论文标题
使用最近的循环器近似实际对称的toeplitz矩阵
Approximating real symmetric Toeplitz matrices using the nearest circulant
论文作者
论文摘要
得出了Frobenius Norm中的真实Toeplitz基质的最近循环近似。该矩阵是对称的。事实证明，对称的循环矩阵是唯一具有所有真实特征值的真实循环矩阵。对于显示指数衰减的情况下，这种近似值和Toeplitz矩阵之间差异的frobenius规范使用$ \ sum_ {n = 1}^n n n^k p^n $的表达式评估了第一个$ k $ seterric moments。与经典近似相比，最近的循环液在任何有限的情况下都表现出明显更好的行为，尽管两者在大型$ m $上共享相同的领先术语。
The nearest circulant approximation of a real Toeplitz matrix in the Frobenius norm is derived. This matrix is symmetric. It is proven that symmetric circulant matrices are the only real circulant matrices with all real eigenvalues. The Frobenius norm of the difference between this approximation and the Toeplitz matrix for the case of a Toeplitz matrix displaying exponential decay is evaluated using an expression of $\sum_{n = 1}^N n^k p^n$ in terms of the first $k$ geometric moments. Compared to a classic approximation the nearest circulant displays dramatically better behaviour in any finite cases, though both share the same leading term for large $M$.
