Sparse matrices, storage schemes

Fig. 9. Sparse matrix storage schemes (a) a square matrix, (b) scheme I, (c) scheme II (linked lists).

Although this algorithm is clear and simple, it presents the most ineffective way of storing the global stiffness matrix since it results in a full sparse matrix. Later in this section we will discuss how the storage space and computation time is minimized by using alternative storing schemes such as banded matrices. 

Another approach to the C matrix construction is a CSF-driven approach proposed by Knowles et al.. With this approach, the density matrix elements dlgrs ars constructed for all combinations of orbital indices p, q, r and s, but for a fixed CSF labeled by n. Each column of the matrix C is constructed in the same way that the Fock matrix F is computed except that the arrays D" and d" are used instead of D and d. As with the F matrix construction described earlier, there are two choices for the ordering of the innermost DO loops. One choice results in an inner product assembly method while the other choice results in an outer product assembly method. The inner product choice, which does not allow the density matrix sparseness to be exploited, results in SDOT operations of length m or about m, depending on the integral storage scheme. The outer product choice, which does allow the density matrix sparseness to be exploited, has an effective vector length of n, the orbital basis dimension. However, like the second index-driven method described above, this may involve some extraneous effort associated with redundant orbital rotation variables in the active-active block of the C matrix. 

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