107 problems

The discretized problem is posed on a grid whose cells may have a large aspect ratio, and the permeability is represented by a tensor with diagonal and off-diagonal components. The…

Let ${\mathcal X}$ and ${\mathcal Y}$ be subspaces of a Hilbert space ${\mathcal H}$ with ${{\rm dim}} \mathcal X={{\rm dim}} \mathcal Y$, and let $A$ be Hermitian. For a vector…

Let $\widetilde{A}$ be an $n\times n$ matrix with $\|\widetilde{A}\|_2\leq 1$, and let $A$ be a Gaussian perturbation of $\widetilde{A}$ with variance $\sigma^2\leq1$. Let $U$ be t…

Generic multiplicity-two conjecture. Under generic assumptions, the algebraic multiplicity of the defective eigenvalue in the nearest defective matrix is exactly $2$. Consequently,…

Let $A=B+iC\in\mathcal A_n$, where $B$ and $C$ are the real and imaginary parts of an accretive-dissipative matrix, and suppose that … Here $\kappa$ denotes the condition number an…

Let $A=B+iC$ be a complex symmetric matrix, where $B,C\in\mathbb R^{n\times n}$ are real symmetric positive definite. Such a matrix is called a Higham matrix. Let $\rho_n(A)$ denot…

Let $\mathbb{F}$ be a field, and consider the result proved in the source for $\mathbb{F}=\mathbb{R}$ concerning a method based solely on singular vector spaces. Complex-field anal…

Maximum growth-factor conjecture. The largest possible growth factor for a $5\times 5$ real matrix under complete pivoting is exactly the lower bound proved in this paper, namely…

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…

Let $A$ be a matrix, let $u$ and $v$ be vectors, and consider the rank-one perturbed linear system $(A+uv^T)x=b$. A system is numerically nonsingular when it is sufficiently nonsin…

Let $A$ be a nonsingular matrix and let $u$ and $v$ be vectors such that $A+uv^T$ is nonsingular. Let $\epsilon_M$ denote the unit roundoff, and let…

Let $A$ be a matrix, let $u$ and $v$ be vectors, and let $\tilde A=A+uv^T$. Let $\epsilon_M$ denote the unit roundoff, and let $\kappa(\cdot)$ denote the matrix condition number. S…

Let $A$ be a matrix, let $u$ and $v$ be vectors, and let $\tilde A=A+uv^T$. Let $\epsilon_M$ denote the unit roundoff, and let $\kappa_2(\cdot)$ denote the matrix 2-norm condition…

Let $A^{(t)}$ denote the matrix produced after $t$ steps of the uniformly random pivoting procedure, and let $A$ be the initial matrix. Write $kappa(A)$ for the condition number of…

Let $\Pi_d$ be the polynomial set associated with degree-$d$ approximation conditions, and let $\overline{\Pi^*_{2^7}}$ denote the closure of the polynomial set obtainable with sev…

Let $\Pi_d$ be the polynomial set associated with degree-$d$ approximation conditions, and let $\overline{\Pi^*_{2^6}}$ denote the closure of the polynomial set obtainable with six…

Let $\Pi_d$ be the polynomial set associated with degree-$d$ approximation conditions, and let $\overline{\Pi^*_{2^5}}$ denote the closure of the polynomial set obtainable with fiv…

Let $m>2$ and let $\Pi^*_{2^m}$ denote the polynomial set associated with evaluation schemes using $2^m$ matrix-matrix multiplications. Numerical computations suggest that generic…

Let $T$ be the transfer operator in the scalar case, with leading eigenvalue $\lambda_0$ and Ritz approximation $\lambda_0^{(m)}$ after $m$ iterations. Let $\theta_0$ be the initia…

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

The hybrid pressure formulation produces a global problem matrix obtained from the discretization of viscous laminar incompressible flows. Negative-definiteness conjecture. The mat…

Let $A_{mp}=Q_{mp}R_{pp}$ be a full QR factorisation, with $Q_{mp}$ orthogonal and $\begin{pmatrix}R_{nn}\\0_{pn}\end{pmatrix}$ the corresponding block structure, and write the der…

Let $n$ be the matrix dimension and let $varepsilon>0$ denote the target accuracy for the Hermitian eigenvalue algorithm. Logarithmic-precision conjecture. $O(\log(n/\varepsilon))$…