Inverting tridiagonal matrices

Table 6-1. Thomas algorithm for inverting tridiagonal matrices...

A submatrix is formed at each point on the grid, relating the concentration of each species to the others (kinetically). The material balance equation for all the species may be written with this submatrix down the diagonal - resulting in a block tridiagonal matrix. This may be solved using a matrix version of the Thomas algorithm which requires each submatrix to be inverted (by LU factorization). 

Invert the tridiagonal matrix either with a spreadsheet or the Thomas algorithm (shown here). The Thomas parameters for each conponent (see Table 6-2 for n-C4 solution) are ... 

Combining Eqs. tl2-45). ( ), and 112-47) results in a tridiagonal matrix, Eq. (6-131 with all terms defined in Eqs. (12-461 ( ), and (12-48). There is one matrix for each conponent. These tridiagonal matrices can each be inverted with the Thomas algorithm (Table 6-11 The results are liquid conponent... 

D35. For Example 13-4 set up the tridiagonal matrix for the mass balances assuming there are 6 stages in the column and the exit raffinate stream concentration is unknowm Develop the values for A, B, C, and D in only the acetic acid matrix using K values determined from Table 13-5. Do not invert the matrix. 

The system matrix of equation (6.108) contains the two diagonal blocks /3X and j3y and the two offdiagonal blocks Ax and Ay which are both banded tridiagonal. Rather than inverting a tridiagonal block as naively suggested earlier, it is much less costly to multiply the two sides of the block matrix equation (6.108) from the left by the nonsingular block matrix... 

For the important case in which the matrices Bij = 0 for jf > i, it follows that the matrix (7 — (/ tj2)Q) is a block triangular matrix, and in the case of one space variable where the diagonal blocks are tridiagonal, (7 — (A /2)Q) can be directly inverted. For two or more space variables, the process of solving (7.5) for large numbers of mesh points would involve again inner iterations. 

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