Polynomials, orthogonal

Orthogonal polynomials are a special category of functions that satisfy the following orthogonality condition with respect to a weighting function w(jc) 0, on the interval [a, fc]  

This orthogonality condition can be viewed as the continuous analog of the orthogonality property of two vectors (see Chap. 2) 

There are many families of polynomials that obey the orthogonality condition. These Me generally known by the name of the mathematician who discovered them Legendre, Chehyshev, Hermite, and Laguerre polynomials are the most widely used orthogonal polynomials. In this section, we list the Legendre and Chehyshev polynomials. 

The Legendre polynomials are orthogonal on the interval f-1, 1] with respect to the weighting function w(a ) = 1. The orthogonality condition is 

With the general polynomial equation discussed above, the value of the first coefficient, a, represents the intercept of the line with the y-axis. The b coefficient is the slope of the line at this point, and subsequent coefficients are the values of higher orders of curvature. A more physically significant model might be achieved by modelling the experimental data with a special polynomial equation a model in which the coefficients are not dependent on the specific order of equation used. One such series of equations having this property of independence of coefficients is that referred to as orthogonal polynomials. 

Bevington presents the general orthogonal polynomial between variables y and X in the form 

As usual, the least squares procedure is employed to determine the values of the regression coefficients a, b, c, d, etc., giving the minimum deviation between the observed data and the model. Also, we impose the criterion that subsequent 

In general, the computation of orthogonal polynomials is laborious but the arithmetic can be greatly simplified if the values of the independent variable, X, are equally spaced and the dependent variable is homoscedastic. In this 

Orthogonal polynomials are particularly useful when the order of the equation is not known beforehand. The problem of finding the lowest-order polynomial to represent the data adequately can be achieved by first fitting a straight line, then a quadratic curve, then a cubic, and so on. At each stage it is only necessary to determine one additional parameter and apply the f-test to estimate the significance of each additional term. 

Source of variation Sum of squares df Mean squares F-ratio 

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Deuflhard P and Wulkow M 1989 Computational treatment of polyreaction kinetics by orthogonal polynomials of a discrete variable Impact of Computing in Science and Engineering vol 1... 

An orientational order parameter can be defined in tenns of an ensemble average of a suitable orthogonal polynomial. In liquid crystal phases with a mirror plane of symmetry nonnal to the director, orientational ordering is specified. 

Spatial Profiles. The cross sections of laser beams have certain weU-defined spatial profiles called transverse modes. The word mode in this sense should not be confused with the same word as used to discuss the spectral Hnewidth of lasers. Transverse modes represent configurations of the electromagnetic field determined by the boundary conditions in the laser cavity. A fiiU description of the transverse modes requires the use of orthogonal polynomials. 

In comparison with the ordinary OZ equation, there is an additional integration with respect to the particle size. To proceed further, this integration can be removed by applying the method proposed by Lado [81,82]. Thus, we expand the (j-dependent functions in orthogonal polynomials 7 = 0,1,2,..., defined such that... 

Bather than using the Chapman-Enskog procedure directly, we shall employ the technique of Burnett,12 which involves an expansion of the distribution function in a set of orthogonal polynomials in particle-velocity space. 

Expansion Polynomials.—The techniques to be discussed here for solving the Boltzmann equation involve the use of an expansion of the distribution function in a set of orthogonal polynomials in particle velocity space. The polynomials to be used are products of Sonine polynomials and spherical harmonics some of their properties will be discussed in this section, while the reason for their use will be left to Section 1.13. 

The and terms above are written in terms of members of the orthogonal polynomial set, and are used in the remaining integrals of Eq. (1-89) (the % and integrals)... 

The random phase error of a wavefront which has passed through turbulence may be expressed as a weighted sum of orthogonal polynomials. The usual set of polynomials for this expansion is the Zernike polynomials, which... 

Kuo, J. E., Wang, H., and Pickup, S., Multidimensional Least-Squares Smoothing Using Orthogonal Polynomials, Anal. Chem. 63, 1991, 630-635. 

In other words, the unknown function is a perturbation of the (Banna distritxition such a perturbation is expressed in terms of orthogonal polynomials and unknown coefficients related to the... 

The orientation is not strictly identical for all structural units and is rather spread over a certain statistical distribution. The distribution of orientation can be fully described by a mathematical function, N(6, q>, >//), the so-called ODF. Based on the theory of orthogonal polynomials, Roe and Krigbaum [1,2] have shown that N(6, generalized spherical harmonics that form a complete set of orthogonal functions, so that... 

