Stored-matrix algorithm

Figure 1. Stored matrix algorithm for hierarchic agglomerative clustering methods.

DDAPLUS will perform better if it is told to use band-matrix algorithms w henever the lower and upper bandwddths of G satisfy 2ml - - mu < Nstvar. The matrix G (and 1 if nondiagonal) wall then be stored more... 

Evaluate the integrals. In a conventional algorithm, they are stored on disk and read in for each iteration. In a direct algorithm, integrals are computed a few at a time as the Fock matrix is formed. 

The resulting weights Wj and scores tj are stored as columns in the matrices W and T, respectively. Note that the matrix W differs now from the previous algorithms because the weights are directly related to X and not to the deflated matrices. Step 2 accounts for the orthogonality constraint of the scores tj to all previous... 

Thus, generally, two matrix transformation algorithms are required, one for B stored triangularly (Ti=r2) and one for B stored rectangularly The transformation could be written as a double sum... 

Fig. 4.4 The match search algorithm creates a matrix with one cell for each pair ofdirected tree edges. The cell stores the overall similarity of the two subtrees. The similarity value is calculated with a dynamic programming scheme shown on the right. First, an extension match (blue ellipsoid) is searched. Then the subtrees are cut and matched in all possible combinations. For each combination, a similarity value can be extracted from the matrix (exemplarily shown by the blue arrows). A maximum-weight bipartite matching solves the assignment of the subtrees.

To use computer storage more efficiently, the vector of unknown temperatures will eventually be stored in the global force vector, f. The next steps in the finite element procedure (Table 9.1) will be to form the global stiffness matrix and force vector, and to solve the resulting linear system of algebraic equations, as presented in Algorithm 5. 

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. 

Fig. 30. Schematic design of a simple but very useful and efficient data reduction algorithm. Data representing the time trajectory of an individual variable are only kept (= recorded, stored) when the value leaves a permissive window which is centered around the last stored value. If this happens, the new value is appended to the data matrix and the window is re-centered around this value. This creates a two-column matrix for each individual variable with the typical time stamps in the first column and the measured (or calculated) values in the second column. In addition, the window width must be stored since it is typical for an individual variable. This algorithm assures that no storage space is wasted whenever the variable behaves as a parameter (i.e. does not change significantly with time, is almost constant) but also assures that any rapid and/or singular dynamic behavior is fully documented. No important information is then lost...

The iterative algorithms that have been proposed for the reconstruction of structures from insufficient information differ from all other methods because they perform in parallel two distinct reconstructions one for the structure matrix, and one for the so-called memory matrix, i.e. for a matrix where any convenient feature can be stored. This is why these algorithms are collectively referred to as the Memory Reconstruction Method (MRM). 

The basic idea in these methods is building up curvature information progressively. At each step of the algorithm, the current approximation to the Hessian (or inverse Hessian, as we shall see) is updated by using new gradient information. The updated matrix itself is not necessarily stored explicitly, as the updating procedure may be defined compactly in terms of a small set of stored vectors. This economizes memory requirements considerably and increases the appeal to large-scale applications. 

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