Chebyshev-Davidson method

Even with standard restart methods such as ARRACK and TRLan, the memory demand can still remain too high in some cases. Hence, it is important to develop a diagonalization method that is less memory demanding but whose efficiency is comparable to ARRACK and TRLan. The Chebyshev-Davidson method [23] was developed with these two goals in mind. Details can be found in [23]. The principle of the method is to simply build a subspace by a procedure based on a form of Block-Davidson approach. The Block-Davidson approach builds a subspace by adding a window of preconditioned vectors. In the Chebyshev-Davidson approach, these vectors are built by exploiting Chebyshev polynomials. 

The first step diagonalization by the block Chebyshev-Davidson method, together with the Chebyshev-filtered subspace method (Algorithm 6.3), enabled us to... 

In order to partition the occupied states into slices, it is necessary to have a guess for the Fermi level. The guess does not need a high degree of accuracy. For this the Fermi level from the previous SCF iteration can be used. This means that the spectrum slicing method will need to be bootstrapped with a regular method similarly to Chebyshev-Davidson. 

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