9 problems

Sharpness conjecture for arbitrary sequences of matrix preconditioners. The preceding assertion should hold for every $0 < \mu \leq \frac{1}{2}$, every nonzero right-hand side vect…

Randomized-BCGS stability conjecture. The overall stability of the two-stage algorithm with randomized block classical Gram–Schmidt pre-processing is ensured when

Worst-case root-convergence factor conjecture. The maximum root-convergence factor of rAA(1) satisfies

Let $\mathcal{X}$ be an input matrix and consider an orthogonalization scheme whose loss of orthogonality is controlled at the level represented by the…

Let ${\bf A}\in\mathbb{C}^{n\times n}$, let $W({\bf A})$ be its field of values, and let $\EuScript P_k$ denote the polynomials of degree at most $k$. Define…

Right-preconditioned GMRES bounds conjecture. Under this condition, right-preconditioned GMRES is backward stable, and similar bounds can be proved for the quantities in Table. The…

Let $\tens{V}_k$ be the Krylov basis computed in the $k$th inner iteration, and suppose that the basis loses linear independence. Here, MGS-GMRES denotes GMRES using modified Gram–…

Let $q$ be a nonlinear fixed-point map with fixed point $x^*$, and consider the linearized fixed-point problem with matrix $I-q'(x^*)$. Suppose GMRES($\infty$) applied to this line…

Calkin-algebra convergence-rate conjecture. $M_{\pi(A)}$ cannot see any eigenspaces of $A$, including eigenspaces of infinite dimension.