6533b860fe1ef96bd12c2f48

RESEARCH PRODUCT

A General Algorithm to Calculate the Inverse Principal $p$-th Root of Symmetric Positive Definite Matrices

Dorothee RichtersMichael LassChristian PlesslThomas D. KühneAndrea Walther

subject

Discrete mathematicsMathematical problemPhysics and Astronomy (miscellaneous)Root (chord)InversePositive-definite matrixMathematics - Rings and AlgebrasNumerical Analysis (math.NA)01 natural sciences010101 applied mathematicsMatrix (mathematics)Quadratic equationRate of convergenceRings and Algebras (math.RA)Convergence (routing)FOS: MathematicsApplied mathematicsMathematics - Numerical Analysis0101 mathematicsMathematics

description

We address the general mathematical problem of computing the inverse p-th root of a given matrix in an efficient way. A new method to construct iteration functions that allow calculating arbitrary p-th roots and their inverses of symmetric positive definite matrices is presented. We show that the order of convergence is at least quadratic and that adaptively adjusting a parameter q always leads to an even faster convergence. In this way, a better performance than with previously known iteration schemes is achieved. The efficiency of the iterative functions is demonstrated for various matrices with different densities, condition numbers and spectral radii.

yearjournalcountryeditionlanguage
2019-01-01
https://dx.doi.org/10.48550/arxiv.1703.02456