Abstract. 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 uppertridiagonal). 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.

Challenges related to development, deployment, and maintenance of reusable software for science are becoming a growing concern. Many scientists' research increasingly depends on the quality and availability of software upon which their works are built. To highlight some of these issues and share experiences, the First Workshop on Sustainable Software for Science: Practice and Experiences (WSSSPE1) was held in November 2013 in conjunction with the SC13 Conference. The workshop featured keynote presentations and a large number (54) of solicited extended abstracts that were grouped into three themes and presented via panels. A set of collaborative notes of the presentations and discussion was taken during the workshop. Unique perspectives were captured about issues such as comprehensive documentation, development and deployment practices, software licenses and career paths for developers. Attribution systems that account for evidence of software contribution and impact were also discussed. These include mechanisms such as Digital Object Identifiers, publication of "software papers", and the use of online systems, for example source code repositories like GitHub. This paper summarizes the issues and shared experiences that were discussed, including cross-cutting issues and use cases. It joins a nascent literature seeking to understand what drives software work in science, and how it is impacted by the reward systems of science. These incentives can determine the extent to which developers are motivated to build software for the long-term, for the use of others, and whether to work collaboratively or separately. It also explores community building, leadership, and dynamics in relation to successful scientific software.

Most commonly used adaptive algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a user-specified absolute error tolerance for a cone, C, of non-spiky input functions in the Sobolev space W 2,∞ [a, b]. Our algorithms automatically determine where to sample the function-sampling more densely where the second derivative is larger. The computational cost of our algorithm for approximating a univariate function f on a bounded interval with L ∞ -error no greater than ε is O f 1 2 /ε as ε → 0. This is the same order as that of the best function approximation algorithm for functions in C. The computational cost of our global minimization algorithm is of the same order and the cost can be substantially less if f significantly exceeds its minimum over much of the domain. Our Guaranteed Automatic Integration Library (GAIL) contains these new algorithms. We provide numerical experiments to illustrate their superior performance.

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.