Quadratic Padé Approximation: Numerical Aspects and Applications
Quadratic Padé Approximation: Numerical Aspects and Applications
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二次 Padé 近似：数值方面和应用
DOI：
10.20537/2076-7633-2019-11-6-1017-1031
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
J. Weideman
中科院分区：
文献类型：
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作者：
M. Fasondini;Nicholas Hale;Rene Spoerer;J. Weideman
pad<s:1>近似法是从幂级数中提取奇异信息的有效工具。线性逼近函数是一种有理函数，可以提供复平面上极点和零点位置的估计。二次逼近具有平方根奇异性，因此可以提供额外的信息，例如分支点位置的估计。在本文中，我们讨论了计算二次逼近的数值方面以及一些应用。讨论了计算近似系数的两种算法：涉及线性系统解的直接方法（在数学界众所周知）和递归方法（在物理界众所周知）。我们比较了这两种方法在浮点运算中实现时的精度，并讨论了它们的优缺点。此外，我们将Luke的线性pad<s:1>近似的摄动分析扩展到二次型情况，并确定了二次型近似中存在虚假分支点的问题，这可能会导致精度的显著损失。对于这个问题的一个可能的补救办法是注意到这些麻烦的点可以用上面提到的递归方法来识别。二次逼近的另一个复杂之处是选择合适的分支。一种可能性是基于线性近似的选择，并结合Stahl的一个例子进行了讨论。我们还知道，二次方法能够对黎曼曲面的二次薄片提供合理的近似，我们在这里通过一个例子来说明这一事实。两个最后的应用显示了二次近似相对于线性近似的优越性：一个涉及特殊函数（朗伯特W函数），另一个涉及非线性偏微分方程（在复平面上的无粘Burgers方程的解的延拓）。
Padé approximation is a useful tool for extracting singularity information from a power series. A linear Padé approximant is a rational function and can provide estimates of pole and zero locations in the complex plane. A quadratic Padé approximant has square root singularities and can, therefore, provide additional information such as estimates of branch point locations. In this paper, we discuss numerical aspects of computing quadratic Padé approximants as well as some applications. Two algorithms for computing the coefficients in the approximant are discussed: a direct method involving the solution of a linear system (well-known in the mathematics community) and a recursive method (well-known in the physics community). We compare the accuracy of these two methods when implemented in floating-point arithmetic and discuss their pros and cons. In addition, we extend Luke’s perturbation analysis of linear Padé approximation to the quadratic case and identify the problem of spurious branch points in the quadratic approximant, which can cause a significant loss of accuracy. A possible remedy for this problem is suggested by noting that these troublesome points can be identified by the recursive method mentioned above. Another complication with the quadratic approximant arises in choosing the appropriate branch. One possibility, which is to base this choice on the linear approximant, is discussed in connection with an example due to Stahl. It is also known that the quadratic method is capable of providing reasonable approximations on secondary sheets of the Riemann surface, a fact we illustrate here by means of an example. Two concluding applications show the superiority of the quadratic approximant over its linear counterpart: one involving a special function (the Lambert W -function) and the other a nonlinear PDE (the continuation of a solution of the inviscid Burgers equation into the complex plane).