Local cubic

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. 

Coefficients for local cubic smoothing differentiation by Savitzky and Golay... 

Local cubic interpolation results in a function whose derivative is not necessarily continuous at the grid points. With a non-local adjustment of the coefficients we can, however, achieve global differentiability up to the second derivatives. Such functions, still being cubic polynomials between each pair of grid points, are called cubic splines and offer a "stiffer" interpolation than the strictly local approach. 

Interpolate the titration curve implementing Akima s method. Compare the interpolating curve with the results of local cubic interpolation and spline interpolation. 

As already briefly mentioned many specimens of quenched carbon steels retain a proportion of the paramagnetic austenitic phase [94]. At room temperature the resonance consists of a single line due to atoms without carbon nearest neighbours, and a quadrupole doublet from those with one carbon neighbour which removes the local cubic symmetry at the iron site [95]. The doublet shows a shift of 0-06 mms relative to the singlet. The same sample (1-6 wt % C) at 895°C showed a much narrower single line because the jump-diffusion time of the carbon atoms becomes less than the excited-state lifetime as the temperature rises. 

In the dhcp crystal structure, which has a stacking sequence ABAC, the Pr ion (with / = 4) experiences a crystal field of approximately local cubic symmetry at the A sites and of approximately local hexagonal symmetry at the B and C sites. The ratio c/(2a) is 1.611, nearer the ideal value of (8/3) = 1.633 than the heavy rare earths. With the c-axis as the quantization direction, the crystal field hamiltonian may be written as... 

Sm exhibits a rhombohedral crystal structure at low temperatures. In the nine layer sequence of close packed planes, two layers of atoms with a locally hexagonal environment alternate with one atomic layer in a locally cubic environment, so that the stacking pattern is of the form hhchhchhc. Hund s rules predict a ground state of the Sm ion with L = 5, 5 = and / = i with a small theoretical saturation moment of f / B/atom. 

As the surface deforms, to maintain good resolution, the grid points on the interface must be reconstructed and this was performed by three steps, namely, node addition, node deletion, and restructuring. For the curvature calculation, they used different methods for two dimensions and three dimensions. In two dimensions, they fit a local, cubic Hermite polynomial to four points on the interface and found the curvature by differentiation with respect to the arc length. The surface tension force was found by evaluating the curvature at the middle of the element. In three dimensions, they used a method described by Todd and McLeod [98]. The details are given in their paper. 

The crystal stracture of olivine materials has been studied by several authors [44, 48]. LiFeP04 crystallizes in the orthorhombic system (No. 62) with Pnma space group. It consists of a distorted hexagonal-close-packed oxygen framework containing Li and Fe located in half the octahedral sites and P ions in one-eighth of the tetrahedral sites [46]. The FeOe octahedra, however, are distorted, lowering their local cubic-octahedral Of, to the symmetry. Comer-shared FeOe octahedra are... 

Comparing Tables 11.3 and 11.4, we find that, in terms of macro iterations, the Newton method converges faster than the quasi-Newton method. The quasi-Newton method works better than the Newton method in the first few iterations, but the local cubic convergence of the Newton method then takes over and ensures that this method gives the smallest number of macro iterations. However, since each Newton iteration is an order of magnitude mote expensive than each quasi-Newton or Davidson iteration, the quasi-Newton and Davidson methods are far more cost-effective. 

The local cubic function can apply to interior data points, but how about the first and last intervals where there are not four surrounding data points For these intervals, the best that can be done is to use a single fitting quadratic function over the first three or last three data points. Thus an LCB algorithm must check for the first and last interval and use a single quadratic there and use the weighted quadratic or local cubic function for interior points. If the data has only two data points, the best that can be done is to fall back on linear interpolation and if there are only three data points, then only a single quadratic function can be used. 

Although it is certainly not an obvious conclusion, the use of these derivative evaluations in Eq. (6.10) through (6.12) will give exactly the same cubic interpolation equation as previously given by the LCB approach of Eq. (6.7). How can one know that this is the case Well they are both cubic polynomials passing through the same data points. Also they have the same identical first derivatives at the end data points. This gives four identical conditions for the cubic polynomials and a cubic polynomial has only 4 parameters. Thus they must be the same cubic polynomial equations. Thus the previous LCB interpolation technique should be more accurately called a cubic Hermite interpolation with end point derivative values evaluated by a three point numerical derivative . For the sake of simplicity the term local cubic or LCB will be used to describe the approach since the cubic equation is determined by the data points at the ends of the interval and the two adjacent data points. 

Listing 6.3. Code for Local Cubic Function interpolation (LCB). 

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