We present an efficient null space free Jacobi–Davidson method to compute the positive eigenvalues of time harmonic Maxwell’s equations. We focus on a class of spatial discretizations that guarantee the existence of discrete vector potentials, such as Yee’s scheme and the edge elements. During the Jacobi–Davidson iteration, the correction process is applied to the vector potential instead. The correction equation is solved approximately as in the standard Jacobi–Davidson approach. The computational cost of the transformation from the vector potential to the corrector is negligible. As a consequence, the expanding subspace automatically stays out of the null space and no extra projection step is needed. Numerical evidence confirms that the proposed scheme indeed outperforms the standard and projection-based Jacobi–Davidson methods by a significant margin.

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Due to vast sequence divergence among different viral groups, sequence alignment is not directly applicable to genome-wide comparative analysis of viruses. More and more attention has been paid to alignment-free methods for whole genome comparison and phylogenetic tree reconstruction. Among alignment-free methods, the recently proposed ‘‘Natural Vector (NV) representation” has successfully been used to study the phylogeny of multi-segmented viruses based on a 12-dimensional genome space derived from the nucleotide sequence structure. But the preference of proteomes over genomes for the determination of viral phylogeny was not deeply investigated. As the translated products of genes, proteins directly form the shape of viral structure and are vital for all metabolic pathways. In this study, using the NV representation of a protein sequence along with the Hausdorff distance suitable to compare point sets, we construct a 60-dimensional protein space to analyze the evolutionary relationships of 4021 viruses by whole-proteomes in the current NCBI Reference Sequence Database (RefSeq). We also take advantage of the previously developed natural graphical representation to recover viral phylogeny. Our results demonstrate that the proposed method is efficient and accurate for classifying viruses. The accuracy rates of our predictions such as for Baltimore II viruses are as high as 95.9% for family labels, 95.7% for subfamily labels and 96.5% for genus labels. Finally, we discover that proteomes lead to better viral classification when reliable protein sequences are abundant. In other cases, the accuracy rates using proteomes are still comparable to that of genomes.

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