论文标题
一种半结合梯度方法，用于求解未对称的正定线性系统
A semi-conjugate gradient method for solving unsymmetric positive definite linear systems
论文作者
论文摘要
共轭梯度（CG）方法是一种经典的Krylov子空间方法，用于求解对称的正定线性系统。我们引入了一种类似的半共轭梯度（SCG）方法，以实现非对称正定线性系统。与CG不同，SCG需要较低三角线性系统的溶液来产生每个半偶联方向。我们证明SCG在理论上等同于完全基于Arnoldi过程的正交方法（FOM），并以有限数量的步骤收敛。由于SCG的三角系统每次迭代的尺寸都会增加，因此我们研究了滑动窗口实现（SWI）以提高效率，并表明所产生的方向仍然是本地半偶联的。反例说明SWI与直接不完整的正交方法（DIOM）不同，该方法是带有滑动窗口的FOM。对流扩散方程和其他应用程序的数值实验表明，SCG是强大的，并且滑动窗口实现SWI允许SCG有效地解决大型系统。
The conjugate gradient (CG) method is a classic Krylov subspace method for solving symmetric positive definite linear systems. We introduce an analogous semi-conjugate gradient (SCG) method for unsymmetric positive definite linear systems. Unlike CG, SCG requires the solution of a lower triangular linear system to produce each semi-conjugate direction. We prove that SCG is theoretically equivalent to the full orthogonalization method (FOM), which is based on the Arnoldi process and converges in a finite number of steps. Because SCG's triangular system increases in size each iteration, we study a sliding window implementation (SWI) to improve efficiency, and show that the directions produced are still locally semi-conjugate. A counterexample illustrates that SWI is different from the direct incomplete orthogonalization method (DIOM), which is FOM with a sliding window. Numerical experiments from the convection-diffusion equation and other applications show that SCG is robust and that the sliding window implementation SWI allows SCG to solve large systems efficiently.
