Pivoting partial

The use of a uniform scale in partial pivoting can also significantly reduce round off eiTors (Gerald and Wheatley, 1984). 

HS, S, HCCU, CO3, RR NH, RR NCOO", H+, OH- and H2O. Hence there are twenty-three unknowns (m and Yj for all species except water plus x ). To solve for trie unknowns there are twenty-three independent equations Seven chemical equilibria, three mass balances, electroneutrality, the use of Equation (6) for the eleven activity coefficients and the phase equilibrium for xw. The problem is one of solving a system of nonlinear algebraic equations. Brown s method (21, 22) was used for this purpose. It is an efficient procedure, based on a partial pivoting technique, and is analogous to Gaussian elimination in linear systems of equations. 

As you learned in the previous sections, LU decomposition with built-in partial pivoting, followed by backsubstitution is a good method to solve the matrix equation Ax = b. You can use, however, considerable simpler technics if the matrix A has some special structure. In this section we assume that A is symmetric (i.e., AT = A), and positive definite (i.e., x Ax > 0 for all x 0 you will encounter the expression x Ax many times in this book, and hence we note that it is called quadratic form.) The problem considered here is special, but very important. In particular, estimating parameters in Chapter 3 you will have to invert matrices of the form A = X X many times, where X is an nxm matrix. The matrix X X is clearly symmetric, and it is positive definite if the columns of X are linearly independent. Indeed, x (x" X)x = (Xx) (Xx) > 0 for every x since it is a sum of squares. Thus (Xx) (Xx) = 0 implies Xx = 0 and also x = 0 if the columns of X are linearly independent. 

The most widely used algorithm is the systematic elimination or Gaussian elimination by partial pivoting. The success of this method is due to its stability, i.e. the algorithm produces small residuals r = Ax — b (x being the numerical solution of the system), despite round-off errors introduced by the computer during computations. The concept of stability of a numerical algorithm will be discussed in more detail in Sect. 4.5. 

A/j for the components (C//4, H O, H2, CO, CO2) at the collocation points (B.41-B.45) are found by a subroutine called FLUX. The subroutine FLUX evaluates ch4 Xcm the collocation points by solving the set of 2N linear algebraic equations (B.41-B.42) — excluding the centre of the pellet where the fluxes are known — by Gauss elimination with partial pivoting using the subroutine called GAUSL (Villadsen and Michelsen, 1978). The rest of the fluxes of the components are found from the stoichiometric equations (5.215). The roots (Uj) of the Jacobi polynomial (w) and the discretization... 

Partial pivoting not only eliminates the problem of zero on the diagonal line, it also reduces the round-off error since the pivot element (i.e., the diagonal element) is the divisor in the elimination process. To demonstrate the pivoting procedure, we use an example of three linear equations. 

ALGORITHM 2. FORWARD ELIMINATION WITH UNSCALED PARTIAL PIVOTING. 

In our previous example, partial pivoting after the first elimination step would change the working array W as follows ... 

In the process of Gaussian elimination (with, say partial pivoting), applied to a sparse system, some zero entries... 

The system of equations Is solved using Gaussian elimination with partial pivoting but since the Newton-Raphson technique requires relatively few iterations to achieve convergence (typically 15 Iterations) this does not require large quantities of CPU time. 

Gauss Elimination Method with Partial Pivoting... 

Figure 8.1 shows an M-file (gauss elim.m) for Gaussian elimination with partial pivoting. 

Figure 8.1 gauss elim M-file for calculating the unknown vector x using the Gaussian elimination method with partial pivoting. 

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