论文标题
代数非理性与矩阵的近似
Approximations of algebraic irrationalities with matrices
论文作者
论文摘要
我们讨论矩阵的使用来提供近似代数非理性的理性序列。特别是，我们研究了代数扩展的定期表示，证明了常规表示矩阵的两个条目之间的比率融合到特定的代数非理性。作为一个有趣的特殊情况，我们专注于立方非理性，从而概括了Khovanskii矩阵，以近似立方非理性。我们讨论了这种近似值的质量，考虑到分母的收敛速度和大小。此外，我们短暂地与井（如牛顿和哈雷的迭代方法）进行了数字比较，表明定期表示提供的近似值似乎更准确地准确，而分母的大小相同。
We discuss the use of matrices for providing sequences of rationals that approximate algebraic irrationalities. In particular, we study the regular representation of algebraic extensions, proving that ratios between two entries of the matrix of the regular representation converge to specific algebraic irrationalities. As an interesting special case, we focus on cubic irrationalities giving a generalization of the Khovanskii matrices for approximating cubic irrationalities. We discuss the quality of such approximations considering both rate of convergence and size of denominators. Moreover, we briefly perform a numerical comparison with well--known iterative methods (such as Newton and Halley ones), showing that the approximations provided by regular representations appear more accurate for the same size of the denominator.
