QR分解法(QR decomposition)

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已于 2024-03-15 10:42:39 修改 · 5.8k 阅读

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于 2017-11-19 10:08:09 首次发布

QR分解将m乘n矩阵A分解为正交矩阵Q和上三角矩阵R的乘积，即A=QR。文章介绍了Householder变换，这是一种找到反射向量u关于与Householder向量v正交的超平面的方法，用于简化QR分解的过程，使得解决Ax=b问题的回代法更加稳定。

QR decomposition divides a m by n matrix $A$ into a product of an orthogonal matrix $Q$ and an upper triangular matrix $R$ :

$A = QR$

Thus
$Ax = b => QRx = b => Q^{T}QRx = Q^{T}b => Rx = Q^{T}b$
QR decomposition can be implemented by several algorithms, such as Gram–Schmidt process, Householder transformations, or Givens rotations. Gram-Schmidt procedure is a sequence of multiplications of A from the right by upper triangular matrices. Householde decomposition $A$ into $QR$ with orthogonal matrices. As orthogonal transformations are stable, using Householder triangularization and back-substitution to slove $A x = b$ is backward stable[9][10].

##Householder transformation
Householder transformation reflects a vector $u$ about a hyperplane which orthogonal to a vector $v$ which is called Householder vector[8].
$u^{f} = u - 2vv^{T}u$

The basic idea of Householder reflection for QR decomposition is to find a linear transformation that changes vector $u$ into a vector which collinear to $e_{i}$ , $u^{f} = ||u||e_{i}$