论文标题
使用稳定的Sanathanan-Koerner迭代的多元合理近似
Multivariate Rational Approximation Using a Stabilized Sanathanan-Koerner Iteration
论文作者
论文摘要
1963年开发的Sanathanan-Koerner迭代是理性近似的经典方法。这种方法将近似的两侧乘以分母多项式产生线性问题，然后在每次迭代中引入一个重量以纠正该线性化。不幸的是，这种重量引入了数值不稳定。我们通过使用Arnoldi迭代的初始向量来构建分子和分母多项式的Vandermonde矩阵来纠正这种不稳定。这种稳定的Sanathanan-Koerner迭代纠正了不稳定性，并产生了任意程度的准确理性近似值。使用Vandermonde与Arnoldi的多元扩展，我们可以将稳定的Sanathanan-Koerner迭代应用于多元理性近似问题。所得的多元近似值通常比现有技术明显好，并且在整个域中显示出更均匀的精度。
The Sanathanan-Koerner iteration developed in 1963 is classical approach for rational approximation. This approach multiplies both sides of the approximation by the denominator polynomial yielding a linear problem and then introduces a weight at each iteration to correct for this linearization. Unfortunately this weight introduces a numerical instability. We correct this instability by constructing Vandermonde matrices for both the numerator and denominator polynomials using the Arnoldi iteration with an initial vector that enforces this weighting. This Stabilized Sanathanan-Koerner iteration corrects the instability and yields accurate rational approximations of arbitrary degree. Using a multivariate extension of Vandermonde with Arnoldi, we can apply the Stabilized Sanathanan-Koerner iteration to multivariate rational approximation problems. The resulting multivariate approximations are often significantly better than existing techniques and display a more uniform accuracy throughout the domain.
