论文标题
GR分解及其与cholesky样因素的关系
GR decompositions and their relations to Cholesky-like factorizations
论文作者
论文摘要
对于给定的矩阵，我们有兴趣计算GR分解$ a = gr $，其中$ g $是相对于给定标量产品的等轴测图。正交QR分解是欧几里德标量产品的代表。对于签名矩阵，将各自的分解作为双曲线QR分解。考虑偏斜的矩阵会导致QR分解。计算GR分解的标准方法是基于连续消除子型矩阵条目。对于双曲线和合成案例，这种方法通常不会导致令人满意的数值准确性。另一种方法通过cholesky的分解计算QR分解，但稳定性也很差。通过第二次重复该过程来改进它。同样，双曲线和符号QR分解与$ ldl^t $和偏斜的cholesky样分解有关。我们表明，利用此连接的方法比基于消除的方法可以提供更好的数值稳定性。
For a given matrix, we are interested in computing GR decompositions $A=GR$, where $G$ is an isometry with respect to given scalar products. The orthogonal QR decomposition is the representative for the Euclidian scalar product. For a signature matrix, a respective factorization is given as the hyperbolic QR decomposition. Considering a skew-symmetric matrix leads to the symplectic QR decomposition. The standard approach for computing GR decompositions is based on the successive elimination of subdiagonal matrix entries. For the hyperbolic and symplectic case, this approach does in general not lead to a satisfying numerical accuracy. An alternative approach computes the QR decomposition via a Cholesky factorization, but also has bad stability. It is improved by repeating the procedure a second time. In the same way, the hyperbolic and the symplectic QR decomposition are related to the $LDL^T$ and a skew-symmetric Cholesky-like factorization. We show that methods exploiting this connection can provide better numerical stability than elimination-based approaches.
