Matrix Storage Schemes

Finite element stiffness matrices are always symmetric and banded. For example, the mesh presented in Fig. 9.8 has a stiffness matrix given by 

The matrix presented in the above equation is clearly a banded matrix, with a bandwidth of 4. Note that the bandwidth is the maximum difference between node numbers of the elements of a given mesh times the number of degrees of freedom per node. Algorithm 8 computes the bandwidth of any mesh with nelem nodes per element. 

Algorithm 8 Compute the bandwidth of a banded Global Stiffness matrix 

By examining the matrix it becomes obvious that within the upper part of the matrix, above or to the right of the bandwidth, all terms are zero and therefore do not need to be stored, or operated on during the solution process. In addition, since the stiffness matrix is symmetric, we do not need to store the stiffness matrix components that are below the 

With a known bandwidth, it is quite straight forward to add the element stiffness matrices into the global system using the scheme presented in Algorithm 9. 

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

Before we proceed to our discussion of global stiffness matrix storage schemes, we will discuss the last aspect of the finite element implementation, namely, the application of the boundary conditions. As discussed earlier, the natural boundary conditions are imbedded in the finite element equation system - it is implied that every boundary node without an... 

We offer these two examples of data storage scheme in order to illustrate the interrelationship between data structure, storage requirement, and the types of operations to be performed. The specific data structure and data manipulation techniques to be used should always be tailored to the structure of the matrix and the requirement of the application. In point of fact both schemes I and II can be modified to overcome some of the stated deficiencies. Gustavson (G9) discussed modifications of scheme I to permit both row- and column-oriented operations and to accommodate fill-ins ... 

Shacham and Kehat s direct method has several advantages over similar methods. The matrix A remains unchanged during the solution process and only row-wise scanning operations have to be carried out. The "systematic packing" storage scheme can be used and the amount of index manipulation is reduced to minimum. The problem of "fill-ins" does not exist since newly-generated elements are included either in the matrix Vf(x ) or in the vector JE(x ). ... 

If the molecular orbitals are organized such that all orbitals of a given irrep are grouped together, the matrix-based storage scheme described above takes on a particularly convenient form. " Using the C2 point group as an example, the T2 matrix of Eq. [238] may be schematically written as... 

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. 

By contrast, Fig. 9c shows an alternative scheme using linked list. In this scheme (scheme II) the information associated with a nonzero element is stored in a triplet containing the row index, the value of the nonzero element, and a pointer to the address of the next element in the same column. The starting addresses of each column are stored in another n locations. Notice that in this scheme the successive elements need not be stored in consecutive locations. To insert or delete an element requires only the change of one or two pointers no rearrangement of the list is necessary. On the other hand, the storage requirement for the same matrix is now 3 N + n and, as it stands, to find a specific nonzero element requires a linear search through the chain. 

The above sequence of steps is what enters a so called conventional MR-CI calculation. This is still a widely used method to solve the MR-CI equations and during the past decades many tricks have been developed to circumvent some of the inherent problems with this approach. These problems are mainly due to the storage of large data sets, in particular the storage of the Hamiltonian matrix elements or even worse the storage of the formula tape. Therefore, if this approach is used the MR-CI expansion has to be drastically truncated. The maximum number of configurations which can be handled is in practice 10 to 20 thousand terms. The Hamiltonian matrix will then contain on the order of a few million non-zero terms. Since an MR-CI expansion without truncation in normal applications is 10s to 107 configurations the adopted truncation scheme has to be extremely efficient if the final result should still be accurate. In the next section we will discuss an alternative approach by which it is possible to handle the non-truncated MR-CI expansion without approximations. 

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. 

The presence of /,/ and components requires an iterative solution of this equation—an approach that necessitates storage of the T3 amplitudes in each iteration This scheme is unreasonable because the number of such amplitudes would rapidly become the computational bottleneck as the size of the molecular system increased. This problem may be circumvented, however, by utilizing the so-called semicanonical molecular orbital basis in which the occupied-occupied and virtual-virtual blocks of the Fock matrix are diagonal. In this basis, the two final terms in the T3 equation above vanish, and the conventional noniterative computational procedure described earlier in the chapter may be employed. 

When information on the stability of analyte in matrix during typical conditions of storage is unavailable from the literature or from proficiency testing (PT) providers, such as the food analysis proficiency assessment scheme (PAPAS), this information shouid be developed in the laboratory as part of the method development and validation. However, a study to obtain information on stability of the analyte first requires the availability of a validated method if results are to be considered reliable. When a method is being developed to introduce a test capability for a new analyte into the laboratory, the new... 

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