# Krylov Iterative Methods for the Geometric Mean of Two Matrices Times a Vector

## TL;DR

This paper introduces an efficient Krylov space-based method for computing the geometric mean of two positive definite matrices times a vector, leveraging matrix-vector products for low-cost approximations.

## Contribution

It presents a novel Krylov iterative approach specifically designed for the geometric mean of matrices times a vector, improving computational efficiency.

## Key findings

- Method achieves accurate approximations with fewer matrix-vector products
- Applicable to large-scale positive definite matrices
- Reduces computational cost compared to traditional methods

## Abstract

In this work, we are presenting an efficient way to compute the geometric mean of two positive definite matrices times a vector. For this purpose, we are inspecting the application of methods based on Krylov spaces to compute the square root of a matrix. These methods, using only matrix-vector products, are capable of producing a good approximation of the result with a small computational cost.