In fact it is essential to balance the Tolasa matirx. In contrast to Figure 1, Figure 4 illustrates the effectiveness of Arnoldi's method at extracting the nearest vector to the sought after eigenvector from a Krylov subspace. Once again for the unbalanced operator the Arnoldi vectors are far from optimal. Figure 5 plots corresponding data for the two-sided Lanczos algorithm. Purple and red still denote to the unbalanced operator, and blue and green still denote the balanced operator.
Residuals are self-explanatory.
Just as with Arnoldi's method, the residuals for the unbalanced operator are misleading. Figure 7 plots the exact and estimated error in the Lanczos values. In words, an advantage of the Lanczos algorithm is that the error estimates are more reliable. In words, Anoldi's method yields much smaller residuals, while the Lanczos algorithm computes more accurate eigenvalues.