×

Approximation schemes for functions of positive-definite matrix values. (English) Zbl 1284.65028

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.
Error bounds for approximating univariate positive definite matrix functions with these schemes are established for all admissible matrix means.

MSC:

65D15 Algorithms for approximation of functions
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
65F60 Numerical computation of matrix exponential and similar matrix functions
15A16 Matrix exponential and similar functions of matrices
26E60 Means
47A64 Operator means involving linear operators, shorted linear operators, etc.
65D17 Computer-aided design (modeling of curves and surfaces)
PDFBibTeX XMLCite
Full Text: DOI Link