Fig. 1. Spectrum of H_w(μ) in Eq. (1) for a 4⁴ lattice (top pane) and a 6⁴ lattice (bottom pane), with μ=0.3 and m_w=−2. Note the difference in scale between real and imaginary axes. The gauge fields were generated using the Wilson plaquette action with gauge coupling β_g=5.1 [7].

Fig. 2. Spectrum of for a 4⁴ lattice (top pane) and a 6⁴ lattice (bottom pane), with μ=0.3 and β_g=5.1.

Fig. 3. Accuracy of the modified Arnoldi method for y=sgn(A)x. The non-Hermitian matrix is A=γ₅D_w(μ), where D_w(μ) is the Wilson–Dirac operator (2) at chemical potential μ=0.3, and x=(1,1,…,1). Top pane: 4⁴ lattice with dim(A)=3072, bottom pane: 6⁴ lattice with dim(A)=15552. The relative error is shown as a function of the Krylov subspace size k for various numbers of deflated eigenvalues m using the LR-deflation. In order to compute the error , the exact solution y was first determined using the spectral definition (4) and the full spectral decomposition computed with LAPACK.

Fig. 4. Comparison of the accuracies achieved with both deflation schemes and the exact projection of y on the composite space . The relative error is shown as a function of the Krylov subspace size k. For both lattice sizes the accuracy of the LR-deflation is slightly better than that of the Schur deflation. Furthermore, the accuracy of the modified Arnoldi approximation is very close to the best possible approximation in the composite subspace.

Ratio of largest deflated eigenvalue over largest eigenvalue for various numbers of deflated eigenvalues m for a 4⁴ and a 6⁴ lattice

CPU time (in seconds) for varying Krylov subspace size k. Top row: 4⁴ lattice with m=32, bottom row: 6⁴ lattice with m=128. Left panes: Schur deflation, right panes: LR-deflation. The time required by the initial calculation of the critical eigenvectors is given in the header of each block. The time needed to construct the Arnoldi basis in the Krylov subspace (column 2) is approximately proportional to Nk(k+2m) for the Schur deflation and Nk² for the LR-deflation. The time used by Roberts' method (34) to compute the sign function of the Hessenberg matrix (column 3) is O(k³). The total time (column 4) also includes the evaluation of Eq. (33) for the Schur deflation and Eq. (41) for the LR-deflation. These timings were measured on an Intel Core 2 Duo 2.33 GHz computer using optimized ATLAS BLAS routines [34]