The dominant cost in solving least-square problems using Newton’s method is often that of factorizing the Hessian matrix over multiple values of the regularization parameter (λ). We propose an efficient way to interpolate the Cholesky factors of the Hessian matrix computed over a small set of λ values. This approximation enables us to optimally minimize the hold-out error while incurring only a...

In [4] the first named author discussed the explicit solutions of the cubic spline interpolation problems. We are now concerned with quintic spline functions. Let 5B[0, n] denote the class of quintic spline functions S(x) defined in the interval [0, n] and having the points 0, 1, • • • , n — 1 as knots. This means that the restriction of S(x) to the interval (*>, J > + 1 ) (P = 0, • • • , n — \...

In this work we show a general procedure to obtain optimal derivative free iterative methods [4] for nonlinear equations f(x) = 0, applying polynomial interpolation to a generic optimal derivative free iterative method of lower order. Let us consider an optimal method of order q = 2n−1, v = φn(x), that uses n functional evaluations. Performing a Newton step w = v − f(v) f ′(v) one obtains a met...

We present an algorithm that incrementally computes a Gröbner basis for the vanishing ideal of any finite set of points in an affine space under any monomial order, and we apply this algorithm to polynomial interpolation in multiple variables. For the case of distinct points, the algorithm is natural generalization of Newton’s interpolation for univariate polynomials. The time complexity in the...

We prove that log-linearly interpolated backoff language models can be efficiently and exactly collapsed into a single normalized backoff model, contradicting Hsu (2007). While prior work reported that log-linear interpolation yields lower perplexity than linear interpolation, normalizing at query time was impractical. We normalize the model offline in advance, which is efficient due to a recur...

It is known that the solutions of characteristic singular integral equations (SIEs) are expressed in terms of singular integrals of Cauchy type with weight functions w (x) = (1 + x)ν (1-x)μ, where ν = ± frac(1, 2), μ = ± frac(1, 2). New quadrature formulas (QFs) are presented to approximate the singular integrals (SIs) of Cauchy type for all solutions of characteristic SIE on the interval [-1, ...

to carry out searching and interpolation efficiently. Brackbill and Ruppert [6] and Westermann [7] have designed A generalized iterative algorithm for searching and locating particles in arbitrary meshes is presented. The algorithm uses Newton schemes to perform searching and bilinear interpolation on method to invert a bijective map of the mesh elements onto a structured meshes composed of non...

This paper presents a new predictor–corrector numerical scheme suitable for fractional differential equations. An improved explicit Atangana–Seda formula is obtained by considering the neglected terms and used as predictor stage of proposed method. Numerical formulas are presented that approximate classical first derivative well Caputo, Caputo–Fabrizio Atangana–Baleanu derivatives. Simulation r...

This paper proposes a new rational Krylov method for solving the nonlinear eigenvalue problem (NLEP): A(λ)x = 0. The method approximates A(λ) by Hermite interpolation where the degree of the interpolating polynomial and the interpolation points are not fixed in advance. It uses a companion-type reformulation to obtain a linear generalized eigenvalue problem (GEP). To this GEP we apply a rationa...