Structure preserving piecewise polynomial interpolation for definite matrices
The problem considered is as follows. We know a smooth Hermitian positive (semi)definite matrix function \(A(t)\) \((t \in \mathbb{R}\), \(A \in \mathbb{C}^{n \times n})\) only at a discrete set of points: \(\{A_ i\}^{N + 1}_{i = 1}\), \(A_ i = A(t_ i)\) where \(t_ i < t_{i + 1}\). We are interested in finding an interpolatory polynomial \(P(t)\) for this sequence (i.e. \(P(t_ i) = A_ i)\) such that: (i) \(P(t)\) is a globally smooth \((C^ k\) with \(k \geq 1)\) function for \(t \in I = [t_ 1, t_{N + 1}]\), (ii) \(P(t)\) is positive (semi)definite for all \(t \in I\), and (iii) if the matrices \(A_ i\) are locally ordered with respect to the natural ordering of (semi)definite matrices, then the interpolant must preserve such an ordering (more precisely, if, for example, \(A_ i \geq A_{i + 1}\) then also \(P(t') \geq P(t'')\) for all \(t'\) and \(t''\) such that \(t_ i \leq t' \leq t'' \leq t_{i + 1})\). The piecewise linear interpolant satisfies (ii) and (iii) but not (i). The author considers piecewise cubic (and higher degree) interpolants which satisfy (i)--(iii). Included in the discussion is the order of approximation of such interpolants as the step size of the subintervals goes to zero. Motivation comes from numerical integration of Hermitian Lyapunov and Riccati differential equations, where the exact solution is known to be positive (semi)definite.
smooth Hermitian positive (semi)definite matrix function
Hermitian Lyapunov and Riccati differential equations
scientific article; zbMATH DE number 614108