Calculation of Gauss quadratures with multiple free and fixed knots

Abstract

Summary. Algorithms are derived for the evaluation of Gauss knots in the presence of fixed knots by modification of the Jacobi matrix for the weight function of the integral. Simple Gauss knots are obtained as eigenvalues of symmetric tridiagonal matrices and a rapidly converging simple iterative process, based on the merging of free and fixed knots, of quadratic convergence is presented for multiple Gauss knots. The procedures also allow for the evaluation of the weights of the quadrature corresponding to the simple Gauss knots. A new characterization of simple Gauss knots as a solution of a partial inverse eigenvalue problem is derived.

Computation of connection coefficients and measure modifications for orthogonal polynomials

“…We emphasize that if one only requires modified recurrence coefficients, then the methods outlined in [11], [15], or more recently [2] are likely more appropriate.…”

“…However, most discussions of Christoffel transformation modifications (when q is a polynomial) center around computing the recurrence coefficients of p [w]. See for instance [6] and [15], with a good overview in [10]. However, in our case we are only tangentially interested in the recurrence coefficients; we instead require the modification coefficients C n,n+k .…”

“…In the case where only the modified recurrence coefficients are desired, another route may be taken: there is a strong connection between Christoffel transformations and numerical linear algebra that allows one to compute the recurrence coefficients for all of the elementary cases [6], [15]. However, if we are interested in the connection coefficients, the method outlined above is more straightforward.…”

“…The modification problem is a classical problem, and the literature boasts deep understanding and fruitful results for practical computation. However, most of these techniques concentrate on the determination of the three-term recurrence coefficients (equivalently, the Jacobi matrix) associated with the modified family p (see [6,8,15]). We explore a slightly different avenue: assuming q is a polynomial of degree K, a result due to Christoffel states that qp n is a linear combination of K + 1 polynomials π m .…”

Error estimates for large-scale ill-posed problems

“…We therefore do not have to compute the latter. The measure dw(t) is obtained by multiplying dw(t) by the factor t. Several derivations of algorithms for modifying the symmetric tridiagonal matrix T ℓ associated with dw(t) to obtain the symmetric tridiagonal matrixT ℓ−1 :=Ȓ T ℓ−1Ȓ ℓ−1 associated with the measure dw(t) are available; see Gautschi [11,12] and Golub and Kautsky [14]. Our derivation is suitable in the context of bounding matrix functionals and differs from the approaches in the references mentioned.…”