V2EX infinity norm

Infinity Norm

定义 Definition

infinity norm(也常称 max norm / maximum norm / Chebyshev norm):在向量或函数空间中,用“最大绝对分量(或最大绝对值)”来衡量大小的范数。
对向量 \(x=(x_1,\dots,x_n)\),无穷范数通常写作
\[ \|x\|_\infty=\max_i |x_i|. \]
(在更一般的函数情形中,可理解为“上确界范数”。)

发音 Pronunciation (IPA)

/nfnti nrm/

例句 Examples

The infinity norm of \((2,-5,1)\) is \(5\).
\((2,-5,1)\) 的无穷范数是 \(5\)。

In optimization, minimizing the infinity norm can reduce the worst-case error across all components.
在优化问题中,最小化无穷范数可以降低各个分量中“最坏情况”的误差。

词源 Etymology

infinity 来自拉丁语 infinitus(意为“无限的、无穷的”);norm 来自拉丁语 norma(“准则、标准”)。在数学里,“无穷范数”之所以带 “infinity”,与 \(L^p\) 范数家族有关:当 \(p \to \infty\) 时,对分量大小的度量趋向于由“最大值”主导,因此得名。

相关词 Related Words

  • Norm
  • Maximum Norm
  • Chebyshev Norm
  • Vector
  • Magnitude
  • L1 Norm
  • L2 Norm
  • Supremum
  • Worst-Case
  • 文学与著作中的用例 Literary Works

    • 《Convex Optimization》(Stephen Boyd, Lieven Vandenberghe):在凸优化建模中讨论包括 \(\|\cdot\|_\infty\) 在内的多种范数与约束形式。
    • 《Matrix Computations》(Gene H. Golub, Charles F. Van Loan):在数值线性代数与误差分析中使用无穷范数/矩阵范数进行界估计。
    • 《Numerical Linear Algebra》(Lloyd N. Trefethen, David Bau III):讲解向量与矩阵范数(含无穷范数)在算法稳定性与误差控制中的作用。
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