On the condition number of Vandermonde matrices with pairs of nearly-colliding nodes
Stefan Kunis, Dominik Nagel
Abstract
Abstract We prove upper and lower bounds for the spectral condition number of rectangular Vandermonde matrices with nodes on the complex unit circle. The nodes are “off the grid,” pairs of nodes nearly collide, and the studied condition number grows linearly with the inverse separation distance. Such growth rates are known in greater generality if all nodes collide or for groups of colliding nodes. For pairs of nodes, we provide reasonable sharp constants that are independent of the number of nodes as long as non-colliding nodes are well-separated.
Topics & Concepts
Vandermonde matrixMathematicsCombinatoricsUpper and lower boundsInverseGeneralityGridUnit (ring theory)PhysicsEigenvalues and eigenvectorsGeometryMathematical analysisQuantum mechanicsPsychotherapistPsychologyMathematics educationMatrix Theory and AlgorithmsRandom Matrices and ApplicationsMathematical Dynamics and Fractals