Sparse matrix defined

Hyper-Harary indices are defined for this matrix and -> Harary indices for the corresponding l -order - sparse matrix applying the Wiener operator. 

It can be observed that P is the adjacency matrix A and that for acyclic graphs each of the above defined path matrices is coincident with the corresponding sparse matrix of the distance matrix. 

These are matrices with relatively few nonzero elements. A binary sparse matrix B is a sparse matrix comprised of elements equal to zero or 1. The geodesic matrix is a binary sparse matrix " B defined as [Harary, 1969a]... 

The edge-Wiener matrix, denoted as W, is a sparse matrix, whose elements different from zero are only those corresponding to pairs of adjacent vertices (i.e., edges) this matrix can thus be considered a weighted adjacency matrix. The edge-Wiener matrix is formally defined as... 

Jacobian is not usually calculated at each iteration, and not even at every timestep. Further time is saved by using sparse matrix techniques to take advantage of the fact that the Jacobian usually possesses many zero elements (cf. equation (2.52) for example). Sparse matrix techniques are similarly used in solving equation (2.78) once the Jacobian has been found. Finally, the integration routine will seek to lengthen the timestep to the maximum extent consistent with a defined accuracy criterion, to take advantage of the strong stability properties of the implicit method. 

MATLAB has extensive sparse matrix facilities. Sparse matrices arise in many engineering problems including those of material and energy balance calculations as illustrated earlier. It is more efficient to define matrices as sparse when appropriate. If you are not sure if a matrix is sparse, then you can use the function issparse (A) which returns a value of 1 if A is sparse and 0 if it is not sparse. To define a matrix as sparse, we use the function sparse (rowpos, colpos, val, m,n) where rowpos are the positions of the nonzero row elements, colpos are the positions of the non-zero column elements, val are the values of the non-zero elements, m is the number of rows, and n the number of columns. For example, if... 

Figure 8.10 gives MATLAB m-file Band.m for implementing the band algorithm. An alternative formulation of the problem is possible in MAT-LAB. Rather than using the band algorithm directly, we can use the sparse matrix capability of MATLAB. The matrix to be inverted can be defined and stored in the sparse or band format (see Chapter 2, section 2.2.3.3). This option is particularly efficient in MATLAB. 

The number of points of the finite difference approximations to the derivatives of the pde and boundary conditions is given by the parameter m. Here, m=5 implies the use of symmetrical M3 (5) and M3(5) formulae for inner grid points and asymmetric five-point formulae close to and at the boundaries. The parameter pscan is the dimensionless scan rate for a UMDE, as defined by Eq. (12.28). The factor fa is for memory allocation of the sparse matrix solver MA2 8. The input list given above was used to simulate the voltammogram for array A4 in Fig. 12.15. 

The TN code in CHARMM uses a preconditioner from the local chemical interactions (bond length, bond angle, and dihedral-angle terms). This sparse matrix is rapid to compute and was found to be effective in practice. Other possibilities of preconditioners in general contexts have also been developed, such as a matrix derived from the BFGS update (defined in Section 6.1). ... 

The Lua language is ideally suited to implement sparse matrix techniques. A table in Lua is an associative array that is efficiently implemented in flie native language in terms of both storage allocation and access speed. A table array can be defined in the language and only the non-zero elements simply be defined. The... 

By the same operation, the unsymmetric P order sparse Cluj-distance matrix CJDu is calculated from the unsymmetric CJDu matrix. Analogously, from the Cluj-detour matrices the corresponding order sparse matrices are defined as CJA and CJAu-For the above-defined sparse Cluj matrices, the following relationships hold for any graph ... 

The augmented matrices P, P and P are obtained by adding two columns to each matrix P, P and P in the first column there is the addition of the square roots of vertex degrees 8 and in the second column the square roots of the van der Waals radii of the atoms. The corresponding sparse path matrices P, P and P of dimension Ax A are defined as the following ... 

The Harary index and hyper-Harary index, defined only for acyclic graphs, are obtained from, respectively, the P -order sparse -> reciprocal Wiener matrix W ... 

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