Approximation schemes for functions of positive-definite matrix values (Q2856628)

This paper investigates matrix means for positive definite matrices in light of approximating matrix functions on this matrix manifold. It studies subdivision schemes such as corner cutting schemes, and Bernstein operators to approximate the matrix exp-log and geometric mean functions of positive definite matrices. It is shown that many of the algebraic and geometric properties of such schemes are derived from the properties of the matrix means such as order preserving or order reversing. In particular, the geometric matrix mean preserves more matrix properties such as monotonicity and convexity than the exp-log matrix mean does.NEWLINENEWLINEError bounds for approximating univariate positive definite matrix functions with these schemes are established for all admissible matrix means.