Tri diagonal matrix

Adesina has shown that it is superfluous to carry out the inversion required by Equation 5-255 at every iteration of the tri-diagonal matrix J. The vector y"is readily computed from simple operations between the tri-diagonal elements of the Jacobian matrix and the vector. The methodology can be employed for any reaction kinetics. The only requirement is that the rate expression be twice differentiable with respect to the conversion. The following reviews a second order reaction and determines the intermediate conversions for a series of CFSTRs. 

SLOPERO REM Subroutine to calculate the coefficient matrix, SLEQ for a tri-diagonal matrix... 

Third, writing the discretized equations in matrix form results in sparse matrices, often of a tri-diagonal form, which traditionally are solved by successive under- or over-relaxation methods using the tri-diagonal matrix algorithm... 

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 system (8-42) should be completed with appropriate equations resulting from the boundary conditions and it can be solved, in principle, by the same factorization method of Crout for systems with tri-diagonal matrix. 

Whereas in the case of the spatially one-dimensional diffusion, the set of difference equations features a tri-diagonal matrix, the system in Eq. (8-68) can be shown to have a block tri-diagonal matrix, which requires the use of special solving methods. 

In Appendix E we show that we can reduce the original problem of calculating the function / of matrix B to a much smaller problem of calculating the same function of the tri-diagonal matrix T,v (expression (E.39))... 

Note that the most expensive part of the numerical calculations is the determination of the matrix Q using the Lanczos method. This matrix depends only on the coefficients of the matrix and the vector fic. Therefore, due to the fact that matrix does not depend on frequency, we should apply this decomposition only once for all frequency ranges (if also vector c does not depend on frequency, which is typical for many practical problems). The calculation of the inverse of the matrix (T+iujfj,(T) is computationally a much simpler problem, because T is a tri-diagonal matrix, and jj, and diagonal matrices. As a result, one application of SLDM allows us to solve forward problems for the entire frequency range. That is why SLDM increases the speed of solution of the forward problem by an order for multifrequency data. This is the main advantage of this method over any other approach. 

SOM Second Order Moments SOR Successive Over-Relaxation SUPERBEE SUPERBEE function TDM A Tri-Diagonal Matrix Algorithm TVD Total Variation Diminishing UDS Upstream Differencing Scheme... 

It is possible to use a more accurate (second-order) derivative, at the cost of extra programming. Equations (F.28) and (F.29) represent a set of / -I- 1 algebraic equations in n -I- 1 unknowns. They are solved using linear algebra techniques. For this simple, one-dimensional problem, a special method is used for a tri-diagonal matrix. See Finlayson (1980) for complete details. 

The above equation may be applied to each of the kmax control volumes Which comprise a column of fluid having a height equal to the local film thickness. By including the no slip velocity conditions which occur at each fluid-solid interface, it is possible to express the kmax unknown Couette velocities in terms of kmax equations. This represents a system of equations which can be solved easily and efficiently by the tri-diagonal matrix algorithm (TDNA). 

A matrix of this form, that is, an upper and lower triangular quadrant for which no value is required (observed by the gray shaded area) is also known as a tri-diagonal matrix. More advanced methods of solving matrices (and in particular tri-diagonal types) are described in Burden and Faires (1997). ... 

A tri-diagonal matrix of this form is easily solved by eliminating die a. elements under the diagonal. The recursion relationships for this reduction are ... 

The complete set of coupled equations for all spatial points forms a tri-diagonal matrix of the form ... 

These equations are valid at each interior point of the spatial variable while separate equations must be specified at the boundaries of the spatial region. The FD method gives an equation for the solution at time step n +1 explicitly in terms of the solution at time step n since only one term in the first of Eq. (12.10) involves the n +1 time point. On the other hand the BD method gives a set of coupled equations that must be solved at each time step for the solution variable since the equation involves spatial points i, i -1 and i +1 at time step n +1. The resulting set of equations for the BD method forms a set of tri-diagonal matrix equations. The solution of such a set of equations has previously been covered in Section 11.5 and the techniques discussed there are applicable to solving the set of BD equations in this application. 

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