We describe algorithm MINRES-QLP and its FORTRAN 90 implementation for solving symmetric or Hermitian linear systems or least-squares problems. If the system is singular, MINRES-QLP computes the unique minimum-length solution (also known as the pseudoinverse solution), which generally eludes MINRES. In all cases, it overcomes a potential instability in the original MINRES algorithm. A positive-definite pre-conditioner may be supplied. Our FORTRAN 90 implementation illustrates a design pattern that allows users to make problem data known to the solver but hidden and secure from other program units. In particular, we circumvent the need for reverse communication. Example test programs input and solve real or complex problems specified in Matrix Market format. While we focus here on a FORTRAN 90 implementation, we also provide and maintain MATLAB versions of MINRES and MINRES-QLP.

中文翻译：

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算法 937：对称和埃尔米特线性方程和最小二乘问题的 MINRES-QLP。

我们描述了算法 MINRES-QLP 及其 FORTRAN 90 实现，用于解决对称或 Hermitian 线性系统或最小二乘问题。如果系统是奇异的，MINRES-QLP 会计算唯一的最小长度解（也称为伪逆解），这通常会避开 MINRES。在所有情况下，它都克服了原始 MINRES 算法中潜在的不稳定性。可以提供正定预调节器。我们的 FORTRAN 90 实现说明了一种设计模式，该模式允许用户将问题数据告知求解器，但对其他程序单元隐藏且安全。特别是，我们避免了反向通信的需要。示例测试程序输入并解决以 Matrix Market 格式指定的真实或复杂问题。虽然我们在这里重点关注 FORTRAN 90 实现，但我们还提供并维护 MINRES 和 MINRES-QLP 的 MATLAB 版本。