This paper is one of a series underpinning the authors’ DAETS code for solving DAE initial value problems by Taylor series expansion. First, building on the second author’s structural analysis of DAEs (BIT 41 (2001) 364–394), it describes and justifies the method used in DAETS to compute Taylor coefficients (TCs) using automatic differentiation. The DAE may be fully implicit, nonlinear, and con...

A treecode is presented for evaluating sums defined in terms of the multiquadric radial basis function (RBF), φ(x) = (|x|2 + c2)1/2, where x ∈ R3 and c ≥ 0. Given a set of N nodes, evaluating an RBF sum directly requires CPU time that scales like O(N2). For a given level of accuracy, the treecode reduces the CPU time to O(N logN) using a far-field expansion of φ(x). We consider two options for ...

For the general nonlinear systems, a universal weighted measurement fusion (WMF) algorithm is presented via the Taylor series expansion method. Based on the proposed fusion algorithm and the well-known Unscented Kalman Filter (UKF), the WMF-UKF is presented. It is proven that the proposed WMF-UKF asymptotically approaches to the centralized measurement fusion UKF (CMF-UKF) with the increase of ...

A new numerical algorithm using quintic splines is developed and analyzed: quintic spline Taylor-series expansion (QSTSE). QSTSE is an Eulerian #ux-based scheme that uses quintic splines to compute space derivatives and Taylor series expansion to march in time. The new scheme is strictly mass conservative and positive de"nite while maintaining high peak retention. The new algorithm is compared ...

An optimized fourth-order staggered-grid finitedifference (FD) operator is derived on a mesh with variable grid spacing and implemented to solve 2-D velocity-stress elastic wave equations. The idea in optimized schemes is to minimize the difference between the effective wave number and the actual wave number. As expected, this optimized variable-grid FD scheme has less dispersion errors than th...

In this paper, we develop and modify Taylor-series expansion method to approximate a solution of nonlinear Volterra integro-differential equations (IDEs) as well as a solution of a system of nonlinear Volterra equations. By means of the nth-order Taylor-series expansion of an unknown function at an arbitrary point, a nonlinear Volterra equations can be converted approximately to a system of non...

We investigate the error surface of the XOR problem for a 2-2-1 network with sigmoid transfer functions. It is proved that all stationary points with finite weights are saddle points with positive error or absolute minima with error zero. So, for finite weights no local minima occur. The proof results from a careful analysis of the Taylor series expansion around the stationary points. For some ...

In this paper, we present an efficient method for determining the solution of the stochastic second kind Volterra integral equations (SVIE) by using the Taylor expansion method. This method transforms the SVIE to a linear stochastic ordinary differential equation which needs specified boundary conditions. For determining boundary conditions, we use the integration technique. This technique give...

An overview of the background of Taylor series methods and the utilization of the differential algebraic structure is given, and various associated techniques are reviewed. The conventional Taylor methods are extended to allow for a rigorous treatment of bounds for the remainder of the expansion in a similarly universal way. Utilizing differential algebraic and functional analytic arguments on ...