Spline function determination

E. B. Smith, The Virial Coefficients of Pure Gases and Mixtures, Clarendon Press, Oxford, 1980. By interpolation (e.g., with cubic spline functions), virial coefficients can be determined for any temperature. 

We are trying to determine the curve S, called spline function, with... 

Table II. Selected Values and Confidence Bands for the Determination of Fenvalerate Estimated by Cubic Spline Functions...

Wegscheider fitted a cubic spline function to the logarithmically transformed sample means of each level. This method obviates any lack of fit, and so it is not possible to calculate a confidence band about the fitted curve. Instead, the variance in response was estimated from the deviations of the calibration standards from their means at an Ot of 0.05. The intersection of this response interval with the fitted calibration line determined the estimated amount interval. 

With the end condition flag EC = 0 on the input, the module determines the natural cubic spline function interpolating the function values stored in vector F. Otherwise, D1 and DN are additional input parameters specifying the first derivatives at the first and last points, respectively. Results are returned in the array S such that S(J,1), J = 0, 1, 2, 3 contain the 4 coefficients of the cubic defined on the I-th segment between Z(I) and ZII+l). Note that the i-th cubic is given in a coordinate system centered at Z(I). The module also calculates the area under the curve from the first point Z(l) to each grid point Z(I), and returns it in S(4,I). The entries in the array S can be directly used in applications, but we provide a further module to facilitate this step. 

The problem of Example 4.1.3 is revisited here. We determine the smoothing spline function and its derivatives assuming identical standard errors d = 0.25 in the measured pH. 

A cubic spline function is mechanically simulated by a flexible plastic strip. Mathematically, a spline function is a cubic in each interval between two experimental points. Thus, for n points, a spline includes n — 1 pieces of cubic each cubic having 4 unknown parameters, there are 4(n — 1) parameters to determine. The following conditions are imposed, (i) Continuity of the spline function and of its first and second derivatives at each of the n — 2 nodes (3n — 6 conditions), (ii) The spline function is an interpolating function (n conditions), (iii) The second derivatives at each extremity are null (2 conditions) this condition corresponds to the natural spline. It may be shown that the natural spline obtained is the smoothest interpolation function. Details concerning the construction of a spline and corresponding programs can be found in Forsythe et al. [127]. Of course, after a spline has been built up, it can be used to calculate derivatives. 

The experimental points in AotlgC) isotherm are calculated by the method of spline approximation (spline function of the polynome (AM) degree where N is the number of experimental measurements) [364,365]. This function allows to determine precisely the initial linear part where the derivative dAo/dlgC becomes constant and corresponds to the maximum... 

An alternative procedure for calculating the spectra involves fitting the experimental results for the viscoelastic functions by means of spline functions. The derivatives of Eqs. (9.81) and (9.82) are determined by means of these functions, and thus the spectra can be obtained. A summary of these and other approximations used to calculate retardation and relaxation spectra from the measured compliance and relaxation functions, respectively, can be found in Refs. 1 and 5. 

K to 10 K, 2 K to 20 K, and 5 K to 30 K. At temperatures higher than 30 K, intervals of 10 K are used. If the data are not reported in the temperature intervals desired by us, the data are interpolated. The spline function technique is used to determine the interpolated heat capacity at desired temperature. If unsmoothed data have been reported by the authors, the data are smoothed by curve-fitting prior to storage. 

At the time t = 0 it is known for the batch reactor that the dimensionless reaction rate 0(X) exactly takes the value of one. If the curve of dX/dh over h(t), which has just been obtained with the help of the previous calculation, is now numerically fitted to a spline function of the first or second order, then the initial gradient dX/dh at h(t) = 0 can be determined by extrapolation. This initial gradient exactly corresponds to ... 

However, the mentioned above single models have some shortages unavoidably. To remedy the defects of single models, hybrid models are actively researched recently. One kind of hybrid models (Qi, 1999) combines part of first principle equations with ANN, in which ANN is used to determine parameters of the first principle models. Fuzzy logic approach (Qian, 1999) is used for representing imprecision and approximation of the relationship among process variables. It is successfully incorporated into conventional process simulators. Several efforts (Baffi, 1999) have been made to combine statistical analysis with non-linear regression, which are polynomial, spline function and ANN. 

In 1968, Sing introduced the as-analysis comparison plot methodology for specific surface area determination [2] the method found a wide application to identify the presence of porosity and evaluate (micro)pore volumes in test adsorbents. No detailed uncertainty analyses exist for pore volumes in porous materials no internationally recognized standard porous materials exist. The comparison of the amount adsorbed by standard and test adsorbents leads to a complex interplay of the combined standard uncertainty (m ) in the amount adsorbed and in the clamped cubic spline functions employed to interpolate common relative pressures and their dependent amounts adsorbed. 

An important feature of the method of lines is selection of the basis functions i (co), which determines the precision of (spatial) curve fitting. The piecewise polynomials known as B splines meet the requirements. Curve fitting by means of spline functions entails division of the solution space into subintervals by means of a series of points called knots. Knots may be either single or multiple, a multiple knot being formed by the coincidence of two or more such points. They are numbered in nondecreasing order of location Si, S2,..., 5i,. A normalized B spline of order k takes nonzero values only over a range of k subintervals between knots, and, for example, Bij (co), the ith normalized B spline of order k for the knot sequence s, is zero outside the interval + nonnegative at = s, and w = Si + j, and strictly... 

Here the B-spline Bim(zf, Xj) is the ith B-spline basis function on the extended partition Xj (which contains locations of the knots in the Zj direction), and is a coefficient. We use cubic splines and sufficient numbers of uniformly spaced knots so that the estimation problem is not affected by the partition. The estimation problem now involves determining the set of B-spline coefficients that minimizes Eq. (4.1.26), subject to the state equations [Eqs. (4.1.24 and 4.1.25)], for a suitable value of the regularization parameter. At this point, the minimization problem corresponds to a nonlinear programming problem. 

In order to ensure successful minimization of the performance index and to enhance our ability to determine the global optimum, we select the corresponding finite-dimensional representation in a different manner than before. We again use B-splines to represent the unknown functions ... 

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