Orthogonal Collocation The orthogonal collocation method has found widespread application in chemical engineering, particularly for chemical reaction engineering. In the collocation method, the dependent variable is expanded in a series of orthogonal polynomials. See "Interpolation and Finite Differences Lagrange Interpolation Formulas. ... 

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). 

Szego, G., Orthogonal Polynomials, [4th edition], American Mathematical Society Providence, 1975... 

Legendre polynomials are one specific variety of a more extended class of orthogonal polynomials called associated Legendre polynomials. An associated Legendre polynomial P,m(x) is defined relative to an ordinary Legendre polynomial P,(x) through... 

Orthogonal Polynomial Expansion of the Spectral Density Operator and the Calculation of Bound State Energies and Eigenfunctions. 

Bigeleisen, J. and Ishida, T. Application of finite orthogonal polynomials to the thermal functions of harmonic oscillators. I. Reduced partition function of isotopic molecules, J. Chem. Phys. 48, 1311 (1968). Ishida, T., Spindel, W. and Bigeleisen, J. Theoretical analysis of chemical isotope fractionation by orthogonal polynomial methods, in Spindel, W., ed. Isotope Effects on Chemical Processes. Adv. Chem. Ser. 89, 192 (1969). 

A.F. Nikiforov, S.K. Suslov and V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer-Verlag Berlin, (1991) S.K. Suslov, Sov. J. Nucl. Phys. 38, 829 (1983). 

A different approach in the use of orthogonal polynomials as a transformation method for the population balance is discussed in (8 2.) Here the error in Equation 11 is minimized by the Method of Weighted Residuals. This approach releases the restrictions on the growth rate and MSMPR operation, however, at the cost of the introduction of numerical integration of the integrals involved, which makes the method computationally unattractive. The applicability in determining state space models is presently investigated and results will be published elsewere. 

A Fourier series is an example of an orthogonal polynomial, meaning that the individual terms which it comprises are independent of each other. It should be possible, therefore, to dissect a complex rotational energy profile into a series of N-fold components, and interpret each of these components independent of all others. For example, the onefold term (the difference between syn and anti conformers) in /7-butane probably reflects the crowding of methyl groups. 

A related approach is the approximation of peak-shaped fimctions by means of orthogonal polynomials, described by Scheeren et al. A function f(t), in this case the chromatographic signal, can be expanded in a series ... 

If the functions 1, t,. .. t are chosen, then the already mentioned moments of f(t) are found. However, convergence is not guaranteed in this case. Moreover, the calculation of the coefficients a requires the solution of N equations with N imknowns and the values a are dependent ofN. The introduction of a specialized set of orthogonal polynomials can be advantageous and circumvents some problems. Suppose that the following integral exists ... 

Approximating a function with an orthogonal polynomial series means that it is not necessary to solve N equations simultaneously. 

The addition of more terms does not influence the values of the already calculated terms. In this aspect, orthogonal polynomials are superior to other polynomials calculation of the coefficients is simple and fast. Moreover, according to the Gram-Schmidt theory every function can be expressed as a series of orthogonal polynomials, using the weighting function w(t). 

The choice of the specific orthogonal polynomial is determined by the convergence. If the signal to be approximated is a bell-shaped function, it is evident to use a polynomial derived from the Gauss function, i.e. one of the so-called classical polynomials, the Hermite polynomial. Widely used is the Chebyschev polynomial one of the special features of this polynomial is that the error will be spread evenly over the whole interval. 

The orthogonal collocation method has several important differences from other reduction procedures. Jn collocation, it is only necessary to evaluate the residual at the collocation points. The orthogonal collocation scheme developed by Villadsen and Stewart (1967) for boundary value problems has the further advantage that the collocation points are picked optimally and automatically so that the error decreases quickly as the number of terms increases. The trial functions are taken as a series of orthogonal polynomials which satisfy the boundary conditions and the roots of the polynomials are taken as the collocation points. A major simplification that arises with this method is that the solution can be derived in terms of its value at the collocation points instead of in terms of the coefficients in the trial functions and that at these points the solution is exact. 

A description of the technique requires appropriate definitions of the orthogonal polynomials Pm(x) as... 

It should be pointed out that this approach using Lagrangian polynomials gives identical results to those that would be obtained using Hermite polynomials since on each element we use orthogonal polynomials of the same order, since the boundary conditions are satisfied by both solutions, since the residuals are evaluated at the same points, and since the first derivatives are continuous across the element boundaries. The only preference for one over... 

M55 Polynomial regression using Forsythe orthogonal polynomials 5500 5628... 

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