We describe how to compute simultaneous Hermite–Padé approximations, over a polynomial ring $\mathsf{K}[x]$ for a field $\mathsf{K}$ using $O^{\sim }(n^{\omega -1}td)$ operations in $\mathsf{K}$, where d is the sought precision, where n is the number of simultaneous approximations using $t<n$ polynomials, and where $O(n^{\omega })$ is the cost of multiplying $n\times n$ matrices over $\mathsf{K}$. We develop two algorithms using different approaches. Both algorithms return a reduced sub-basis that generates the complete set of solutions to the input approximation problem that satisfy the given degree constraints. Previously, the cost $O^{\sim }(n^{\omega -1}td)$ has only been reached with randomized algorithms finding a single solution for the case $t<n$. Our results are made possible by recent breakthroughs in fast computations of minimal approximant bases and Hermite–Padé approximations for the case $t\ge n$.

我们描述了如何在多项式环上同时计算Hermite-Padé近似值 $\mathsf{ķ}[X]$ 对于一个领域 $\mathsf{ķ}$ 使用 ${Ø}^{〜}（{ñ}^{\omega -\mathrm{1个}}Ťd）$ 在 $\mathsf{ķ}$，其中d是寻求的精度，其中n是使用的同时逼近数$Ť<ñ$ 多项式，以及在哪里 $Ø（{ñ}^{\omega }）$ 是倍增的成本 $ñ\times ñ$ 矩阵超过 $\mathsf{ķ}$。我们使用不同的方法开发了两种算法。两种算法都返回一个简化的子基，该子基可生成满足给定度约束的输入近似问题的完整解集。以前，费用${Ø}^{〜}（{ñ}^{\omega -\mathrm{1个}}Ťd）$ 只有通过随机算法找到针对该案例的单一解决方案才能实现 $Ť<ñ$。案例的最小近似基和Hermite-Padé近似快速计算的最新突破为我们的结果提供了可能$Ť\ge ñ$。