1. Explore applications of the recent ”Quantum Algorithm for Linear Systems of Equations” by Harrow, Hassidim and Lloyd [5]. This algorithm can solve a linear system of the form Ax = b, in time logarithmic in the dimensions of A, an exponential improvement over classical algorithms. However, the quantum algorithm cannot output the vector solution x explicitly, but only give an approximation to ...

In this article, we prove optimal convergence rates results for regularization methods for solving linear ill-posed operator equations in Hilbert spaces. The results generalizes existing convergence rates results on optimality to general source conditions, such as logarithmic source conditions. Moreover, we also provide optimality results under variational source conditions and show the connect...

We consider finite volume methods for the numerical solution of conservation laws. In order to achieve high-order accurate numerical approximation to non-linear smooth functions, we introduce a new class of limiter functions for the spatial reconstruction of hyperbolic equations. We therefore employ and generalize the idea of double-logarithmic reconstruction of Artebrant and Schroll [R. Artebr...

Using power series methods, Harris and Sibuya [3, 4] recently showed that if A: is an ordinary differential field of characteristic zero and y 5¿ 0 is an element of a differential extension of fc such that y and l/y satisfy linear differential equations with coefficients in fc, then y'¡y is algebraic over fc. Using differential galois theory, we generalize this and characterize those polynomial...

the index of matrix a in cn.n is equivalent to the dimension of largest jor-dan block corresponding to the zero eigenvalue of a. in this paper, indicialequations and normal equations for solving inconsistent singular linear systemof equations are investigated.

Logarithmic norms are often used to estimate stability and perturbation bounds in linear ODEs. Extensions to other classes of problems such as nonlinear dynamics, DAEs and PDEs require careful modifications of the logarithmic norm. With a conceptual focus, we combine the extension to nonlinear ODEs [15] with that of matrix pencils [10] in order to treat nonlinear DAEs with a view to cover certa...

We present a memory efficient parallel algorithm for the solution of tridiagonal linear systems of equations that are diagonally dominant on a very large number of processors. Our algorithm can be viewed as a parallel partitioning algorithm. We illustrate its performance using some examples. Based on this partitioning algorithm, we introduce a recursive version that has logarithmic communicatio...

This article presents algorithms for temporal parallelization of Bayesian smoothers. We define the elements and operators to pose these problems as solutions all-prefix-sums operations which efficient parallel scan-algorithms are available. present general filtering smoothing equations, specialize them linear/Gaussian models. The advantage proposed is that they reduce linear complexity standard...

in this article we decide to define a modified homotopy perturbation for solving non-linear integral equations. almost, all of the papers that was presented to solve non-linear problems by the homotopy method, they used from two non-linear and linear operators. but we convert a non-linear problem to two suitable non-linear operators also we use from appropriate bases functions such as legendre ...