Diagonal system

The canon () function by default will return the diagonalized system, and in thise case, in the system object sd. For example, we should find sd. a to be identical to the matrix l that we obtained a few steps back. 

Now I shall show how the nearly diagonal system can easily be modified to incorporate additional interacting species. In this illustration I shall add the calculation of the stable carbon isotope ratio specified by 813C. All of the parameters that affect the concentrations of carbon and calcium are left as in program SEDS03, so that the concentrations remain those that were plotted in Section 8.4. I shall not repeat the plots of the concentrations but present just the results for the isotope ratio. 

The flux problem can now be easily solved in the diagonalized system using Eq. 6.31 the solution can then be transformed back to the original concentration coordinates by using the inverse relationships... 

Solution. Assume solutions in the diagonalized system having the standard point-source form... 

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. 

The equation systems commonly found in the simulation of electrochemical problems have a diagonal structure with a bandwidth 2n+l such that a j = 0 for j > i + n and for i > j n. By particularising the LU decomposition for diagonal systems [extended Thomas algorithm) the above expressions turn into... 

Note that we obtain two equations of the same type as Eq. (9.248) for our coupled system of differential equations, which can be combined into a single hepta-diagonal matrix equation that further allows us to determine the two radial functions simultaneously To derive this hepta-diagonal system we apply Eq. (G.25) (see appendix G) to the transformed coupled Eqs. (9.227) and (9.228) and arrange all 2 x n discretized equations into a matrix equation. The first... 

A fj,] value of zero indicates a completely diagonal system while values tending toward one indicate a lack of diagonal dominance in the system and, therefore, a strongly coupled system. 

The matrix of this system is called the three-diagonal matrix. For solving three-diagonal system. 

Separation factors greater than rmity result in x-y curves that lie entirely above the 45 diagonal. The higher the value of a, the greater the distance between the two lines. This implies that separation becomes easier as tiie x-y curve bulges out and away from the diagonal. Systems with x-y curves close to the diagonal are by contrast difficult to separate. 

The governing equations are discretized by using the finite difference method. The Reynolds equation solution leads to solving a tri-diagonal system of linear equations. Using the semi-implicit scheme of Crank-Nicholson solves the energy equations in the film and in the rings. 

Certain engineering problems yield sets of simultaneous linear algebraic equations that are predominantly diagonal systems. A predominantly diagonal system of linear equations has coefficients on the diagonal that are larger in absolute value than the sum of the absolute values of the other coefficients. For example, the set of equations ... 

Example 2.3 Solution of Chemical Reaction and Material Balance Equations Using the Jacobi Iteration for Predominantly Diagonal Systems of Linear Algebraic Equations. 

Eq. (6.34) and the appropriate boundary conditions constitute a set of linear algebraic equations, so the Gauss methods for the solution of such equations may be used. Eq. (6.34) is actually a predominantly diagonal system therefore, the Gauss-Seidel method (see Sec. 2.7) is especially suitable for the solution of this problem. Rearranging Eq. (6.34) to solve for m,- ... 

---