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Calculation of band structures of three dimensional photonic crystals amounts to solving large-scale Maxwell eigenvalue problems, which are notoriously challenging due to high multiplicity of zero eigenvalues. In this paper, we try to address this problem in such a broad context that band structures of three dimensional isotropic photonic crystals in all $14$ Bravais lattices can be efficiently computed in a unified framework. In this work, we uncover the delicate machinery behind several key results of our framework and on the basis of this new understanding we drastically simplify the derivations, proofs and arguments. Particular effort is made on reformulating the Bloch condition for all $14$ Bravais lattices in the redefined orthogonal coordinate system, and establishing eigen-decomposition of discrete partial derivative operators by identifying the hierarchical structure of the underlying normal (block) companion matrix, and reducing the eigen-decomposition of the double-curl operator to a simple factorization of a $3$-by-$3$ complex skew-symmetric matrix. With the validity of the novel nullspace free method in the broad context, we perform some calculations on one benchmark system to demonstrate the accuracy and efficiency of our algorithm to solve Maxwell eigenvalue problems.

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The Lanczos method is often used to solve a large scale symmetric matrix eigenvalue problem. It is well-known that the single-vector Lanczos method can only find one copy of any multiple eigenvalue and encounters slow convergence towards clustered eigenvalues. On the other hand, the block Lanczos method can compute all or some of the copies of a multiple eigenvalue and, with a suitable block size, also compute clustered eigenvalues much faster. The existing convergence theory due to Saad for the block Lanczos method, however, does not fully reflect this phenomenon since the theory was established to bound approximation errors in each individual approximate eigenpairs. Here, it is argued that in the presence of an eigenvalue cluster, the entire approximate eigenspace associated with the cluster should be considered as a whole, instead of each individual approximate eigenvectors, and likewise for approximating clusters of eigenvalues. In this paper, we obtain error bounds on approximating eigenspaces and eigenvalue clusters. Our bounds are much sharper than the existing ones and expose true rates of convergence of the block Lanczos method towards eigenvalue clusters. Furthermore, their sharpness is independent of the closeness of eigenvalues within a cluster. Numerical examples are presented to support our claims.

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