experiments, finding a relation between X and the ratio of fr Jf0. The following equations are a curve-fit of his curve, relating these two factors ... 

Consequently, the initial nonstationary partial pressure of the adsorbate in the gas stream is accounted for with the help of the parameter p [87], Cutting the series included in Equation 5.107, and using only the first four terms, an approximation to Equation 5.107 is obtained, which is numerically fitted to the experimental data [88,90], In Figure 5.32, some examples of the uptake curves fitted with Equation 5.107 are reported (see Ref. [88] for more details). 

Some MR manufacturers use numerical algorithms to process the concentration-time curves. In such algorithms, user interaction is mandatory for definition of the end of the first pass. The numerical algorithms, however, are much faster than algorithms based on curve fitting, because they avoid the time-consuming non-linear fitting procedure. 

Schalla, M. and Weiss, M., Pharmacokinetic curve fitting using numerical inverse Laplace transformation, European Journal of Pharmaceutical Sciences, Vol. 7, 1999, pp. 305-309. 

The importance of this result is that it leads to an overall objective criterion for sample size determination that averages criteria based on specific model assumptions. Thus it provides a solution that is robust to model uncertainty. Closed-form calculations of (8) are intractable, so we have developed numerical approximations to the conditional entropies Ent(6k n, yk, MLk) and Ent(9k n, yk, MGk). The computations of the expected Bayes risk are performed via stochastic simulations and the exact objective function is estimated by curve fitting as suggested by Miiller and Parmigiani (1995). These details are available on request from the authors. 

Using Eqs. (5-55), (5-81), and (5-82) for the local heat transfer in their respective ranges, obtain an expression for the average heat transfer coefficient, or Nusselt number, over the range 5 x 105 < Re < 10 with Recri, = 5 x 10s. Use a numerical technique to perform the necessary integration and a curve fit to simplify the results. 

Several graphical curve-fitting techniques have been developed (see Padday [53] for details) that can be used in conjunction with the numerical integration of the Laplace equation by Bashforth and Adams (and by subsequent workers) to determine d and to obtain y v. Smolders [54,55] used a number of coordinate points of the profile of the drop for curve fitting. If the surface tension of the liquid is known and if 0 > 90, a perturbation solution of the Laplace equation derived by Ehrlich [56] can be used to determine the contact angle, provided the drop is not far from spherical. Input data are the maximum radius of the drop and the radius at the plane of contact of the drop with the solid surface. The accuracy of this calculation does not depend critically on the accuracy of the interfacial tension. 

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