Interpolations

When a value is required from a table at conditions which lie between listed values, interpolation is necessary. If M, the quantity sought, is a function of a single independent variable X and if linear interpolation is appropriate, as in the tables for saturated steam, then a direct proportionality exists between corresponding differences in M and in X. When M, the value at X, is intermediate between two given values. Mi at X and M2 at X2, then  

Forexample, theenthalpy of saturated vapor steam at413.95 K (140.8°C) is intermediate between the following values taken from Table F. 1  

When M is a function of two independent variables X and Y and if linear interpolation is appropriate, as in the tables for superheated steam, then double linear interpolation is required. Data for quantity M at values of the independent variables X and Y adjacent to the given values are represented as follows  

Double linear interpolation between Are given values of M is represented by  

The aim of this part of the book is to present the main and current numerical techniques that are used in polymer processesing. This chapter presents basic principles, such as error, interpolation and numerical integration, that serve as a foundation to numerical techniques, such as finite differences, finite elements, boundary elements, and radial basis functions collocation methods. 

The analysis of polymer processing is reduced to the balance equations, mass or continuity, energy, momentum and species and to some constitutive equations such as viscosity models, thermal conductivity models, etc. Our main interest is to solve this coupled nonlinear system of equations as accurately as possible with the least amount of computational effort. In order to do this, we simplify the geometry, we apply boundary and initial conditions, we make some physical simplifications and finally we chose an appropriate constitutive equations for the problem. At the end, we will arrive at a mathematical formulation for the problem represented by a certain function, say / (x, T, p, u.), valid for a domain V. Due to the fact that it is impossible to obtain an exact solution over the entire domain, we must introduce discretization, for example, a grid. The grid is just a domain partition, such as points for finite difference methods, or elements for finite elements. Independent of whether the domain is divided into elements or points, the solution of the problem is always reduced to a discreet solution of the problem variables at the points or nodal pointsinxxnodes. The choice of grid, i.e., type of element, number of points or nodes, directly affects the solution of the problem. 

We must keep in mind that, when using numerical techniques, we are only finding an approximate solution to the problem for the given nodes inside the domain. The error of this approximate solution is defined as the difference between the solution and the true value, i.e., 

Typically, the error in equation (7.1) is called truncation error. This is a hypothetical definition of the error, because it excludes gross errors, which are caused by unpredictable mistakes (for example, human or mechanical mistakes in an experiment), and round off errors, which are a result of having only a finite number of digits. The truncation error is the error that is implied in an approximation. More in-depth definitions on error can be found in the literature, as presented by Achieser [2], Hildebrand [10] or Davis [8], 

Every time we need a specific value of a mathematical function or a property in a table, such as specific heat in a cooling problem, we perform linear interpolation to obtain the desired value. Such a linear interpolation is the simplest way to compute data between known values. The given data can be a result of a physical measurement or numerical 

The difference between interstitial species and interpolated species is one of scale. Interstitial sites are thought of as normally unoccupied positions in relatively closely packed crystal structures. One tends to speak of interpolation when foreign atoms enter larger normally unoccupied volumes, cages, or tunnels in a structure, which otherwise remains topotactically unchanged. 

The likelihood of finding that a nonstoichiometric composition range is due to the presence of interpolated atoms in a crystal will depend upon the openness of the 

The principal composition variation in these bronzes is due to the interpolation of large metal atoms into the tunnels. There is considerable evidence to suggest that the tunnel filling can be ordered both within an individual tunnel and between tunnels. 

Not all analytical data can be recorded on a continuous basis discrete measurements often have to be made and they may not be at regular time or space intervals. To predict intermediate values for a smooth graphic di lay, or to perform many mathematical manipulations, e.g. Savitzky-Golay smoothing, it is necessary to evaluate regularly spaced intermediate values. Such values are obtained by interpolation. 

Obviously, if the true underlying mathematical relationship between the independent and dependent variables is known then any value can be computed exactly. Unfortimately, this information is rarely available and any required interpolated data must be estimated. 

