Matrix diagonal dominance

In general, no simple, consistent set of analytical expressions for the resonance condition of all intradoublet transitions and all possible rhombicities can be derived with the perturbation theory for these systems. Therefore, the rather different approach is taken to numerically compute all effective g-values using quantum mechanics and matrix diagonalization techniques (Chapters 7-9) and to tabulate the results in the form of graphs of geff,s versus the rhombicity r = E/D. This is a useful approach because it turns out that if the zero-field interaction is sufficiently dominant over... 

The initial matrix K is diagonally dominant in columns, but its rows can include elements that are much bigger than the correspondent diagonal elements. 

The matrix in (1.59) is diagonally dominant, and we can use module M17 to solve the equation. As in example 1.3.3, we separate the coefficients from the right hand side by the character "=" in each DATA line. 

From the standpoint of using multicomponent diffusion in a numerical simulation, it can be beneficial to pose the multicomponent diffusion in terms of an equivalent Fickian diffusion process [72,422]. To do this, imagine that a new mixture diffusion coefficient can be defined such that the first term (summation) in Eq. 12.166 can be replaced with the right-hand side of Eq. 12.162. An advantage of the latter is that the diffusion of the fcth species depends on its own mole fraction gradient, rather than on the gradients of all the other species the Jacobian matrix is more diagonally dominant, which can sometimes facilitate numerical solution. 

If this condition is fulfilled (this occurs for a well tessellated surface) the charges obtained are fully reliable, as a strictly diagonal dominated matrix is not singular [30], If there are pairs of very close tesserae (for example tessera i and j), a simple safety measure is to annihilate the corresponding diagonal elements, Ty and T, . Note that the methods based on tesserae weights are implicitly not affected by this problem. 

K. Briggs, Diagonally Dominant Matrix, From MathWorld, A Wolfram Web Resource, created by E. W. Weisstein, http //mathworld.wolfram.com/DiagonallyDominantMatrix.html... 

To make ( (s) diagonally dominant, it is necessary to select a specific type of controller. Rosenbrock has proposed that K(s) K.qKh(s)Kr (s), i.e., a product of three controller matrices. Ka is a permutation matrix, which scales the elements in G(s)K(s) and makes some preliminary assignment of single loop connections, usually to assure integrity. This step can be used to make (G(s)K(s)diagonally dominant as s 0. K s) can be chosen to meet stability criteria. Finally the elements of Kc(s), a diagonal matrix, can be selected to improve performance of the system. The proper selection of and Kt>(s) are the most difficult parts of the design process, and this step should be considered iterative, especially for an inexperienced user. 

It is apparent from the first and last rows of this matrix, that again the simple Dirichlet boundary conditions, Eq. (8-3), have been considered. Since X > 0, the matrix A is positive definite and diagonally dominant. For solving system (8-28), the very efficient Crout factorization method for linear systems with tri-diagonal matrix can be applied (see Press et al. 1986, Section 2.4). 

The Thomas algorithm will always converge if the tridiagonal matrix is diagonally dominant. In other words, the matrix is such that... 

As in our last illustration, this matrix is dominated by the diagonal elements. Since the mixture is almost pure nitrogen with only traces of the condensable vapors remaining we may let the mole fraction of nitrogen approach unity, 1, and the mole fractions of... 

The (E) matrix was found to be diagonally dominant so that following the eigenvalues did not prove at all difficult for FH2 and FD2. 

A diagonally dominant matrix is a matrix such that the absolute value of the diagonal term is larger than the sum of the absolute values of other elements in the same row, with the diagonal term larger than the corresponding sum for at least one row that is. 

This condition of diagonal dominant matrix is required in the solution of a set of linear equations using iterative methods, details of which are seen in Section B.6. 

There are a number of methods available to solve for the solution of a given set of linear algebraic equations. One class is the direct method (i.e., requires no iteration) and the other is the iterative method, which requires iteration as the name indicates. For the second class of method, an initial guess must be provided. We will first discuss the direct methods in Section B.5 and the iterative methods will be dealt with in Section B.6. The iterative methods are preferable when the number of equations to be solved is large, the coefficient matrix is sparse and the matrix is diagonally dominant (Eqs. B.8 and B.9). 

The presented algorithm works if and only if the pivot entry k, k) is zero in none step k,k =, ..., n-, or, equivalently, if and only if for none fc the (n - k) x ( - k) principal submatrix of the coefficient matrix A is singular. (Apx p submatrix of A is said to be the principal if it is formed by the first p rows and by the first p columns of A.) That assumption always holds (so the validity of the above algorithm is assured) in the two important cases where A is row or column diagonally dominant and where A is Hermitian (or real symmetric) positive definite (compare the list of special matrices in Section II.D). (For example, the system derived in Section II.A is simultaneously row and column diagonally dominant and real symmetric multiplication of all the inputs by -1 makes that system also positive definite. The product V V for a nonsingular matrix y is a Hermitian positive definite matrix, which is real symmetric if V is real.)... 

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