By means of the Matlab symbolic/variable-precision facilities, routines are developed that generate an arbitrary number of recurrence coefficients to any given precision for polynomials orthogonal with respect to weig...
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By means of the Matlab symbolic/variable-precision facilities, routines are developed that generate an arbitrary number of recurrence coefficients to any given precision for polynomials orthogonal with respect to weight functions of Laguerre and Jacobi type containing logarithmic factors. The vehicle used is a symbolic modified chebyshev algorithm based on ordinary as well as modified moments, executed with sufficiently high precision. The results are applied to Gaussian quadrature of integrals involving weight functions of the type mentioned.
In the scalar case, computation of recurrence coefficients of polynomials orthogonal with respect to a nonnegative measure is done via the modified chebyshev algorithm. Using the concept of matrix biorthogonality, we ...
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In the scalar case, computation of recurrence coefficients of polynomials orthogonal with respect to a nonnegative measure is done via the modified chebyshev algorithm. Using the concept of matrix biorthogonality, we extend this algorithm to the vector case.
modified moments of a nonnegative measure determine uniquely the recurrence coefficients of the polynomials orthogonal with respect to the given nonnegative measure. The sensitivity of the underlying nonlinear maps wi...
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modified moments of a nonnegative measure determine uniquely the recurrence coefficients of the polynomials orthogonal with respect to the given nonnegative measure. The sensitivity of the underlying nonlinear maps with regard to perturbations in the modified moments has been studied by Gautschi and Fischer, and is further analyzed in this paper. In particular, we discuss the question of how (not) to choose the support of the underlying two measures. (C) 1998 Elsevier Science B.V. All rights reserved.
We consider the problem of generating the three-term recursion coefficients of orthogonal polynomials for a weight function v(t) = r(t)w(t), obtained by modifying a given weight function w by a rational function r. Al...
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We consider the problem of generating the three-term recursion coefficients of orthogonal polynomials for a weight function v(t) = r(t)w(t), obtained by modifying a given weight function w by a rational function r. algorithms for the construction of the orthogonal polynomials for the new weight v in terms of those for the old weight w are presented. All the methods are based on modified moments. As applications we present Gaussian quadrature rules for integrals in which the integrand has singularities close to the interval of integration, and the generation of orthogonal polynomials for the (finite) Hermite weight e(-t2), supported on a finite interval [-b, b].
The nonsymmetric Lanczos algorithm reduces a general matrix to tridiagonal by generating two sequences of vectors which satisfy a mutual bi-orthogonality property. The process can proceed as long as the two vectors ge...
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We consider the problem of numerically generating the recursion coefficients of orthogonal polynomials, given an arbitrary weight distribution of either discrete, continuous, or mixed type. We discuss two classical me...
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We consider the problem of numerically generating the recursion coefficients of orthogonal polynomials, given an arbitrary weight distribution of either discrete, continuous, or mixed type. We discuss two classical methods, respectively due to Stieltjes and chebyshev, and modern implementations of them, placing particular emphasis on their numerical stability properties. The latter are being studied by analyzing the numerical condition of appropriate finite-dimensional maps. A number of examples are given to illustrate the strengths and weaknesses of the various methods and to test the theory developed for them.
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