Tridiagonal systems of equations

Another special case of the matrix equation Ax=b is the one with A beeing tridiagonal, i.e., having nonzero elements only on the diagonal plus or minus one column. For example, the equations 

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  

1700 REt 1702 1 LINEAR EQUATIONS KITH TRIDIA6CNAL MATRIX t 1704 PEN llltllllltIHIIItlllimillllltlllintlllimitlltl 

The 3 nonzero entries of the i-th row of the coefficient matrix occupy the variables A(I), B(I) and C(I), so that A(l) and C(N) are not used. We need an auxiliary vector P() of N elements. Since there is no pivoting, you may experience overflow (i.e., division by a too small pivot) even for a nonsingular matrix, if it is not diagonally dominant. 

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. 

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The sets of equations can he solved using the Newton-Raphson method. The first form of the derivative gives a tridiagonal system of equations, and the standard routines for solving tridiagonal equations suffice. For the other two options, some manipulation is necessary to put them into a tridiagonal form (see Ref. 105). 

Coefficients a, a2 and b are obtained by the Fourier analysis and the relatively rapid solution of the resulting tridiagonal system of equations, due to the implicit nature of (2.31). A typical set is a = 22, a% = 1, and b = 24. To comprehend their function, let us observe Figure 2.2 that assumes the computation of dHy/dx and dEz/dx at i = 0. For the first case, constraint Ey = Ez = Hx = 0 at i = 0 indicates that dHy/dx (likewise for all H derivatives) must also be zero. In the second case, to calculate dEz/dx at i = one needs its values at i = —, . Nonetheless, point i = — is outside the domain and to find a reliable value for the tridiagonal matrix, the explicit, sixth-order central-difference scheme is selected... 

Note that the first of the two equations has terms in both variables, Ca and Cb- This is not a problem when using an explicit method, but these are practically useless when hers are involved and implicit methods are normally chosen for their greater efficiency. However, while these produce tridiagonal systems of equations for single species or uncoupled multireactions, they produce more complicated systems for coupled reaction systems, which are not amenable to the usual solution techniques. This problem was solved recently by Rudolph [7] and references therein, who introduced the technique known as hlock-tridiagonal solution into electrochemical simulations, (see Ref. [2], 3rd edn. only) and Sect. 1.3.9. 

This rearranges, for all j, into a tridiagonal system of equations for... 

Equation 8.17 rearranges to a tridiagonal system of equations which may be solved very conveniently using the Thomas algorithm (Aziz and Settari, 1979). 

Note that the unknowns are at level j + 1, so all terms on the right-hand side are known. When the finite difference equation is applied for all /, a tridiagonal system of equations results. Such systems can be solved very efficiently (as compared to full-matrix linear systems) using a method called the Thomas algorithm (see below). The Matrix.xla function SYSLINT can also be used. 

Now consider the solution of the resulting tridiagonal system of equations as given by Eq. (11.37) or the matrix form as in Eq. (11.38). One of flie simplest methods of solution is to assume that there exist some functions e,. and such that ... 

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