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In the literature, most known sequence transformations can be written as a ratio of two determinants. But, it is not always this case. One exception is that the sequence transformation proposed by Brezinski, Durbin, and Redivo-Zaglia cannot be expressed as a ratio of two determinants. Motivated by this, we will introduce a new algebraic tool—pfaffians, instead of determinants in the paper. It turns out that Brezinski-Durbin-Redivo-Zaglia’s transformation can be expressed as a ratio of two pfaffians. To the best of our knowledge, this is the first time to introduce pfaffians in the expressions of sequence transformations. Furthermore, an extended transformation of high order is presented in terms of pfaffians and a new convergence acceleration algorithm for implementing the transformation is constructed. Then, the Lax pair of the recursive algorithm is obtained which implies that the algorithm is integrable. Numerical examples with applications of the algorithm are also presented. [ABSTRACT FROM AUTHOR]

In some approximation problems, sampling from the target function can be both expensive and time-consuming. It would be convenient to have a method for indicating where approximation quality is poor, so that generation of new data provides the user with greater accuracy where needed. In this paper, we propose a new adaptive algorithm for radial basis function (RBF) interpolation which aims to assess the local approximation quality, and add or remove points as required to improve the error in the specified region. For Gaussian and multiquadric approximation, we have the flexibility of a shape parameter which we can use to keep the condition number of interpolation matrix at a moderate size. Numerical results for test functions which appear in the literature are given for dimensions 1 and 2, to show that our method performs well. We also give a three-dimensional example from the finance world, since we would like to advertise RBF techniques as useful tools for approximation in the high-dimensional settings one often meets in finance. [ABSTRACT FROM AUTHOR]

The first 60 coefficients in the three-term recurrence relation for monic polynomials orthogonal with respect to cardinal B-splines φ as the weight functions on [0, m] ( m ∈ ℕ) are obtained in a symbolic form. They enable calculation of parameters, nodes, and weights, in the corresponding Gaussian quadrature up to 60 nodes. The efficiency of these Gaussian quadratures is shown in some numerical examples. Finally, two interesting conjectures are stated. [ABSTRACT FROM AUTHOR]

We present a new smaller upper bound for all the elements in the associate generalized Hessian used in the smoothing quadratic regularization (SQR) algorithm proposed by Bian and Chen (SIAM J. Optim. 23: 1718-1741, ). We modify the SQR algorithm by making use of the new upper bound. Numerical results show that our new upper bound improves the performance of the SQR algorithm significantly. [ABSTRACT FROM AUTHOR]