Quadratic interpolation

In this way the child spectrum is transformed into a spectrum as if measured on the parent instrument. In a more refined implementation one establishes the highest correlating wavelength channel through quadratic interpolation and, subsequently, the corresponding intensity at this non-observed channel through linear interpolation. In this way a complete spectrum measured on the child instrument can be transformed into an estimate of the spectrum as if it were measured on the parent instrument. The calibration model developed for the parent instrument may be applied without further ado to this spectram. The drawback of this approach is that it is essentially univariate. It cannot deal with complex differences between dissimilar instruments. 

In the ICVGT, corrections (a), (e) and (f) are not taken into account. When using the established quadratic interpolation curve, these corrections (within 0.3 mK) are taken into account through the factor B of eq. (9.1). Details of construction and operation for 3He and 4He ICVGT are reported in ref. [22,23] for 3He and [24-26] for 4He. 

With regard to sensor sensitivity, Fig. 3.26 shows the wavelength shift and amplitude changes vs. chloroform concentration with quadratic interpolation for the 160 and 260 nm coated LPG compared with the uncoated one. As expected, without the sensing overlay negligible variations have been observed in both the measured parameters. Sensitivities of —0.130 nm/ppm and 0.163 dB/ppm were observed for the thinner in the range of 0 10 ppm. Sensitivities of —0.85 nm/ppm and 0.26 dB/ppm were measured for the thicker overlay, in the same range. 

Two different formulas for quadratic interpolation can be compared Equation (5.8), the finite difference method, and Equation (5.12). 

Note that a solution on the first iteration seems to be remarkable, but keep in mind that the function is quadratic so that quadratic interpolation should be good even if approximate formulas are used for derivatives. 

Each wafer has 100 chip sites with 0.25 cm2 active area. The daily production level is to be 2500 finished wafers. Find the resist thickness to be used to maximize the number of good chips per hour. Assume 0.5 < f < 2.5 as the expected range. First use cubic interpolation to find the optimal value of t, t. How many parallel production lines are required for t, assuming 20 h/day operation each How many iterations are needed to reach the optimum if you use quadratic interpolation ... 

If quadratic interpolation is used with starting points of t = 1,2, and 3, the following iterative sequence results ... 

Determine the relative rates of convergence for (1) Newton s method, (2) a finite difference Newton method, (3) quasi-Newton method, (4) quadratic interpolation, and (5) cubic interpolation, in minimizing the following functions ... 

We can reach the minimum of fix) in two stages using first s° and then s1. Can we use the search directions in reverse order From x° = [1 l]T we can carry out a numerical search in the direction s° = [—4 —2]T to reach the point x1. Quadratic interpolation can obtain the exact optimal step length because /is quadratic, yielding a = 0.27778. Then... 

For the next stage, the search direction is s1 = [1 —4]r, and the optimal step length calculated by quadratic interpolation is a1 = 0.1111. Hence... 

A quadratic interpolation subroutine was used to minimize in each search direction. Table E6.5 lists the values of/(x), x, V/(x), and the elements of [H(x) + /3I]-1 for each stage of the minimization. A total of 96 function evaluations and 16 calls to the gradient evaluation subroutine were needed. 

Solution. Based on the data in Table E12.4A we minimized / with respect to R using a quadratic interpolation one-dimensional search (see Chapter 5). The value of Rm from Equation (a) was 11.338. The initial bracket was 12 < R < 20, and R = 16, 18, and 20 were selected for the initial three points. The convergence tolerance on the optimum required that/ should not change by more than 0.01 from one iteration to the next. 

The iterative program incorporating the quadratic interpolation search yielded the results in Table E12.4B. The optimum reflux ratio was 17.06 and the cost,/, was 3870/day. Table E12.4C shows the variation in/ for 10 percent change in R. The profit function changes 100/day or more. 

A remarkable fact is that Equation 2.19 is the same as Equation 2.14, and Equation 2.20 is the same as Equation 2.17 if one ignores the Dirac delta function, although Equations 2.14 and 2.17 result from the ensemble approach, while Equations 2.19 and 2.20 result from the smooth quadratic interpolation. Thus, the expressions given by Equations 2.19 and 2.20 are fundamental to evaluate the chemical potential (electronegativity) and the chemical hardness. 

Now, as in the case of the energy, up to this point, we have worked with the nonsmooth expression for the electronic density. However, in order to incorporate the second-order effects associated with the charge transfer processes, one can make use of a smooth quadratic interpolation. That is, with the two definitions given in Equations 2.23 and 2.24, the electronic density change Ap(r) due to the electron transfer AN, when the external potential v(r) is kept fixed, may be approximated through a second-order Taylor series expansion of the electronic density as a function of the number of electrons,... 

The values of the two derivatives, /°(r) and A/(r), at the reference point N0, may be approximated through this smooth quadratic interpolation between the points pWn (r), pNo(r), and Pnm(r), when combined with the two conditions,... 

Also, it is interesting to note that in the smooth quadratic interpolation, the curve of the total energy as a function of the number of electrons shows a minimum for some value of N beyond N0 (see Figure 2.1). This point has been associated by Parr et al. [49] with the electrophilicity index that measures the energy change of an electrophile when it becomes saturated with electrons. Together with this global quantity, the philicity concept of Chattaraj et al. [50,51] has been extensively used to study a wide variety of different chemical reactivity problems. 

The crucial step in the univariate search procedure is undoubtedly minimizing along the line ei. In situations where the gradient of the function along the line is not readily available, direct search procedures along the given line (i.e. one-dimensional direct search procedures) must be employed to find the minimum. Many such procedures are available (see, e.g. Cooper and Steinberg6 pp. 136-151), but one of the more efficient procedures seems to be quadratic interpolation. This may briefly be described as follows. 

This method has been found to reduce the overall computational effort in approaching the optimum by up to 30% compared to that obtained by accepting the first point satisfying an Armijo cone condition on the merit function while allowing step reductions of up to a factor of 10 at each line search iteration and using a quadratic interpolation formula to estimate the step reduction. 

Curves A and B are alternative interpretations of the experimental situation. Curve B is a plot of the 6 term Cauchy dispersion formula derived by Zeiss and Meath, while curve A is a simple quadratic interpolation (2-term Cauchy formula) between the static value of Cuthbertson40 and the Zeiss-Meath39 value at 514.5 nm (the only point where the polarizability anisotropy has been measured). Theoreticians appear to have taken these two values to heart. Curves C and D are plots of similar formulae [a(co) = 4(1 + Bofi) derived theoretically by Christiansen et al.44 and Kongsted et al.45 respectively, using the methods shown in Table 6 with suitable time-dependent procedures. The points obtained from the MCSCF46 work and the DFT/SAOP method48 are also plotted. The ZPVA correction of 0.29 au has been added at all theoretical points at all frequencies. 

The quadratic upstream interpolation for convective kinetics (QUICK) scheme of Leonard [106] uses a three-point upstream-weighted quadratic interpolation for the cell face values. In the third order QUICK scheme the variable profile between P and E is thus approximated by a parabola using three node values. At location e on a uniform Cartesian grids, tpe is approximated as ... 

The quadratic interpolation has a third order truncation error on both uniform and non-uniform grids [114, 49]. However, when this interpolation scheme is used in conjunction with the midpoint rule approximation of the surface integral, the overall approximation is still of second order accuracy (i.e., the accuracy of the quadrature approximation). Although the QUICK approximation is slightly more accurate than CDS, both schemes converge asymptotically in a second order manner and the difference are rarely large [49]. 

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