Structure preserving piecewise polynomial interpolation for definite matrices

AbstractThe purpose of this note is the following: Suppose we only know a smooth Hermitian positive (semi)definite matrix function A(t), t ϵ R, A ϵ Cn×n, at a discrete set of points: {Ai}N+1i=1, Ai = A(ti), where ti < ti+1. We are interested in finding an interpolatory polynomial P(t) for this sequence [i.e., P(ti) = Ai] satisfying the following conditions: (i) P(t) is a globally smooth function ∀t ϵ I ≔ [t1, tN+1], (ii) P(t) is positive (semi)definite ∀t ϵ I, (iii) if the matrices Ai are locally ordered with respect to the natural ordering of (semi)definite matrices, then the interpolant must preserve such an ordering. In this note, we restrict to piecewise polynomial interpolants. The simplest choice of the piecewise linear interpolant is easily seen to satisfy (ii)–(iii), but obviously not (i). Here we consider piecewise cubic (and higher degree) interpolants which satisfy (i)–(iii) above. We also briefly discuss some applications and possible extensions of the results herein