Diagonal dominance

Notice that the matr ix has been arranged so that it is as nearly diagonal dominant as the data permit. 

Usually the INA is a very conservative measure of stability. Compensators are found by trial and error to reshape the INA plots so that the circles are small and the system is diagonally dominant. The INA method strives for the elimination of interaction among the loops and therefore has limited usefulness in process control where load rejection is the most important question. 

Appendix 1. Estimates of Eigenvectors for Diagonally Dominant Matrices with Diagonal Gap Condition Appendix 2. Time Separation and Averaging in Cycles... 

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

APPENDIX 1. ESTIMATES OF EIGENVECTORS FOR DIAGONALLY DOMINANT MATRICES WITH DIAGONAL GAP CONDITION... 

The Gaussian elimination can be used without pivoting because of diagonal dominance (ref. 1). Due to the many zeros the algorithm (sometimes called Thomas algorithm) is very easy to implement ... 

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. 

Both schemes are also used as SCF convergence accelerators. The DIIS scheme is particularly efficient when used in conjunction with CPCM and IEFPCM schemes, in which the diagonal dominancy of T is less prominent than in DPCM. DIIS is very efficient from the point of view of CPU times, but it requires the storage of several sets of intermediate charges. DAMP is less efficient but requires the storage of two sets of intermediate charges only. 

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). 

When p = 0.5, the method is the Crank-Nicholson implicit method. The expansion point should be taken at (i+l/2,j). The truncation error is of the order (Ax)2 plus order (Ay)2. No stability criterion comes out of the von Neumann analysis, but difficulties can come about if diagonal dominance is not kept for the tridiagonal algorithm. 

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

When A is large, a is diagonally dominant, and the method approaches that described in equation (19.31). When A is small, the method approaches that described in equation (19.31). 

Diagonal dominance and all positive coefficients ensure boundedness. Special procedures are invoked to ensure the boundedness of many higher order schemes, which otherwise, may produce wiggles and unbounded solutions. Some of these methods are discussed in the following. 

The diagonal dominance of such a modified discretized equation set is increased since the coefficient of modified equation is larger than that in Eq. (6.31) while other coefficients remain the same. This formulation has a positive effect on many iterative solution methods and is, therefore, recommended. 

In the Davidson method, one approximates A by the current iteration s eigenvalue, and H is assumed to be diagonally dominant so that 6 can be approxi-... 

Another improvement suggested by these authors and others54,56,173 is to lift the assumption of strict diagonal dominance of the Hamiltonian in the preconditioner. One selects a subspace of the most important A-electron functions... 

Khosla PK, Rubin SG (1974) A diagonally dominant second order accurate implicit scheme. J Comput Fluids 2 207-209... 

Owing to the presence of the terms D" and e , which contain the unknown C , Eq. (39) has to be solved iteratively. Diagonal dominance is guaranteed if this is done in canonical MO basis, i.e. by writing C as... 

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