Interpolation nearest

The octane value of an unknown fuel sample is determined by comparing its knocking tendency to various primary reference fuels. Its measured octane is equal to the octane of the PRE which has the same knocking intensity. Knock intensity is controlled to an average value by varying the compression ratio of the CER engine. In practice, the exact value of a fuel s octane number is determined to the nearest 0.1 octane number by interpolation from two PREs that are no more than two octane numbers apart. 

The scope of the early papers was to use the SK approach to accurately interpolate the results of first principles calculations of the energy bands and densities of states. An important characteristic of these calculations is that the first, second, and third nearest neighbor interactions are treated as independent parameters, which is advantageous for minimizing the rms deviation from the first principles bands. 

Mix 400 mL of pure concentrated hydrochloric acid with 250-400 mL of distilled water so that the specific gravity of the resultant acid is 1.10 (test with a hydrometer). Insert a thermometer in the neck of a 1 L Pyrex distillation flask so that the bulb is just opposite the side tube, and attach a condenser to the side tube use an all-glass apparatus. Place 500 mL of the diluted acid in the flask, distil the liquid at a rate of about 3-4 mL min-1 and collect the distillate in a small Pyrex flask. From time to time pour the distillate into a 500 mL measuring cylinder. When 375 mL has been collected in the measuring cylinder, collect a further 50 mL in the small Pyrex flask watch the thermometer to see that the temperature remains constant. Remove the receiver and stopper it this contains the pure constant boiling point acid. Note the barometric pressure to the nearest millimetre at intervals during the distillation and take the mean value. Interpolate the concentration of the acid from Table 10.5. 

Here the summation is over molecules k in the same smectic layer which are neighbours of i and 0 is the angle between the intermolecular vector (q—r ) projected onto the plane normal to the director and a reference axis. The weighting function w(rjk) is introduced to aid in the selection of the nearest neighbours used in the calculation of PsCq). For example w(rjk) might be unity for separations less than say 1.4 times the molecular width and zero for separations greater than 1.8 times the width with some interpolation between these two. The phase structure is then characterised via the bond orientational correlation function... 

In order to increase the accuracy of the approximation to the convective term, not only the nearest-neighbor nodes, but also more distant nodes can be included in the sum appearing in Eq. (37). An example of such a higher order differencing scheme is the QUICK scheme, which was introduced by Leonard [82]. Within the QUICK scheme, an interpolation parabola is fitted through two downstream and one upstream nodes in order to determine O on the control volume face. The un-... 

Nearest neighbor distance inlier, n - a spectrum residing within a significant gap in the multivariate calibration space, the result for which is subject to possible interpolation error across the sparsely populated calibration space. 

Fig. 5 Spatial distribution of the average annual precipitation during the control period 1961-1990, computed using ordinary kriging interpolation with cells of 1 km2 and 40 nearest neighbours. Crosses represent the location of the gauging stations...

If no exactly matching sample can be found, the concentration which might lead to a match is approximated through visual interpolation from the two nearest reference samples. If it is impossible to arrive at an exact match for any concentration, then the concentration is found which experience shows to be equivalent. The corresponding tinctorial strength is supplemented by a verbal description of the remaining differences in hue and cleanness. 

This oscillation may have no relation at all to the behavior of the "true" function. Therefore, we cannot recommend global interpolation except for small samples. In large samples interpolation is rarely needed. For medium size samples low order local interpolation considering 3-6 nearest neighbors of the point x of interest does the job in most cases. The most popular method is local cubic interpolation in the Aitken form programmed in the following module. 

The module selects the four nearest neighbors of X and evaluates the cubic interpolating polynomial. 

Interpolation Estimate all readings to the nearest tenth of the distance between scale divisions. 

The correlations from which the equilibrium ratio data in Appendix A were taken include charts for convergence pressures of 800, 1000, 1500, 2000, 3000, 5000, and 10,000 psia. When the convergence pressure for the mixture is between the values for which charts are provided, interpolate between charts. Interpolation is necessary when the operating pressure is near the convergence pressure. At low pressure, simply use the chart with convergence pressure nearest the value for the mixture. 

EXAMPLE A sugar solution has the sp. gr. 1-1832 at 17 5°. The nearest value in the table, 1-18305, corresponds with 40-7° Brix or 40-7% by weight of gugar, and interpolation shows the exact concentration to be 40-73%. 

Every model has limitations. Even the most robust and best-validated regression model will not predict the outcome for all catalysts. Therefore, you must define the application domain of the model. Usually, interpolation within the model space will yield acceptable results. Extrapolation is more dangerous, and should be done only in cases where the new catalysts or reaction conditions are sufficiently close to the model. There are several statistical parameters for measuring this closeness, such as the distance to the nearest neighbor within the model space (see the discussion on catalyst diversity in Section 6.3.5). Another approach uses the effective prediction domain (EPD), which defines the prediction boundaries of regression models with correlated variables [105]. 

Interpolation for discrete known components appears easy and obvious in applying Table 1.10. The molecular weight of any component may easily be matched to the nearest tabled component of same molecular weight or interpolated between any two molecular weights. Here molecular weight is stated however, the boiling points may also be used equally for accurate interpolation. 

Two transformations are applied to the data of fig 4. First, the abscissa is transformed to the mean coordination number z (= mean number of nearest neighbours). The interpolation formula proposed by Bhatt and Rice, ... 

Some commonly used interpolation practices are discussed here. A simple and straightforward approximation to the value at the CV face center is linear interpolation between the two nearest nodes. At location e on a Cartesian grid (Fig. 6.2), general interpolation coefficients (yS) for such a scheme can be written ... 

Another classical approximation for the value at GCV face center is obtained by linear interpolation between the two nearest nodes. The linear interpolation corresponds to the central difference approximation of the first derivative in FDMs. At location e on a non-uniform Cartesian grid we have [49, 202] ... 

We will begin by assuming that, for a particular query point [, an algorithm exists (discussed below) for finding the nearest point in composition space , L It then remains to determine if 09 lies in the EOA of , L The definition of the EOA requires aparameter toi which controls the interpolation error (and consequently the number of tabulated points). Denoting the singular values of A(0q] At) by o, (j e 1,..., Ns), the initial principal axes matrix for the EOA can be defined by... 

A buret is used for the accurate delivery of a variable amount of solution. Its principal use is in titrations, where a standard solution is added to the sample solution until the end point (the detection of the completion of the reaction) is reached. The conventional buret for macrotitrations is marked in 0.1-mL increments from 0 to 50 mL one is illustrated in Figure 2.13. The volume delivered can be read to the nearest 0.01 mL by interpolation (good to about 0.02 or 0.03 mL). Burets... 

A comparison of the difference table, page 309, with Newton s formula will show that the interpolated term yx is built up by taking the algebraic sum of certain proportions of each of the terms employed. The greatest proportions are taken from those terms nearest the interpolated term. Consequently we should expect more accurate results when the interpolated term occupies a central position among the terms employed rather than if it were nearer the beginning or end of the given series of terms. 

Let us take the series y0, yv y2, ys, y4 so that the term, yx, to be interpolated lies nearest to the central term y2. Hence, with our former notation, Newton s expression assumes the form... 

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