Fig. 1. Generic relaxed CG.

Fig. 2. Spectrum of the Wilson fermion matrix M for our 4⁴ configuration (left), spectrum of D_u for μ=0.3 (right).

Fig. 3. Relaxed and non-relaxed CG for (8). Left: μ=0.3. Right: μ=0.1.

Fig. 4. Overview of our recursive preconditioning computational scheme.

Fig. 5. Relaxed GMRESR.

Fig. 6. Convergence history for unrelaxed CG, relaxed CG and relaxed GMRESR for one inversion of the squared equation (2). We plot the norm of the residual vs. the number of calls to the Wilson operator Q. The tics indicate each (outer) iteration: On the 4⁴ lattice relaxed GMRESR needs 5 iterations to converge, whereas unrelaxed and relaxed CG both require 11 iterations.

Fig. 7. Convergence history for unrelaxed SUMR, relaxed SUMR and relaxed GMRESR(SUMR), two pass strategy for (2).

Advised Krylov subspace method and corresponding strategy for tuning the precision of the matrix–vector products as a function of the properties of the matrix A

Times (in seconds) for one inversion on the five 4⁴ configurations with β=5.4, run on 1 processor of ALiCE. The number in brackets is the gain from the unrelaxed and unpreconditioned (CG) inversion

Times (in seconds) for one inversion on the three 8⁴ configurations with β=5.6, run on 16 processors of ALiCE

Times (in seconds) for one inversion on the quenched 16⁴ configuration at β=6.0, run on 16 processors of ALiCE

Times (in seconds) for the inversion of chiral fermions on the 8⁴, μ=0.3 ensemble, run on 16 processors of ALiCE

Times (in seconds) for two SUMR inversions on the five 4⁴ configurations at β=5.4, run on one processor of ALiCE

Times (in seconds) for two SUMR inversions on the 8⁴ configurations at β=5.6, run on 16 processors of ALiCE

Times (in seconds) for two SUMR inversions on the quenched 16⁴ configuration, run on 16 processors of ALiCE

The times (in seconds) needed to calculate one relGMRESR(CG) inversion of the overlap operator, and to calculate n_p eigenvalues of the Wilson operator for different values of n_p, on the 8⁴ configuration 1 with μ=0.1

The times (in seconds) needed to calculate one relGMRESR(CG) inversion of the overlap operator, and to calculate n_p eigenvalues of the Wilson operator for different values of n_p, on the 4⁴ configuration 1, with μ=0.1