B-spline function

Figure 7. Part of a typical set of cubic B-spline functions calculated using equi-spaced knots...

Ho also computed series of high-lying Feshbach-type resonances converging to the TV = 3,..,9 hydrogen threshols (133). Several tens of resonances were reported. Chen (134) applied a basis set of B-spline functions within the saddle-point complex rotation method to compute parameters of twelve and low lying Feshbach resonances. 

However, Shea (1985) has indicated that in practice exponential splines do not produce more stable estimates of the term structure than polynomial splines. He also recommended using basis spUnes or B-splines, functions that are identically 0 over most of the approximation space, to prevent loss of accuracy due to the lack of observations at the long end of the curve. 

Least-squared methods used to derive the current interbank curve are very similar to those used to derive the current nondefault Treasury curve. After converting market data into equivalent zero-coupon rates, the zero-coupon yield curve is derived using a two-stage process, first writing zero-coupon rates as a B-spline function, and then fitting them through an ordinary least-squared method. 

Matrix B represents the BIRE associated to the displacements of the BEM nodes due to concentrated unit forces applied in the directions of the degrees of freedom that vary in time according to a B-Spline function, and captures the traveling wave characteristics. Subsequently, the response, u, of the elastodynamic system subjected to arbitrary external forces f = P applied at the nodes are computed at time step N as ... 

A sculptured surface is obtained by interpolation with two-dimensional B-spline-functions. A B-spline surface is considered as a collection of surface patches and the whole surface is a mosaic of these patches linked together with proper continuity (Figure 9). Due to its computational efficiency a uniform bicubic B-spline surface has been implemented. The bicubic B-spline patch can be written in matrix form as... 

The description of the photoionization process by means of a method based on the Density Functional Theory (DFT) is reviewed. The present approach is based on a basis set expansion in B-spline functions, which are particularly suited to deal with the boundary conditions of the continuum states. Both Kohn-Sham (KS) and its extension to the Time Dependent (TD-DFT) formalism are considered. The computational aspects of the method are described the implementations for atoms, for molecules in One Centre Expansion (OCE) and for molecules with the Linear Combination of Atomic Orbital (LCAO) scheme. The applications of the method are discussed, from atoms to large fullerenes, with comparison with available experimental data. 

FDA is a multi-step process in which the existing data is converted to a functional form, smoothed using Fourier functions or B-spline functions, and then modelled using a functional one-way ANOVA (fANOVA). In other words, FDA uses information along curves (or functions). This technique is therefore useful to examine adaptive behaviour that clearly changes over space and time (Figure 18.3). This differs from the time-series analysis previously described, in that a FDA does not require equally spaced time intervals. 

The7 th derivative of a B spline function of order fc, with coefficients a,-, is given (de Boer, 1978) by... 

We now return to Eqs. (6.1) and (6.2). If the right-hand sides of these equations are taken to be B spline functions of order k, then it can be shown (Margolis, 1978) that the spatial truncation error, that is, the amount by which the approximate solution fails to solve the partial differential equations (4.12) and (4.13) at time t, is 0((5 ). Next, if the number of continuity conditions to be satisfied at each interior (i = 2. .. /) breakpoint is v, then only the first m = kl — v l — 1) knots of the sequence form the origins of fresh polynomial pieces. It follows that the number of pp coefficients and hence the... 

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