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Algorithm 919: A Krylov Subspace Algorithm for Evaluating the ψ-Functions Appearing in Exponential Integrators

机译:算法919:用于评估指数积分器中出现的ψ函数的Krylov子空间算法

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We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector representing the initial condition. The matrix function is a linear combination of the matrix exponential and other functions related to the exponential (the so-called ψ-functions). Such computations are the major computational burden in the implementation of exponential integrators, which can solve general ODEs. Our approach is to compute the action of the matrix function by constructing a Krylov subspace using Arnoldi or Lanczos iteration and projecting the function on this subspace. This is combined with time-stepping to prevent the Krylov subspace from growing too large. The algorithm is fully adaptive: it varies both the size of the time steps and the dimension of the Krylov subspace to reach the required accuracy. We implement this algorithm in the MATLAB function phipm and we give instructions on how to obtain and use this function. Various numerical experiments show that the phipm function is often significantly more efficient than the state-of-the-art.
机译:我们开发了一种算法,用于计算具有多项式不均匀性的线性常微分方程(ODE)大型系统的解。这等效于计算表示初始条件的矢量上某个矩阵函数的作用。矩阵函数是矩阵指数和其他与指数相关的函数(所谓的ψ函数)的线性组合。这样的计算是指数积分器实现中的主要计算负担,可以解决一般的ODE。我们的方法是通过使用Arnoldi或Lanczos迭代构造一个Krylov子空间并将该函数投影到该子空间上来计算矩阵函数的作用。这与时间步长相结合,以防止Krylov子空间变得太大。该算法是完全自适应的:它会改变时间步长和Krylov子空间的大小,以达到所需的精度。我们在MATLAB函数phipm中实现了该算法,并给出了如何获取和使用此函数的说明。各种数值实验表明,phipm函数通常比现有技术有效得多。

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