ARPACK relies heavily upon a number of basic operations and algorithms provided by the BLAS and LAPACK . These have contributed greatly to the robustness, accuracy, and computational performance of ARPACK. The most important of these with respect to performance is the BLAS subroutine _gemv . For a fixed number nev of requested eigenvalues and a fixed length ncv Arnoldi basis, the computational cost scales linearly with n, the order of the matrix. The rate of execution (in FLOPS) for the IRA iteration is asymptotic to the rate of execution of _gemv.
In the outline of the implementation described in Figure 5.1, is a upper Hessenberg matrix, , and the residual vector is -orthogonal to the columns of . For each j,where OP and are defined with respect to one of the computational modes described in Chapter 3.
The integer k denotes the desired number of eigenvalue approximations and this may, at times, be greater than the users request for nev approximations. The integer will be the largest size factorization constructed. An eigenvalue of is a Ritz value of OP and is the corresponding Ritz vector when (A formal definition of Ritz value and vector is given in Chap. 4). The normalization is assumed throughout.
Update to get the new length Arnoldi factorization (See Algorithm 3, Chapter 4 ).