# How to compute Green's Functions for entire Mass Trajectories within Krylov Solvers

## TL;DR

This paper introduces a novel numerical method integrated within Krylov solvers to efficiently compute multiple Green's functions and their derivatives for matrices of the form A=D-m, enabling rapid evaluation across a range of mass parameters in lattice QCD.

## Contribution

The paper presents a new general approach to compute many Green's functions simultaneously within one iteration, applicable to matrices of a specific structure, and demonstrates its effectiveness in lattice QCD applications.

## Key findings

- Efficient computation of Green's functions for multiple masses using a single inversion.
- Ability to derive derivatives and Taylor expansions of solutions with respect to parameters.
- Application to lattice QCD shows significant computational savings.

## Abstract

The availability of efficient Krylov subspace solvers play a vital role for the solution of a variety of numerical problems in computational science. Here we consider lattice field theory. We present a new general numerical method to compute many Green's functions for complex non-singular matrices within one iteration process. Our procedure applies to matrices of structure $A=D-m$, with $m$ proportional to the unit matrix, and can be integrated within any Krylov subspace solver. We can compute the derivatives $x^{(n)}$ of the solution vector $x$ with respect to the parameter $m$ and construct the Taylor expansion of $x$ around $m$. We demonstrate the advantages of our method using a minimal residual solver. Here the procedure requires $1$ intermediate vector for each Green's function to compute. As real life example, we determine a mass trajectory of the Wilson fermion matrix for lattice QCD. Here we find that we can obtain Green's functions at all masses $\geq m$ at the price of one inversion at mass $m$.