59 problems

Kung–Traub conjecture. The order satisfies

Let $N\geq4$, let $1\leq K\leq N-2$, and let $F>0$. Denote by $L^{{\rm qcf}}_F$ and $L^{{\rm qnl}}_F$ the force-based and quasinonlocal QC operators, respectively. Spectral equival…

Optimality conjecture for Krylov information. For the class $\Lambda$ of matrix-vector multiplications and every orthogonally invariant class $F$, Krylov information is optimal.

Let the inverse participation ratio (IPR) measure the concentration of the residual, and consider power-weighted sampling in an asynchronous randomized Jacobi method. Dual-role con…

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…

Forsythe's conjecture. Oscillatory limiting behavior occurs for every fixed restart length $s$.

Let $T_F(\alpha,\omega)$ be the forward-error iteration matrix, let $\lambda_F^{(k)}$ denote the normwise error-reduction factor at iteration $k$, and let $\rho$ and $\kappa$ denot…

Optimal threshold conjecture. The optimal limit for $s$ in the convergence theorem is actually $s>1$, which would yield the scaling $n\sim \omega^{1^+}$ as the frequency $\omega$ g…

Let $\alpha_{\mathrm{d}}$ denote the dissipative parameter in the DN-coupled fluid-structure-interaction problem. In the reported TC2--5 settings, non-monotone convergence behavior…

Consider fluid-structure-interaction problems with closed or nearly closed fluid configurations, where nearly closed configurations contain an arbitrarily narrow connection produci…

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

Small-micro-density conjecture. When $utotext{large}$, $n{g^tau}$ should be sufficiently small that the convergence rate of the MM-L method is not harmed.

Let $A_kx_k=b_k$ and $A_{k+1}x_{k+1}=b_{k+1}$ be successive linear systems, and let the systems be sufficiently close that the matrix differences $4A_{k+1}-A_k4_2$ are small. Let t…

Fourfold extrapolation conjecture. The quantity $\kappa””_{18}$ is an upper bound for $\kappa$, while $\kappa””_{19}$ is a lower bound for $\kappa$. This is a numerical refinement…

Monotonicity conjecture. For all $m\in\mathbb N$, one has

Let $\boldsymbol{A}$ be the coefficient matrix of the linear system and let $\boldsymbol{M}$ be an inaccurate factorization used to form the initial solution…

Let $A$ be an $n\times n$ real symmetric matrix. The standard implementation of the Jacobi method is an iterative algorithm for diagonalizing $A$ by orthogonal plane rotations. Ope…

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…

Finite convergence conjecture. If $T$ is symmetric and positive definite, then the semi-smooth Newton iteration converges after finitely many steps.

Let $U\in\mathbb{C}^{n\times n}$ be a scaled upper-triangular factor of an incomplete LU factorization, written as $U=I+U_s$, where $I$ is the identity matrix and $U_s$ is strictly…

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–…

The Discrete ALIF inner-loop convergence conjecture asserts that if

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…

Let Newton-Anderson denote the iterative method studied in the paper, and call a problem degenerate when its solution does not satisfy the usual nonsingularity assumptions for quad…

Let $\omega$ denote the frequency, and consider a Helmholtz problem discretized sufficiently accurately with a perfectly matched layer (PML) region whose thickness is kept constant…