Abstract In this paper, third-order 3-stage diagonally implicit Runge–Kutta–Nystrom method embedded in fourthorder 4-stage for solving special second-order initial value problems is constructed. The method has the property of minimized local truncation error as well as the last row of the coefficient matrix is equal to the vector output. The stability of the method is investigated and a standar...

We develop an improved bound for the approximation error of the Nyström method under the assumption that there is a large eigengap in the spectrum of kernel matrix. This is based on the empirical observation that the eigengap has a significant impact on the approximation error of the Nyström method. Our approach is based on the concentration inequality of integral operator and the theory of mat...

Lemma 8. Given an m × m symmetric matrix A and a target rank k, we let C1 contain the c1 columns of A selected by a column sampling algorithm such that the following inequality holds: ∥∥A− PC1A∥∥2F ≤ f∥∥A−Ak∥∥2F . Then we select c2 = kf −1 columns to construct C2 and c3 = (c1+ c2) −1 columns to construct C3, both using the adaptive sampling according to the residual B1 = A − PC1A and B2 = A − P...

Spectral methods for dimensionality reduction and clustering require solving an eigenproblem defined by a sparse affinity matrix. When this matrix is large, one seeks an approximate solution. The standard way to do this is the Nyström method, which first solves a small eigenproblem considering only a subset of landmark points, and then applies an out-of-sample formula to extrapolate the solutio...

Convergence results are proved for a class of quadrature methods for integral equations of the form y(t) = fit) + /ô° k(t, s)y(s) ds. An important special case is the Nystrom method, in which the integral term is approximated by an ordinary quadrature rule. For all of the methods considered here, the rate of convergence is the same, apart from a constant factor, as that of the quadrature approx...