Polynomials piecewise

Various global and piecewise polynomials can be used to fit the data. Most approximations are to be used with M < N. One can sometimes use more and more terms, and calculating the value of for each solution. Then stop increasing M when the value of no longer increases with increasing M. 

When the underlying distribution is not known, tools such as histograms, probability curves, piecewise polynomial approximations, and general techniques are available to fit distributions to data. It may be necessary to assume an appropriate distribution in order to obtain the relevant parameters. Any assumptions made should be supported by manufacturer s data or data from the literature on similar items working in similar environments. Experience indicates that some probability distributions are more appropriate in certain situations than others. What follows is a brief overview on their applications in different environments. A more rigorous discussion of the statistics involved is provided in the CPQRA Guidelines. ... 

Note that state variable profiles are one order higher than the controls because they have explicit interpolation coefficients defined at the beginning of each element. With this representation of Z(t) and U(t), we can extend this approach to piecewise polynomials and apply orthogonal collocation on NE finite elements (of length Aoc,). This leads to the following nonlinear algebraic equations ... 

X + i(f) = piecewise polynomial approximation of Z(t) over element i, = piecewise polynomial approximation of U(t) over element i, Zii = coefficient of polynomial approximation, + i(f),... 

As mentioned in Section IV. A, a straightforward way to deal with optimal control problems is to parameterize them as piecewise polynomial functions on a predefined set of time zones. This suboptimal representation has a number of advantages. First, the approaches developed in the previous subsection can be applied directly. Secondly, for many process control applications, control moves are actually implemented as piecewise constants on fixed time intervals, so the parameterization is adequate for this application. 

Another approach is to replace the table-lookup contents by a piecewise polynomial approximation. While less general, good results have been obtained in practice [Cook, 1992, Cook, 1996], For example, one of the SynthBuilder clarinet patches employs this technique using a cubic polynomial [Porcaro et al., 1995],... 

In actual computations, the numerical differentiation can introduce a large error and should be avoided. A simple solution to this would be to fit spline functions, or a piecewise polynomial and overall smooth function of E, to the numerically calculated eigenphase sum 5(E), and then to differentiate the spline functions analytically [52, 53]. In the E-matrix method [44], the analytic E dependence of the R matrix and associated matrices can be taken advantage of in the direct differentiation of these quantities. This technique was found to be useful for automatic and fast analysis of the results of E-matrix method calculations [54-56]. 

In the sequential strategy, a control (manipulated) variable profile is discretized over a time interval. The discretized control profile can be represented as a piecewise constant, a piecewise linear, or a piecewise polynomial function. The parameters in such functions and the length of time subinterval become decision variables in optimization problem. This strategy is also referred to a control vector parameterization (CVP). 

For many applications, interpolations of functions of two or three variables defined in two-and three-dimensional domains must be considered. For example, global interpolations in two- and three-dimensional systems are analogous to polynomial interpolation in onedimensional systems however, global interpolants do not exist in 2- and 3D. This is a big drawback in numerical analysis because a basic tool available for one variable is not available for multivariable approximation [21], The best developed aspect of this theory is that of piecewise polynomial approximation, associated with finite element and finite volume approximations for partial differential equations, which will be examined in detail in Chapters 9 and 10. 

Pascal s triangle is often used to generate piecewise polynomial interpolations for various domains (triangles and rectangles in 2D and tetrahedrons, cubes and shells in 3D). In fact, most of the families of elements that are commonly used in finite elements, finite volumes and boundary elements, come from expansions of this triangle (more detail can be found in [67, 68]). 

The answer to this difficulty lies in the use of piecewise approximants, such as cubic splines, which are in general use in the mathematics literature (11). Carey and Finlayson (12) have introduced a finite-element collocation method along these lines, which uses polynomial approximants on sub-intervals of the domain, and apply continuity conditions at the break-points to smooth the solution. It would seem more straight-forward, however, to use piecewise polynomials which do not require explicit continuity... 

The use of standard piecewise polynomials, together with an appropriate transformation of the infinite domain, overcomes these difficulties, provided some care is taken in the transformation. ... 

A standard operation on piecewise polynomial curves is that of knot insertion where we pretend that there are additional knots (junctions between pieces) at which the discontinuity of nth derivative happens to be of zero magnitude. 

A piecewise polynomial will typically have / = 1 at the places where the pieces meet, so that the cubic B-spline is C2+1 at its knots. It is, of course C°° over the open intervals between the knots. 

The only solution is k = 1, so that the only binary schemes with piecewise polynomial limit curves are the B-splines. 

C. de Boor A Practical Guide to Splines. Springer 1978 J.Lane and R.Riesenfeld A theoretical development for the computer generation and display of piecewise polynomial surfaces. IEEE Trans Pattern Anal. Machine Intel 2(1), pp35-46, 1980... 

We observe better predictions as the number of terms increase. However, we need more terms at t = 0. For values of t > 0, N = 20 is enough. Hence, a piecewise polynomial can be used to define the initial condition at t = 0 and the separation of variables solution obtained can be used for values of t > 0. 

Another structure for expressing a nonlinear relationship between X and Y is splines [333] or smoothing functions [75]. Splines are piecewise polynomials joined at knots (denoted by Zj) with continuity constraints on the function and all its derivatives except the highest. Splines have good approximation power, high flexibility and smooth appearance as a result of continuity constraints. For example, if cubic splines are used for representing the inner relation ... 

Finite Element Methods In using a finite element method, one typically divides the spatial domain in zones or elements and then requires that the approximate solutions have the form of a specified polynomial over each of the elements. Discrete equations are then derived by requiring that the error in the piecewise polynomial approximate solution (i.e., the residual when the polynomial is substituted into the basic partial differential equation) be minimum. The Galerkin approach is one of the most popular, requiring that the error be orthogonal to the piecewise polynomial space itself. Characteristically, finite element methods lead to implicit approximation equations to be solved, which are usually more complicated than analogous finite difference methods. Their computational expense is comparable to, or can exceed, global implicit methods. 

The function Bj,fc(r) is a piecewise polynomial of degree k-1 inside the interval U[Pg.141]

The set of B-splines of order k on the knot sequence f< forms a complete basis for piecewise polynomials of degree fc — 1 on the interval spanned by the knot sequence. We represent the solution to the radial Dirac equation as a linear combination of these B-splines and work with the B-spline representation of the wave functions rather than the wave functions themselves. 

The cavity spectrum is complete in the space of piecewise polynomials of degree A — 1 and, therefore, can be used instead of the real spectrum to evaluate correlation corrections to states confined to the cavity. The spectrum splits into two distinct equal heJves, the lower energy half of the spectrum with < —mc represents the positron states and the upper half represents electron bound and continuum states. In evaluating the sums over states in the MBPT expressions for E > and in Eqs. (102) and (103), we of course omit contributions from the lower half of the spectrum. 

Unlike the traditional Taylor s series expansion method, the Galerkin approach utilizes basis functions, such as linear piecewise polynomials, to approximate the true solution. For example, the Galerkin approximation to the sample problem Equation 23.1 would require evaluating Equation 23.13 for the specific grid formation and specific choice of basis function ... 

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