The difference between values of adjacent points is assumed to be linear function of the distance separating them. The closer a point is to an observation, the closer its value is to that of the observation. Despite the simplicity of the calculation, linear interpolation should be used with care as the abrupt changes in slope that may occur at recorded values are unlikely to reflect accurately the more smooth transitions likely to be observed in practice. A better, and graphically more acceptable, result is achieved by fitting a smooth curve to the data. Suitable polynomials offer an excellent choice. 

Polynomial interpolation is simply an extension of the linear method. The polynomial is formed by adding extra terms to the model to represent curved regions of the spectrum and using extra data values in the model. 

If only one pair of measurements had been made, say ( i,Xi), then a zeroth order equation of the type = yi, for all would be the only possible solution. With two pairs of measurements, ( i,Xi) and (y2,X d then a first-order linear model can be proposed. 

the experimental points are smoothed by a continuous line, with regular curvature. Parallel chords AjBj, A2B2. of middle points M, M2,.. . are drawn. The intersection of the locus of points Mx, M2,.. . with the smoothed curve provides a tangent point M. The tangent MT is parallel to chords. 

Here too, the experimental points are first smoothed. A planar mirror is placed perpendicular to the sheet of paper, the base of the mirror intercepting the curve at the point where the tangent is to be determined. By rotating the mirror around a vertical axis, it is led to a position normal to the curve this position is attained when the curve and its image appear continuous. The knowledge of a normal provides the tangent. 

A one-dimensional interpolation consists of constructing an interpolation function f such that f(f,) — xit whatever i is. The unknown parameters of function f are determined from these relationships. It is worth noting that the interpolating curve passes rigorously through the experimental points, although these points are not free from stochastic errors. For this reason, and also to get equally spaced points, a smoothing of experimental points is often done before interpolation. 

For n + 1 experimental points th, (i = 1, 2,. . . , n + 1) given, it can be proved that there exists a single polynomial of order n interpolating the n + 1 points. The uniqueness of the interpolation polynomial does not imply any specific form. If a usual polynomial 

Solving this system for unknowns a - can be achieved by the methods of Sect. 4.2 but, in most cases, the equations are very ill conditioned and, thus, the use of such polynomials is limited to low orders (less than about 10). 

The result that gets printed out is 11. However, a more succinct way to express this would be to combine the first three lines into a single expression passed to print  

Note that the arithmetic expression is processed as 3+(4 2) rather than (3 + 4) 2. When evaluating numeric expressions, Perl uses the standard rules of precedence. The precedence can be changed by explicitly using parentheses  

Another interesting difference between double- and single-quoted strings is what happens when a variable is embedded inside a string. In double-quoted strings, the variable is expanded to its contents, a process known as string interpolation. This can aid readability considerably  

Assuming that x again contains Mary had a little lamb, the above statement outputs 

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For systems containing four components, most previous attempts for calculating LLE use geometrical correlations of ternary data (Branckner, 1940), interpolation of ternary data (Chang and Moulton, 1953), or empirical correlations of ternary data (Prince, 1954 Henty, 1964). These methods all have two... 

Experimental values were interpolated from the data of Brown et al. (1964). ... 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

Twenty percent is too low, since S A cf is positive at the end of year 5. Thirty percent is too large, since SAdcf is negative at the end of year 5, and is the case with 25 percent. The answer must be between 20 and 25 percent. Interpolating on the basis of 2 Adcf, the DCFRR = 23 percent. 

The factor enabling interpolation of reduced properties of a pure compound or mixture between two reduced properties calculated on two reference fluids merits attention in order to understand its meaning. 

This method utilizes essentially the concept developed by Fitzer in 1955. According to the principle of three-parameter corresponding states, the compressibility factor z, for a fluid of acentric factor w, is obtained by interpolating between the compressibilities Zj and Z2 for the two fluids having acentric factors w, and (p -... 

At low temperature and pressure, the conductivity of a pure hydrocarbon is obtained by linear interpolation between two known conductivities ... 

The functions giCdT) and g2([Pg.166]

The can be Interpolated using Table 4.18 and the fugacities calculated by the Soave model. 

