Polynomials approximation methods

Based on the criterion used to obtain the formula Taylor series expansion or polynomial approximation methods. 

We have tested polynomial approximation methods for a finite spatial domain, using several variations on the polynomial form. In this example, we will show how the polynomial approximation can also be carried out on problems having semi-infinite domain. 

Numerical methods are often used to find the roots of polynomials. A detailed discussion of these techniques is given under Numerical Analysis and Approximate Methods. ... 

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

Another class of methods of unidimensional minimization locates a point x near x, the value of the independent variable corresponding to the minimum of /(x), by extrapolation and interpolation using polynomial approximations as models of/(x). Both quadratic and cubic approximation have been proposed using function values only and using both function and derivative values. In functions where/ (x) is continuous, these methods are much more efficient than other methods and are now widely used to do line searches within multivariable optimizers. 

Hwang, J.-T., Sensitivity analysis in chemical kinetics by the method of polynomial approximations, Int. J. Chem. Kinetics 15, 959 (1983). 

This cubic can be factored (but in general polynomial equations require numerical approximation methods) ... 

It can be proved that this numerical method is of order 1 in h = max diam(7j). As mentioned above, higher order methods can be obtained by first using curved tesserae instead of planar triangles and then increasing the degree of the polynomial approximation on each tessera (/ , or P2 BEM [2]). 

The evaluation of these elements and the underlying theoretical support for the method can be found in Villadsen and Michelsen [38] who also provided subroutine listings that were used in this study. The boundary condition for the adsorbent particles is 0li>L+1 = collocation points that corresponds to a particular L-th-order polynomial approximation. The boundary condition for the capsule core is 0ci> M+1 = 0mi>o where M is the number of internal collocation points that correspond to a particular M-th-order polynomial approximation, and the boundary condition for the hydrogel membrane is bl where N is the number of internal collocation points that corresponds to a particular N-th-order polynomial approximation. Since the boundary conditions for the adsorbent and capsule core are coupled, and that of the capsule core and hydrogel membrane are also coupled, the boundary... 

Note that the results of the maximum profit problem obtained using the techniques presented above will be close to those determined by rigorous optimisation method (using the techniques presented in Mujtaba and Macchietto, 1993, 1996) only if the polynomial approximations are very good as were the case for the example presented here. Mujtaba et al. (2004) presented Neural Network based approximations of these functions. 

NUMERICAL METHODS FOR SCIENTISTS AND ENGINEERS, Richard Hamming. Classic (ext stresses frequency approach in coverage of algorithms, polynomial approximation, Fourier approximation, exponential approximation, other topics. Revised and enlarged 2nd edition. 721pp. 55 854. 

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... 

An element for the stress components composed of 16 sub-elements (4x4) on which bilinear (continuous) polynomials are used, was introduced by Marchal and Crochet in [28]. This leads to a continuous C° approximation of the three variables. The velocity is approximated by biquadratic polynomials while the pressure is linear. Fortin and Pierre ([17]) made a mathematical analysis of the Stokes problem for this three-field formulation. They conclude that the polynomial approximations of the different variables should satisfy the generalized inf-sup (Brezzi-Babuska) condition introduced by Marchal and Crochet and they proved it was the case for the Marchal and Crochet element. In order to take into account the hyperbolic character of the constitutive equation, Marchal and Crochet have implemented and compared two different methods. The first is the Streamline-Upwind/Petrov-Galerkin (SUPG). Thus a so-called non-consistent Streamline-Upwind (SU) is also considered (already used in [13]). As a test problem, they selected the "stick-slip" flow. With SUPG method applied to this problem, wiggles in the stress and the velocity field were obtained. In the SU method, the modified weighting function only applies to the convective terms in the constitutive equations. 

There exist (4, 5, 8, 9, 27) simple direct relations, between isotope effect, structure, and force field, which do not necessarily require a complete knowledge of all molecular parameters and avoid the solution of the secular equation. These relations are, however, approximations restricted to limited ranges of temperature. [Newer approximation methods, based on expansions in Jacobi polynomials, are applicable over wide ranges of temperatures (6, i6).] In the past, before the ready availability of fast digital computers, tests of the validity of these approximations were usually fairly limited in nature, but recent extensive tests on model calculations of kinetic isotope effects have been carried out 23, 28). In addition, extensive tests of power-series approximations (not considered in the present paper) have now been performed (6,16). 

The polynomial expressions for Cp in Table B.2 are based on experimental data for the listed compounds and provide a basis for accurate calculations of enthalpy changes. Several approximate methods follow for estimating heat capacities in the absence of tabulated formulas. 

If the Laplace transform of a function /(/) is f s), then f(t) is the inverse Laplace transform of f(s). Although an integral inversion formula can be used to obtain the inverse Laplace transform, in most cases it proves to be too complicated. Instead, a transform table (1), is used to find the image function f f). For more complicated functions, approximate methods are available. In many cases the inverse of a ratio of two polynomials must be... 

When this method is applied to the polynomial approximation of mechanisms, the function G is the response of the kinetic model calculated using the original detailed reaction mechanism, and (/> is a series of orthonormal polynomials constructed by a Gram-Schmidt orthonormalization process using the data set. The function F, defining the final algebraic model, is constructed in such a way that only the significant members of the summation are considered. 

The non existence of oscillation in the proposed discretization allied with the robustness of the computer code indicate that the association of these techniques seems to be adequate to solve the mathematical model of gasoil cracking. Other discretization methods, such as global polynomial approximation [17], could bring about oscillatory profiles through the bed length. 

For every experimental P-T curve obtained for the three-component system, the data were fitted to the corresponding equations using the least squares method. The coefficients of the polynomial approximation to the decomposition curves for the whole investigated composition range of the three-component system are shown in Table 2. 

The polynomial approximation of / is a local model which is only valid in the explored experimental domain. It is not possible to extrapolate and draw any conclusions outside this domain. It is therefore important to determine a good experimental domain prior to establishing a response surface model with many parameters involved. By the methods of Steepest ascent. Chapter 10, and Sequential... 

Any approximation of the derivative of a function in terms of values of that function at a discrete set of points is called a finite difference approximation. There are several ways of constructing such approximations, the Taylor series approach illustrated above is frequently used in numerical analysis because it supplies the added benefit that information about the error is obtained. Another method uses interpolation to provide estimates of derivatives. In particular, we use interpolation to fit a smooth curve through the data points and differentiate the resulting curve to get the desired result. A collection of low order approximations (i.e., first to fourth order polynomial approximations) of first and second order derivative terms can be found in textbooks like [49, 50, 167]. 

Villadsen J, Michelsen ML (1978) Solution of Differential Equations Models by Polynomial Approximation. Prentice-Hall, Englewood Cliffs Warming RF, Hyett BJ (1974) The modified equation approach to the stability and accuracy analysis of finite-difference methods. J Comp Phys 14 159-179 Waterson NP, Deconinck H (2007) Deign principles for bounded higher-order convection schemes-a unified approach. J Comput Phys 224 182-207 Wesseling P (1992) An Introduction to Multigrid Methods. John Wiley Sons, New York... 

Simpson s Rule This method is based on a second-order polynomial approximation of the function. For equally spaced points, the integral of the function between xq and X2 is... 

---