V2EX back-substitution

Back-Substitution

定义 Definition

back-substitution(回代法/回代):在线性代数中,先把方程组化为上三角形式(通常通过高斯消元),再从最后一个未知数开始,逐步向前代入求出所有未知数的求解步骤。常见于解上三角线性方程组与高斯消元法的最后阶段。(在更广义语境里也可指“把已求出的结果再代回去验证或求前面量”的做法。)

发音 Pronunciation (IPA)

/bk sbsttjun/(美式常见)
/bk sbsttjun/(英式亦常见)

例句 Examples

We use back-substitution to find \(x\) and \(y\) after elimination.
消元之后,我们用回代法求出 \(x\) 和 \(y\)。

After reducing the system to an upper triangular matrix, the algorithm computes the solution efficiently by back-substitution, starting from the last row.
把方程组化为上三角矩阵后,算法从最后一行开始用回代法高效计算解。

词源 Etymology

该词由 back-(向后、倒回) + substitution(代入、替换) 组成,字面意思是“向后代入”:先求出后面的未知数,再“倒着”把结果代回前面的方程中,逐步得到全部解。作为线性方程组求解流程中的术语,它通常与高斯消元、上三角矩阵等概念一起出现。

相关词 Rlated Words

文学与典籍出现 Literary Works

  • **Gilbert Strang, Introduction to Linear Algebra**(线性代数教材中讲解高斯消元后求解步骤时常出现 back-substitution)
  • **Lloyd N. Trefethen & David Bau III, Numerical Linear Algebra**(在三角线性系统求解与数值算法部分常用该术语)
  • **Gene H. Golub & Charles F. Van Loan, Matrix Computations**(矩阵计算与解线性方程组章节中常出现回代相关描述)
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