ABSTRACT We consider the fundamental problem of estimating the difference between the exact value and approximations that depend on a single real parameter . It is well‐known that if the error satisfies an asymptotic expansion, then we can use Richardson extrapolation to approximate . In this paper, our primary concern is the accuracy of Richardson's error estimate , that is, the size of the relative error . In practice, the computed value is different from the exact value . We show how to determine when the computational error is irrelevant and how to estimate the accuracy of Richardson's error estimate in terms of Richardson's fraction . We establish monotone convergence theorems and derive upper and lower bounds for in terms of and . We classify asymptotic error expansions according to their practical value rather than the order of the primary error term. We present a sequence of numerical experiments that illustrate the theory. Weierstrass's function is used to define a sequence of smooth problems for which it is impractical to apply Richardson's techniques.

Summary We analyze the convergence of quasi‐Newton methods in exact and finite precision arithmetic using three different techniques. We derive an upper bound for the stagnation level and we show that any sufficiently exact quasi‐Newton method will converge quadratically until stagnation. In the absence of sufficient accuracy, we are likely to retain rapid linear convergence. We confirm our analysis by computing square roots and solving bond constraint equations in the context of molecular dynamics. In particular, we apply both a symmetric variant and Forsgren's variant of the simplified Newton method. This work has implications for the implementation of quasi‐Newton methods regardless of the scale of the calculation or the machine.

Summary In this paper, we present the StarNEig library for solving dense nonsymmetric standard and generalized eigenvalue problems. The library is built on top of the StarPU runtime system and targets both shared and distributed memory machines. Some components of the library have support for GPU acceleration. The library currently applies to real matrices with real and complex eigenvalues and all calculations are done using real arithmetic. Support for complex matrices is planned for a future release. This paper is aimed at potential users of the library. We describe the design choices and capabilities of the library, and contrast them to existing software such as LAPACK and ScaLAPACK. StarNEig implements a ScaLAPACK compatibility layer which should assist new users in the transition to StarNEig. We demonstrate the performance of the library with a sample of computational experiments.

Summary Triangular linear systems are central to the solution of general linear systems and the computation of eigenvectors. In the absence of floating‐point exceptions, substitution runs to completion and solves a system which is a small perturbation of the original system. If the matrix is well‐conditioned, then the normwise relative error is small. However, there are well‐conditioned systems for which substitution fails due to overflow. The robust solvers xLATRS from LAPACK extend the set of linear systems which can be solved by dynamically scaling the solution and the right‐hand side to avoid overflow. These solvers are sequential and apply to systems with a single right‐hand side. This paper presents algorithms which are blocked and parallel. A new task‐based parallel robust solver (Kiya) is presented and compared against both DLATRS and the non‐robust solvers DTRSV and DTRSM. When there are many right‐hand sides, Kiya performs significantly better than the robust solver DLATRS and is not significantly slower than the non‐robust solver DTRSM.