19 problems

Renegar's conjecture. Computing ${\mathscr C}(a)$ for a given data $a$ is at least as difficult as solving the problem for which $a$ is a data.

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

Let $A$ be a Gaussian perturbation of a square matrix $\widetilde{A}$, in the setting of the preceding smoothed condition-number analysis. Logarithmic-factor conjecture. The factor…

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…

Condition-number monotonicity conjecture. For the Gaussian or inverse multiquadric RBF,

Let $m\geq 4$ be an even integer, let $\boldsymbol{E}_m$ denote the vector set defined in the paper, and let $\beta_{\boldsymbol{A}}$ be the condition number associated with…

Condition-number equality conjecture.

Well-conditioning conjecture. The matrix $S_n$ is well conditioned.

Semisimple-eigenvalue conjecture. In the case of polynomial eigenvalue problems, problems with semisimple eigenvalues are not ill-conditioned in this framework; moreover, much of t…

Finite expected angular condition number conjecture.

Infinite expected regular condition number conjecture.

Let $n_1,n_2,n_3$ satisfy $n_1 \ge n_2 \ge n_3 \ge 3$, and let random rank-$1$ tuples of length $r$ in $\mathbb{R}^{n_1 \times n_2 \times n_3}$ have their expected condition number…

Let $\mathcal{M}_{n}$ be the set of complex $n\times n$ matrices acting on the Banach space $\mathbb{C}^{n}$ equipped with a norm. For any invertible $T\in\mathcal{M}_{n}$ and any…

For frequency indices $k_1$ and $k_2$, let $\hat L^{k_1,k_2}$ be the corresponding block of $\hat L$, and write $\kappa(\hat L^{k_1,k_2})$ for its condition number. Condition-numbe…

Let $f$ be a polynomial system, let $\kappa(f)$ denote its condition number, and let the zero counting problem for $f$ be the problem of computing the number of its real zeros. Com…

Let $A$ be the random matrix considered in the paper, let $\kappa(A)$ denote its condition number, and let $\lambda$ be the parameter appearing in the bound … The numerical experim…

Let $H_{\psi\text{-div}}$ be the Hessian matrix of the $\psi$-divergence M-estimator, and let $H_{\psi\text{-KMM}}$ be the Hessian matrix of the corresponding $\psi$-KMM loss funct…

Structured condition-number computability conjecture. The structured condition number $\kappa_{struct}$ can only be computed accurately if the original problem $p$ can be computed…

Reciprocal structured condition-number conjecture. For traditional arithmetic, (1) the reciprocal of the structured condition number is an approximation of the relative distance fr…