Nonzero element

If the symmetries of the two adiabatic functions are different at Rq, then only a nuclear coordinate of appropriate symmeti can couple the PES, according to the point group of the nuclear configuration. Thus if Q are, for example, normal coordinates, xt will only span the space of the totally symmetric nuclear coordinates, while X2 will have nonzero elements only for modes of the correct symmetry. 

A diagonal matrix has nonzero elements only on the principal diagonal and zeros elsewhere. The unit matrix is a diagonal matrix. Large matrices with small matrices symmetrically lined up along the principal diagonal are sometimes encountered in computational chemistry. 

A tridiagonal matr ix has nonzero elements only on the pr incipal diagonal and on the diagonals on either side of the pr incipal diagonal. If the diagonals on either side of the principal diagonal are the same, the matrix is a symmetr ic tr idiagonal matr ix. 

Procedure. Subtraet xl from the input matrix above. Load the resulting upper semimatrix into MOBAS. The first element is 1,1,0.5,0. Reeall that MOBAS requires 600 of only the nonzero elements in the upper semimatrix. Obtain the eigenvalues and eigenveetors. 

This has zero value unless both sets n") and are identical with the set so the matrix is a diagonal one with only one nonzero element, and its trace is obviously unity. Such a matrix is called an elementary matrix, see Chapter 7, Eq. (7-92). 

In the case of boundary conditions of the second or third kinds its order is TV + 1, while for the system (6) with the supplementary conditions (8) the order is At — 1. All the matrices of interest possess the main feature they have nonzero elements only on the three diagonals (the main and tw o adjacent ones). 

Such a triangle additive scheme will be economical once we involve economical diagonal operators a = 1,2,..., m. Economical schemes arising in practical implementations of multidimensional mathematical-physics problems turn out to be triangle additive schemes (usually lower, but sometimes upper), whose matrices are of a special structure. As a rule, nonzero elements of the matrix (C ap) stand only on one or two diagonals adjacent to the main diagonal. With this in mind, the scheme... 

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. 

Next, take the first row with a nonzero element in the first column and divide through by the leading element. If vn / 0, this will give a new first row of... 

I then show that most of the elements of the sleq array for this problem are zero. Nonzero elements are present only on the diagonal and immediately adjacent to the diagonal. The array has this property because each differential equation for temperature in a latitude band is coupled only to temperatures in the adjacent latitude bands. The subroutine SLOPER, which calculates the elements of the sleq array, can be modified so that it does not waste time calculating elements that are known in advance to be zero. Similarly, the subroutine GAUSS need not take the time to convert to zero elements that are zero already. I present suitably modified versions of both these subroutines. The new solver is a lot faster than the old one. 

The next stage of the solution consists of subtracting a multiple of the first row from the lower rows, to make the first element in each of these lower rows equal to zero. But the only nonzero elements in the first column are in the first two rows, so this manipulation need extend no further than the second row. The values in the first row are multiplied by the first element in the second row, and the resultant values are subtracted from the second row to convert the first element in the second row to zero. But... 

The next step is to divide the nonzero elements in the second row by the first nonzero element in the second row—the elements in the third and last columns. The second element in the second row is set to one. The calculation proceeds in this way from row to row with many fewer divisions and subtractions required to convert the matrix into the form with zero values below the diagonal and ones on the diagonal. When the conversion has finally been achieved, there are ones on the diagonal and nonzero values only in the last column and for the elements immediately to the right of the diagonal. 

Like the climate system described in Chapter 7, this diagenetic system consists of a chain of identical reservoirs that are coupled only to adjacent reservoirs. Elements of the sleq array are nonzero close to the diagonal only. Unnecessary work can be avoided and computational speed increased by limiting the calculation to the nonzero elements. The climate system, however, has only one dependent variable, temperature, to be calculated in each reservoir. The band of nonzero elements in the sleq array is only three elements wide, corresponding to the connection between temperatures in the reservoir being calculated and in the two adjacent reservoirs. The diagenetic system here contains two dependent variables, total dissolved carbon and calcium ions, in each reservoir. The species are coupled to one another in each reservoir by carbonate dissolution and its dependence on the saturation state. They also are coupled by diffusion to their own concentrations in adjacent reservoirs. The method of solution that I shall develop in this section can be applied to any number of interacting species in a one-dimensional chain of identical reservoirs. 

The array is nearly diagonal, with NUMSPEC nonzero elements off the diagonal FOR jrow = 1 TO nrow - 1 diag = sleq(jrow, jrow) jrns = jrow + numspec IF jrns > nrow THEN jrns = nrow jrp1 = jrow + 1 FOR jcol = jrp1 TO jrns... 

Like the climate system of Chapter 7, this system yields nonzero elements of the sleq array only close to the diagonal. Much computation can be eliminated by modifying the solution subroutines, SLOPER and GAUSS. I presented the modified subroutines SLOPERND and GAUSSND, which differ from the equivalent routines of Chapter 7 in that they can accommodate an arbitrary number of interacting species. To illustrate how the computational method can be applied to more species, I added to the system a calculation of the stable carbon isotope ratio, solving finally for the three... 

The nonzero elements are those with both indices the same and are called the diagonal elements. The sum of the diagonal elements of a square matrix is called the trace... 

Modem LP solvers can solve very large LPs very quickly and reliably on a PC or workstation. LP size is measured by several parameters (1) the number of variables n, (2) the number of constraints m, and (3) the number of nonzero entries nz in the constraint matrix A. The best measure is the number of nonzero elements nz because it directly determines the required storage and has a greater effect on computation time than n or m. For almost all LPs encountered in practice, nz is much less than mn, because each constraint involves only a few of the variables jc. The problem density 100(nz/mn) is usually less than 1%, and it almost always decreases as m and n increase. Problems with small densities are called sparse, and real world LPs are always sparse. Roughly speaking, a problem with under 1000 nonzeros is small, between 1000 and 50,000 is medium-size, and over 50,000 is large. A small problem probably has m and n in the hundreds, a medium-size problem in the low to mid thousands, and a large problem above 10,000. 

If each row of T 1 has only one nonzero element, then physically this means that in the new coordinates xc = [xr xg r], where xr is a 7-dimensional vector, the subsystem... 

If some rows of T-1 have more than one nonzero element, there are linear combinations between variables in xr and variables in xg r. Thus, the estimable portion of the system is of dimension ob less than 7 (ob < 7) and the nonestimable one is of dimension (g — ob). 

Equation (118) provides the conceptual basis for all subsequent considerations The nonzero elements of the matrices A and 0% define the new parameter space of the system, that is, the possible dynamic behavior of the system is evaluated in terms of these new parameters. Crucial to the analysis, the elements of both matrices have a well-defined and straightforward interpretation in biochemical terms, making their evaluation possible even in the face of incomplete knowledge about the detailed kinetic parameters of the involved enzymes and membrane transporters. Any further evaluation now rest on a careful interpretation of the two parameters matrices. 

To form the adjacency matrix in a simpler way, note that the elements of the first column of the adjacency matrix in Fig. 7b correspond exactly to the elements of the first column of the occurrence matrix in Fig. 6 if the output element is deleted. Similarly, the elements of column 2 of the adjacency matrix correspond exactly to the elements of the fourth column of the occurrence matrix if the output element is deleted. Therefore, if the columns of the occurrence were permuted until all of the output elements appeared on the main diagonal, as in Fig. 7a, the nonzero elements of the occurrence matrix... 

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