22 problems

Let $l,g$ be two even functions with $g>0$ on $(0,\pi)$, and suppose that $f\equiv l/g$ is monotone increasing over $(0,\pi)$. Set $X_n\equiv T_n^{-1}(g)T_n(l)$ for all $n$. Let th…

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…

The discussion concerns plane-wave Trefftz discretizations on meshes and the preconditioners $\mathtt{P}_2$, $\mathtt{P}_5$, and $\mathtt{P}_7$, where $\mathtt{P}_2$ and…

Rank-preservation conjecture. The rank requirement for a valid preconditioner holds with exponentially-high probability for suitable SASOs and for SRFTs.

Let $\boldsymbol{P}_{\mathrm{upp}}$ be the preconditioner for the linear tangent problems in Newton's method that retains the NS–CH submatrix, and let…

Let $l,g$ be two even functions with $g>0$ on $(0,\pi)$, and suppose that $f\equiv l/g$ is monotone increasing over $(0,\pi)$. Set $X_n\equiv T_n^{-1}(g)T_n(l)$ for all $n$. Let…

Let $l$ and $g$ be even, real-valued functions on $(-\pi,\pi)$, with $g$ nonnegative and not identically zero almost everywhere, and let $f=l/g$ be even and monotonic on $[0,\pi]$.…

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 $A$ be the discretization of the first-kind Fredholm equation for Laplace's equation in three dimensions, and let $T^{-1}$ be the preconditioner defined from the domain decompo…

Let $A$ be the discretized integral operator in the referenced two- or three-dimensional models, and let $T^{-1}_{\text{Schwarz}}$ be the Schwarz preconditioner associated with the…

Let ${\rm dom}$ be a convex domain, let $s$ satisfy $0<s<1$, and let ${\mathcal H}_h^s$ and ${\mathcal G}_h^s$ denote the fine-grid and two-grid operators, respectively, with mesh…

A polynomial preconditioner is constructed using the power basis method, and LAPACK may issue a warning when the computation of its coefficients encounters an ill-conditioned or no…

Let $S_{k,\omega}$ and $N_{k,\omega}$ be the single-layer and hypersingular operators, respectively, and let a preconditioner have a remainder whose smoothing order can be compared…

Let $k(x)$ be a uniformly positive, bounded, and measurable function on $\Omega\subset\mathbb{R}^d$, with $d=1,2,$ or $3$, and consider a finite-element discretization of the gener…

Field-of-values conjecture. The field of values $W(J_h)$ can be bounded away from $0$, independently of $N_x$.

Uniform boundedness conjecture. For the truncated problem,

Consider stochastic reformulations of a linear system in which the distribution ${\cal D}$ and matrix ${\bf B}$ are designed to reflect the structure of ${\bf A}$. Such design choi…

Uniform condition-number conjecture. There exist $s<0$ and a constant $C$ such that

Let $V_h\subset V$ and $Q_h\subset Q$ be inf-sup compatible finite-dimensional spaces, and let ${\mathsf{A}}$ and ${\mathsf{T}}$ be the corresponding stiffness and trace matrices.…

Let an ILU(0) preconditioner be applied to a sparse linear system with coefficient matrix, and measure the approximation accuracy of its preconditioning matrix as in the paper. Coe…

Let ${\widetilde W}$ denote the matrix of eigenvectors for the symmetrically preconditioned force-based QC operator ${L}^{-1/2}L^{{\rm qcf}}_F{L}^{-1/2}$. Assume $A_F>0$. Symmetric…

Let $N\geq4$, let $1\leq K\leq N-2$, and let $F>0$. Let ${L}^{-1}$ be the inverse of the discrete Laplacian operator $L$, and consider the preconditioned operators…