The European regulations have set SO2 emission limits for industrial combustion systems. They range from 1700 mg/Nm for power generation systems of less than 300 MW and to 400 mg/Nm for those exceeding 500 MW between 300 and 500 MW, the requirements are a linear interpolation (Figure 5.24). To give an idea how difficult it is to meet these requirements, recall that for a fuel having 4% sulfur, the SO2 emissions in a conventional boiler are about 6900 mg/Nm this means that a desulfurization level of 75% will be necessary to attain the SO2 content of 1700 mg/Nm and a level of 94% to reach 400 mg/Nm. ... 

One way of calculating the IRR is to plot the NPV against discount rate, and to extrapolate/ interpolate to estimate the discount rate at which the NPV becomes zero, as in the Present Value Profile in Figure 13.16. The alternative method of calculating IRR is by... 

The data volume, which can be imported as data block or single slices, can be cutted or rebinned (e.g. if the data set is very large) and interpolated (e.g. interpolating intermediate slices between measured CT cuts in the case of 2D-CT). 

The critical surface tension concept has provided a useful means of summarizing wetting behavior and allowing predictions of an interpolative nature. A schematic summary of 7 values is given in Fig. X-10 [123]. In addition, actual contact angles for various systems can be estimated since )3 in Eq. X-38 usually has a value of about 0.03-0.04. 

There are many large molecules whose mteractions we have little hope of detemiining in detail. In these cases we turn to models based on simple mathematical representations of the interaction potential with empirically detemiined parameters. Even for smaller molecules where a detailed interaction potential has been obtained by an ab initio calculation or by a numerical inversion of experimental data, it is usefid to fit the calculated points to a functional fomi which then serves as a computationally inexpensive interpolation and extrapolation tool for use in fiirtlier work such as molecular simulation studies or predictive scattering computations. There are a very large number of such models in use, and only a small sample is considered here. The most frequently used simple spherical models are described in section Al.5.5.1 and some of the more common elaborate models are discussed in section A 1.5.5.2. section Al.5.5.3 and section Al.5.5.4. 

Sometimes it may be convenient to use an even simpler interpolation fomuila that coimects the different rate coefficient limits [4]... 

In simple relaxation (the fixed approximate Hessian method), the step does not depend on the iteration history. More sophisticated optimization teclmiques use infonnation gathered during previous steps to improve the estimate of the minunizer, usually by invoking a quadratic model of the energy surface. These methods can be divided into two classes variable metric methods and interpolation methods. 

The coefficients p. are chosen so that, on a quadratic surface, the interpolated gradient becomes orthogonal to all Aq. This condition is equivalent to minimizing the energy in the space spaimed by the displacement vectors. In the quadratic case, a further simplification can be made as it can be shown that all p. with the... 

An alternative, and closely related, approach is the augmented Hessian method [25]. The basic idea is to interpolate between the steepest descent method far from the minimum, and the Newton-Raphson method close to the minimum. This is done by adding to the Hessian a constant shift matrix which depends on the magnitude of the gradient. Far from the solution the gradient is large and, consequently, so is the shift d. One... 

The full dynamical treatment of electrons and nuclei together in a laboratory system of coordinates is computationally intensive and difficult. However, the availability of multiprocessor computers and detailed attention to the development of efficient software, such as ENDyne, which can be maintained and debugged continually when new features are added, make END a viable alternative among methods for the study of molecular processes. Eurthemiore, when the application of END is compared to the total effort of accurate determination of relevant potential energy surfaces and nonadiabatic coupling terms, faithful analytical fitting and interpolation of the common pointwise representation of surfaces and coupling terms, and the solution of the coupled dynamical equations in a suitable internal coordinates, the computational effort of END is competitive. 

The first technique is very intuitive. Out of the few proteins that could be crystallized in a number of different conformations, adenylate kinase is probably the best-studied example. By combining nine observed crystal structures and interpolating between them, a movie was constructed that visualized a hypothetical path of its hinge-bending transition (jVonrhein et al. 1995]). 

G. Benettin and A. Giorgilli. On the Hamiltonian interpolation of near to the identity symplectic mappings with applications to symplectic integration algorithms. J. Stat. Phys. 74 (1994)... 

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