7 problems

Let $A=A^T\in\mathbb{R}^{n\times n}$ be symmetric. For an integer $s$ satisfying $1\leq s<d(A)$, let $v_0\in\mathbb{R}^n$ be a unit-norm initial vector with $d(A,v_0)\geq s+1$, and…

Let $A$ be a matrix, and consider the Arnoldi cross iteration, which alternately uses $A$ and $A^T$ in the Arnoldi projection process; for symmetric matrices this is the correspond…

Let the conjugate gradient method be restarted every $s$ steps, and consider its normalized error vectors or normalized residual vectors. Forsythe's conjecture. The normalized erro…

Let $\bm{y}(t)$ be the solution of the second-order differential equation and let $\bm{y}_m(t)$ be its Krylov subspace approximant, with residual $\bm{r}_m(t)$ and error…

Let ${\bf B}_{m+\ell}$ be partitioned as … and define … Here $\widetilde{{\bf B}}_\ell$ is the corresponding projected bidiagonal block produced by the recycled hybrid projection m…

The adaptive $s$-step conjugate gradient method selects block sizes using a bound on the residual gap. More generally, consider $s$-step variants of recursively computed residual m…

Conjecture on improved error-estimate factors. $\gamma_1$ and $\gamma_3$ can be replaced by some other better forms, at least for $A$ real symmetric semipositive or negative, whose…