SCIENTIA SINICA Mathematica, https://doi.org/10.1360/SSM-2019-0336

On zero-error computation

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  • ReceivedDec 31, 2019
  • AcceptedApr 13, 2020
  • PublishedJul 9, 2020


It is important both in theory and in practice to study how to obtain exact results via numeric computation, which we call zero-error computation.In this paper, we firstly indicate which kind of numbers are suitable for zero-error computation: One can compute the exact value from its approximate values for every element in a uniformly discrete set, in which there exists a nonzero separation bound between two distinct elements. Based on this observation, we give such a separation bound for algebraic numbers, which can be seen as a necessary condition on error control for zero-error computation of algebraic numbers. However, this condition may not be sufficient, depending on different algorithms. For the PSLQ (partial-sum-LQ-decompsition)-based algorithm, we give a sufficient condition on the precision that is quasi-linear in the degree of the algebraic number to be recovered, while the corresponding condition for the LLL (Lenstra-Lenstra-Lovász)-based algorithm is quadratic. We also suggest several potential research areas in the future.

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    Algorithm 1 有理数恢复

    Require:正整数 $N$ 和一个浮点数 $r\in(0,~1)$.

    Output:一个有理数 $b$.

    令 $t$ 为 $r$ 的有理数表示, $h_0~:=~0$, $h_1~:=~1$, $k_0~:=~1$, $k_1~:=~0$; $a~:=~\lfloor~t\rfloor$, $b:=~t~-~a$; $k_2~:=~a\cdot~k_1~+~k_0$.

    while $k_2\le~N$ do

    $\binom{h_2}{k_2}~:=~a\cdot\binom{h_1}{k_1}~+\binom{h_0}{k_0}$, $\binom{h_0}{h_1}~:=\binom{h_1}{h_2}$, $\binom{k_0}{k_1}~:=\binom{k_1}{k_2}$.

    if $b~=~0$ then



    更新 $t~:=~\frac{1}{b}$, $a~:=~\lfloor~t\rfloor$, $b:=~t~-~a$.

    end if

    end while

    return $b~:=~h_0/k_0$.

  • Table 1   基于浮点 LLL 算法的代数数极小多项式重构复杂度上界
    算法 比特复杂度上界
    L$^2$ (参见文献[-1])/H-LLL (参见文献[-1]) $\CO(d^{7+\varepsilon}~+~d^{6+\varepsilon}\log~N~+~n^{5+\varepsilon}\log^2N)$
    渐近 LLL (参见文献[-1]) $\CO(d^{6+\varepsilon}~+~d^{4+\varepsilon}\log^2N)$
    L$^1$ (参见文献[-1]) $\CO(d^{6+\varepsilon}~+~d^{5+\varepsilon}\log~N~+ d^{\omega+1+\varepsilon}{\log^{1+\varepsilon}N})$
  • Table 2   代数数 bm$\alpha~=~{1}/(\sqrt[r]{2}+\sqrt[s]{3})$ 极小多项式近似重构中的误差控制 (bm$n$ 为次数, bm$N$ 为高度)
    $r$ $s$ $n$ $N$ (11) 中的输入误差 (18) 中的输入误差
    $2$ $4$ $8$ $104$ $3.9537\times~10^{-67}$ $7.2936\times~10^{-48}$
    $2$ $6$ $12$ $552$ $~7.6741\times~10^{-134}$ $5.2144\times~10^{-88}$
    $4$ $6$ $24$ $32364$ $~1.0465\times~10^{-445}$ $~1.9748\times~10^{-260}$
    $4$ $8$ $32$ $823984$ $~1.2404\times~10^{-765}$ $~2.8489\times~10^{-438}$
    $6$ $8$ $48$ $400286016$ $~~1.7439\times~10^{-1647}$ $~5.8168\times~10^{-919}$

    Algorithm 2 基于 LLL 算法重构模不大于 $1$ 的代数数的极小多项式

    Require:代数数次数的上界 $d$, 高度的上界 $N$, LLL 约化基合法参数 $\Xi$, 使得 (10) 成立的最小正整数 $s$, 以及满足 \begin{align} |\alpha-\bar{\alpha}|<\frac{1}{2^{s+2}d} \tag{11} \end{align} 的代数数的近似值 $\bar{\alpha}$.

    Output:代数数 $\alpha$ 的极小多项式 $P_\alpha(X)$.

    对 $i=1,\ldots,~d$ 计算 $\bar{\alpha}_i$ 使其满足 $\abs{\bar{\alpha}^i~-\bar{\alpha}_i}<2^{-s-1/2}$.

    for $n$ from $1$ to $d$

    构造格 $L_s$.

    stp:lll 调用任一 LLL 算法计算格 $L_s$ 的一组 LLL 约化基, 基中的第一个向量记为 $\tilde{\boldsymbol{v}}$.

    if $\norm{\tilde{\boldsymbol{v}}}<2^{d/2}(d+1)N$ then

    return $\tilde{\boldsymbol{v}}$ 对应的多项式 $v(X)$.

    end if

    end for


    Algorithm 3 基于 $\PSLQ_{\varepsilon}$ 算法重构模不大于 $1$ 的代数数的极小多项式

    Require:代数数次数的上界 $d$, 高度的上界 $N$, 满足 \begin{align} |\alpha-\bar{\alpha}|< \frac{1}{128(d+1)^{d+11/2}N^{2d}} \tag{18} \end{align} 的代数数的近似值 $\bar{\alpha}$.

    Output:代数数 $\alpha$ 的极小多项式 $P_\alpha(X)$.

    for $n$ from $1$ to $d$

    令 $\bar{\boldsymbol{\alpha}}:=(\bar{\alpha}^n,~\bar{\alpha}^{n-1},~\ldots,~1)$, 并令 $\bar{\boldsymbol{x}}~:=~\bar{\boldsymbol{\alpha}}/{\norm{\bar{\boldsymbol{\alpha}}}}$.

    按 (12) 构造 $\bar{\boldsymbol{x}}$ 的超平面矩阵 $H:=H_{\bar{\boldsymbol~x}}$.

    按 (17) 设置 $\varepsilon_2$ 的值, 并以 $\abs{h_{n,~n}}<\varepsilon_2$ 为终止条件调用 $\PSLQ_{\varepsilon}$ 算法 (参见文献[3]) 返回非零整向量 $\boldsymbol{m}\in\integer^{n+1}$.

    if $\norm{\boldsymbol{m}}<N$ then

    return $\boldsymbol{m}$ 对应的多项式 $m(X)=\sum_{i=0}^nm_iX^{n-i}$.

    end if

    end for

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