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We extend the proof in [M. Crouzeix and C. Palencia, The numerical range is a(1 + √2̅) -spectral set, SIAM Jour. Matrix Anal. Appl., 38 (2017), pp. 649-655] to show that other regions in the complex plane areK -spectral sets. In particular, we show that various annular regions are(1 + √2̅ ) -spectral sets and that a more general convex region with a circular hole in it is a(3 + 2 √3̅ ) -spectral set.

We provide numerical bounds on the Crouzeix ratio for KMS matrices A which have a line segment on the boundary of the numerical range. The Crouzeix ratio is the supremum over all polynomials p of the spectral norm of p ( A ) divided by the maximum absolute value of p on the numerical range of A . Our bounds satisfy the conjecture that this ratio is less than or equal to 2. We also give a precise description of these numerical ranges.

Many conjugate gradient-like methods for solving linear systems $Ax=b$ use recursion formulas for updating residual vectors instead of computing the residuals directly. For such methods it is shown that the difference between the actual residuals and the updated approximate residual vectors generated in finite precision arithmetic depends on the machine precision $\\epsilon$ and on the maximum norm of an iterate divided by the norm of the true solution. It is often observed numerically, and can sometimes be proved, that the norms of the updated approximate residual vectors converge to zero or, at least, become orders of magnitude smaller than the machine precision. In such cases, the actual residual norm reaches the level $\\epsilon \\| A \\| \\| x \\|$ times the maximum ratio of the norm of an iterate to that of the true solution. Using exact arithmetic theory to bound the size of the iterates, we give a priori estimates of the size of the final residual for a number of algorithms.

For any nonsingular matrix $A$ and any positive integer $m$, the $m$th root, $A{1/m}$, can be defined using any branch cut of the $m$th root function that does not pass through an eigenvalue of $A$. Cleary, $A{1/m}$ approaches the identity as $m \\rightarrow \\infty$, but we are interested in how it approaches the identity. It is shown that $\\lim_{m \\rightarrow \\infty} [W( A{1/m} ) ]m = \\exp [ W( \\log A ) ]$, where $W$ denotes the field of values and $\\log A$ is defined using the same branch cut. It is also shown that $\\| A{1/m} \\|m$ approaches $\\exp( \\alpha ( \\log A ) )$, where $\\| \\cdot \\|$ denotes the spectral norm and $\\alpha ( \\cdot )$ is the numerical abscissa. Implications for the convergence rate of the GMRES algorithm are discussed, especially when $W(A)$ contains the origin. Additionally, it is shown that if $A$ is a strict contraction and $\\rho \\in ( \\| A \\| , 1 )$, then the matrices $[ ( I + \\rho A ){-1} (A + \\rho I ) ]{1/m}$, $m=1,2, \\ldots,$ are all strict contractions. The significance of results of this sort in a possible approach to proving Crouzeix's conjecture is discussed.

Given a nonincreasing positive sequence $f ( 0 ) \\geq f ( 1 ) \\geq \\cdots \\geq f ( n - 1 ) > 0$, it is shown that there exists an $n$ by $n$ matrix $A$ and a vector $r^0 $ with $ \\| r^0 \\| = f ( 0 ) $ such that $f ( k ) = \\| r^k \\|,\\,k = 1, \\cdots ,n - 1$, where $r^k $ is the residual at step $k$ of the GMRES algorithm applied to the linear system $Ax = b$, with initial residual $r^0 = b - Ax^0 $. Moreover, the matrix $A$ can be chosen to have any desired eigenvalues.

Let W ( A ) denote the field of values (numerical range) of a matrix A . For any polynomial p and matrix A , define the Crouzeix ratio to have numerator max | p ( ζ ) | : ζ ∈ W ( A ) and denominator ‖ p ( A ) ‖ 2 . Crouzeix’s 2004 conjecture postulates that the globally minimal value of the Crouzeix ratio is 1 / 2, over all polynomials p of any degree and matrices A of any order. We derive the subdifferential of this ratio at pairs ( p ,  A ) for which the largest singular value of p ( A ) is simple. In particular, we show that at certain candidate minimizers ( p ,  A ), the Crouzeix ratio is (Clarke) regular and satisfies a first-order nonsmooth optimality condition, and hence that its directional derivative is nonnegative there in every direction in polynomial-matrix space. We also show that pairs ( p ,  A ) exist at which the Crouzeix ratio is not regular.