MINRES-QLP：求解不定或奇异对称系统的Krylov子空间方法
MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems
摘要：
CG、SYMMLQ和MINRES是求解对称线性方程组的Krylov子空间方法。当这些方法被应用到一个不相容的系统（即，一个奇异的对称最小二乘问题），CG可能会崩溃和SYMMLQ的解决方案可能会爆炸，而MINRES将提供一个最小二乘解，但不一定是最小长度（伪逆）的解决方案。这种理解促使我们设计一个MINRES算法来计算奇异对称系统的最小长度的解决方案。MINRES使用来自Lanczos过程的三对角矩阵的QR因子（其中R是上三对角）。MINRES-QLP使用QLP分解（其中右侧的旋转将R减少到下三对角形式）。对于病态系统（奇异或非奇异），MINRES-QLP可以给出比MINRES更精确的解。我们得到预处理MINRES-QLP，新的停止规则，更好的估计的解决方案和剩余的范数，矩阵范数，和条件数。
英文摘要：
CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric systems of linear equations. When these methods are applied to an incompatible system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a least-squares solution but not necessarily the minimum-length (pseudoinverse) solution. This understanding motivates us to design a MINRES-like algorithm to compute minimum-length solutions to singular symmetric systems. MINRES uses QR factors of the tridiagonal matrix from the Lanczos process (where R is upper-tridiagonal). MINRES-QLP uses a QLP decomposition (where rotations on the right reduce R to lower-tridiagonal form). On ill-conditioned systems (singular or not), MINRES-QLP can give more accurate solutions than MINRES. We derive preconditioned MINRES-QLP, new stopping rules, and better estimates of the solution and residual norms, the matrix norm, and the condition number.
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