Interpolation error

From Figure 12-65, by trial-and-error interpolation, when e = 0.7, Cp = 0.735. 

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 thermocouple measured liquid temperature to within 1 K. The helium line pressure was measured to within 4.6 kPa (0.667 psia). The height of liquid on top of the screen was measured to within 0.318 cm (1/8 in.). The uncertainty in the raw DPT measurement was less than 34.5 kPa (0.005 psia). However, due to read off errors, interpolating between recorded values, and due to uncertainty in liquid head pressure, the total imcertainty in reported bubble point values was 62 kPa (0.009 psia), which was no greater than 1.2% at the smallest measured value. 

All numerical computations inevitably involve round-off errors. This error increases as the number of calculations in the solution procedure is increased. Therefore, in practice, successive mesh refinements that increase the number of finite element calculations do not necessarily lead to more accurate solutions. However, one may assume a theoretical situation where the rounding error is eliminated. In this case successive reduction in size of elements in the mesh should improve the accuracy of the finite element solution. Therefore, using a P C" element with sufficient orders of interpolation and continuity, at the limit (i.e. when element dimensions tend to zero), an exact solution should be obtaiiied. This has been shown to be true for linear elliptic problems (Strang and Fix, 1973) where an optimal convergence is achieved if the following conditions are satisfied ... 

The scatter of the points around the calibration line or random errors are of importance since the best-fit line will be used to estimate the concentration of test samples by interpolation. The method used to calculate the random errors in the values for the slope and intercept is now considered. We must first calculate the standard deviation Sy/x, which is given by ... 

Comparison of Alignment Charts and Cartesian Graphs. There are typically fewer lines on an alignment chart as compared to Cartesian plots. This reduces error introduced by interpolation and inconsistency between scales. For example, to find a point (x,j) on a Cartesian graph one draws two lines, one perpendicular to each axis, and these reference lines intersect at the point x,j). This point (x,j) may correspond to some finite value found by rea ding a contour map represented by a family of curves corresponding to different values of the function. 

Average errors at low pressures for compounds with tabulated m and C are within a few percent. When values of m and C are calculated from only two vapor pressure points, the method should be used only for interpolation and limited extrapolation. The method is usable from about 220 K (so long as it is above the freezing point of the compound) to the critical point of water (about 647 K). 

Results and parameters may be interpolated between temperatures. Average errors are between 2 and 6 percent, with the higher errors at the lower temperatures. 

NOTE pj p f Pc) of 1/2, -A, and 14 may be used to cover the ranges 0.2—0.4, 0.4—0.6, 0.6-0.7, and 0.7—0.8, respectively, with a maximum error in .q of 5 percent at pL = 6.5 m atm, less at lower pLs. Linear interpolation reduces the error generally to less than 1 percent. Linear interpolation or extrapolation on T introduces an error generally below 2 percent, less than the accuracy of the original data. 

The calculation of (DCFRR) usually requires a trial-and-error solution of Eq. (9-57), hut rapidly convergent methods are avadahle [N. H. Wild, Chem. Eng, 83, 15.3-154 (Apr. 12, 1976)]. For simplicity linear interpolation is often used. 

The interpolated geometry and gradient are generated by requiring that the nonn of an error vector is minimum, subject to a normalization condition. 

Note that in reading the chart, values are off the scales, hut hy approximate interpolation, a value of F = 1.0 is not unreasonable. In any case the error in using this value will be small. 

Interpolation of this type may be extremely unreliable toward the center of the region where the independent variable is widely spaced. If it is possible to select the values of x for which values of f(x) will be obtained, the maximum error can be minimized by the proper choices. In this particular case Chebyshev polynomials can be computed and interpolated [11]. 

Neville s algorithm constructs the same unique interpolating polynomial and improves the straightforward Lagrange implementation by the addition of an error estimate. 

In addition to the chemical inferences that can be drawn from the values of AS and AH, considered in Section 7.6, the activation parameters provide a reliable means of storing and retrieving the kinetic data. With them one can easily interpolate a rate constant at any intermediate temperature. And, with some risk, rate constants outside the experimental range can be calculated as well, although the assumption of temperature-independent activation parameters must be kept in mind. For archival purposes, values of AS and AH should be given to more places than might seem warranted so as to avoid roundoff error when the exponential functions are used to reconstruct the rate constants. 

Concerning the numerical accuracy, the closed form solutions of normal surface deformation have been compared to the numerical results calculated through the three methods of DS, DC-FFT, and MLMI. The influence coefficients used in the numerical analyses were obtained from three different schemes Green function, piecewise constant function, and bilinear interpolation. The relative errors, as defined in Eq (39), are given in Table 2 while Fig. 4 provides an illustration of the data. 

TABLE 2—Relative errors for DS, FFT-based method and MLMI over different grids (%) (Green, Constant and Bilinear stand, respectively, for the schemes based on Green s function, constant function, and linear interpolation in determining the influence coefficients). ... 

Fig. 4—Comparison of relative error for different schemes, (a) A comparison of relative errors for a uniform pressure on a rectangle area 2a X 2b, in which the multi-summation is calculated via DS, FFT, and MLMI, and 1C is determined through bilinear interpolation based scheme, (b) A comparison of relative errors for a uniform pressure on a rectangle area 2ax2fa, in which the multisummation is calculated via DS and 1C is determined through the Green, constant, and bilinear-based schemes, (c) A comparison of relative errors for a Hertzian pressure on a circular region in radius a, in which the multi-summation is calculated via DS, and 1C is determined through the Green, constant, and bilinear-based schemes.

Critical ( -values for p - 0.05 are available. " - In lieu of using these tables, the calculated -values can be divided by the appropriate Student s t(f, 0.05) and V2 and compared to the reduced critical -vdues (see Table 1.12), and data file QRED TBL.dat. A reduced -value that is smaller than the appropriate critical value signals that the tested means belong to the same population. A fully worked example is found in Chapter 4, Process Validation. Data file MOISTURE.dat used with program MULTI gives a good idea of how this concept is applied. MULTI uses Table 1.12 to interpolate the cutoff point for p = 0.05. With little risk of error, this table can also be used fo = 0.025 and 0.1 (divide q by t(/, 0.025) /2 respectively t f, 0.1) V 2, as appropriate. 

Figure 4.30. Back-calculated results for file VALID2.dat. The data from the left half of Fig. 4.29 are superimposed to show that the day-to-day variability most heavily influences the results at the lower concentrations. The lin/lin format is perceived to be best suited to the upper half of the concentration range, and nearly useless below 5 ng/ml. The log/log format is fairly safe to use over a wide concentration range, but a very obvious trend suggests the possibility of improvements (a) nonlinear regression, and (b) elimination of the lowest concentrations. Option (b) was tried, but to no avail While the curvature disappeared, the reduction in n, logf.t) range, and Sxx made for a larger Pres and. thus, larger interpolation errors.

The difference, e.g., 5.0 - 1.4 in the eolumn marked 20 ng/ml, must be attributed to the interpolation error, which in this case is due to the uneer-tainties associated with the four Rodbard parameters. For this type of analysis, the FDA-accepted quantitation limit is given by the lowest ealibration concentration for which CV < 15%, in this case 5 ng/ml the eross indicates... 

These equations may be used by interpolation to estimate, with -10% error, sound absorption in melts. 

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