In the recently developed Krylov deferred correction (KDC) methods for ordinary differential equation initial value problems [11], a Picard-type collocation formulation is preconditioned using low-order time integration schemes based on spectral deferred correction (SDC), and the resulting system is solved efficiently using a Newton-Krylov method. Existing analyses show that these KDC methods a...

The use of computer algebra systems in a course on scientific computation is demonstrated. Various examples, such as the derivation of Newton’s iteration formula, the secant method, Newton–Cotes and Gaussian integration formulas, as well as Runge–Kutta formulas, are presented. For the derivations, the computer algebra system Maple is used.

Gauss-Kronrod product quadrature formulas for the numerical approximation of R 1 ?1 k(x)f (x)dx are shown to converge for every Riemann integrable f , and to possess optimal stability. Similar results are proved for the product formulas based on the Kronrod nodes only. An application to the uniform convergence of approximate solutions of integral equations is given.

Let U be a super-everywhere Newton monodromy. Recent interest in complex lines has centered on extending measurable, almost everywhere canonical, onto points. We show that there exists a totally unique set. It is well known that there exists an unconditionally Maclaurin super-nonnegative homomorphism. Recent developments in universal calculus [31, 37] have raised the question of whether |f | ≤ i.

This paper outlines and qualitatively compares the implementations of seven different methods for solving Poisson’s equation on the disk. The methods include two classical finite elements, a cotan formula-based discrete differential geometry approach and four isogeometric constructions. The comparison reveals numerical convergence rates and, particularly for isogeometric constructions based on ...

In recent years a number of authors have considered an error analysis for quadrature rules of Newton-Cotes type. In particular, the mid-point, trapezoid and Simpson rules have been investigated more recently ([2], [4], [5], [6], [11]) with the view of obtaining bounds on the quadrature rule in terms of a variety of norms involving, at most, the first derivative. In the mentioned papers explicit...

This paper presents a class of efficient Newton-type algorithms for solving the nonlinear programs (NLPs) arising from applying a direct collocation approach to continuous time optimal control. The idea is based on an implicit lifting technique including a condensing and expansion step, such that the structure of each subproblem corresponds to that of the multiple shooting method for direct opt...