Tri diagonal

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... 

This subroutine assumes that the matrix is tri-diagonal diag = sleq(1, 1)... 

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... 

For implicit schemes, we will obtain a system of linear algebraic equations that must be solved. As mentioned in Example 8.1, one-dimensional diffusion problems generate tri-diagonal matrices, that can be solved for using the Thomas algorithm or other fast matrix routines. Equation (8.83) can be written as... 

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. 

For the uniform grid, the coefficients of the three nodal values involved in the interpolation become 3/8 for the downstream point, 6/8 for the first upstream node and —1/8 for the second upstream node. This scheme is more complex than CDS and it extends the computational molecule by one more node in each direction (the conventional tri-diagonal methods are, therefore, not directly applicable. See the discussion in the following subsection). The scheme has a third-order truncation error and was made popular by Leonard (1979). The transportiveness property is built into the scheme by considering two upstream and one downstream node. However, the main coefficients of the discretized equations are not guaranteed to be positive. This may lead to instability and may lead to unbounded (wiggles) solutions under certain conditions. 

The matrices derived from partial differential equations are always sparse, i.e. most of their elements are zero. For one-dimensional systems the discretization process leads to tri-diagonal systems, a system with only three non-zero coefficients per equation. Since the systems are often very large we find that iterative methods are generally much more economical than direct methods. 

For convenience in presenting the TDMA algorithm, a somewhat different nomenclature will be used [141]. Suppose the grid points were numbered 1, 2, 3,. .., N, with points 1 and N denoting the boundary points. Consider a system of equations that has a tri-diagonal form ... 

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