Methods Cubic

Simple linear interpolation is not always adequate for data tables with extensive curvature. Cubic interpolation uses the values of four adjacent data points to evaluate the coefficients of the cubic equation y = a + bx + cx + dx.  

A compact and elegant implementation of cubic interpolation in the form of 

William J. Orvis, Excel 4 for Scientists and Engineers, Sybex Inc., Alameda, CA, 1993. 

The cubic interpolation function forces the curve to pass through all the known data points. If there is any experimental scatter in the data, the result will not be too pleasing. A better approach for data with scatter is to find the coefficients of a least-squares line through the data points, as described in Chapter 11 or 12. 

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The first processor is used to calculate the function in the point selected by the efficient method (cubic or parabolic interpolation), whereas the remaining np — 1 processors evaluate the function in the points opportunely positioned in the three subintervals into which the interval is split. 

Gregory-Newton forward interpolation method. Lagrange polynomial interpolation method. Cubic splines interpolation method. 

The first part of the method involves sorting all the atoms into their appropriate cells. This sorting is rapid, and may be perfonned at every step. Then, within the force routine, pointers are used to scan tlirough the contents of cells, and calculate pair forces. This approach is very efficient for large systems with short-range forces. A certain amount of unnecessary work is done because the search region is cubic, not (as for the Verlet list) spherical. 

Hach molecular mechanics method has its own functional form MM+. AMBER, OPL.S, and BIO+. The functional form describes the an alytic form of each of th e term s in th e poteri tial. For exam pie, MM+h as both a quadratic and a cubic stretch term in th e poten tial whereas AMBER, OPES, and BIO+ have only c nadratic stretch term s, I h e functional form is referred to here as the force field. For exam pie, th e fun ction al form of a qu adratic stretch with force constant K, and equilibrium distance i q is ... 

Both the Coulomb cmd Lennard-Jones potentials can be considered examples of this type. In the cell multipole method the simulation space is divided into uniform cubic... 

Several basis schemes are used for very-high-accuracy calculations. The highest-accuracy HF calculations use numerical basis sets, usually a cubic spline method. For high-accuracy correlated calculations with an optimal amount of computing effort, correlation-consistent basis sets have mostly replaced ANO... 

Epitaxial crystal growth methods such as molecular beam epitaxy (MBE) and metalorganic chemical vapor deposition (MOCVD) have advanced to the point that active regions of essentially arbitrary thicknesses can be prepared (see Thin films, film deposition techniques). Most semiconductors used for lasers are cubic crystals where the lattice constant, the dimension of the cube, is equal to two atomic plane distances. When the thickness of this layer is reduced to dimensions on the order of 0.01 )J.m, between 20 and 30 atomic plane distances, quantum mechanics is needed for an accurate description of the confined carrier energies (11). Such layers are called quantum wells and the lasers containing such layers in their active regions are known as quantum well lasers (12). 

Alkali Metal Perchlorates. The anhydrous salts of the Group 1 (lA) or alkah metal perchlorates are isomorphous with one another as well as with ammonium perchlorate. Crystal stmctures have been determined by optical and x-ray methods (38). With the exception of lithium perchlorate, the compounds all exhibit dimorphism when undergoing transitions from rhombic to cubic forms at characteristic temperatures (33,34). Potassium perchlorate [7778-74-7] KCIO, the first such compound discovered, is used in pyrotechnics (qv) and has the highest percentage of oxygen (60.1%). 

With the Monte Carlo method, the sample is taken to be a cubic lattice consisting of 70 x 70 x 70 sites with intersite distance of 0.6 nm. By applying a periodic boundary condition, an effective sample size up to 8000 sites (equivalent to 4.8-p.m long) can be generated in the field direction (37,39). Carrier transport is simulated by a random walk in the test system under the action of a bias field. The simulation results successfully explain many of the experimental findings, notably the field and temperature dependence of hole mobilities (37,39). 

Air Permeability. Air permeabiUty is an important parameter for certain fabric end uses, eg, parachute fabrics, boat sails, warm clothing, rainwear, and industrial air filters. Air permeabiUty of a fabric is related to its cover, or opacity. Both of these properties are related to the amount of space between yams (or fibers in the case of nonwovens). The most common method for specifying air permeabiUty of a fabric involves measuring the air flow per unit area at a constant pressure differential between the two surfaces of the fabric. This method, suitable for measuring permeabiUty of woven, knitted, and nonwoven fabrics, is described in ASTM D737. Units for air permeabiUty measured by this method are generally abbreviated as CFM, or cubic feet per square foot per minute. 

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