Gaussian elimination partial pivoting

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. 

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. 

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. 

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. 

In Gaussian elimination with partial pivoting, when moving to column i, we first examine all elements in this column at or below the diagonal, and select the row j > i with the largest magnitude element. 

To demonstrate Gaussian elimination with partial pivoting, consider the system of equations (1.70) with the augmented matrix... 

With Gaussian elimination and partial pivoting, we have a method for solving linear systems that either finds a solution or fails under conditions in which no unique solution exists. In this section, we consider at more depth the question of when a linear system possesses a real solution (existence) and if so, whether there is exactly one (uniqueness). These questions are vitally important, for linear algebra is the basis upon which we build algorithms for solving nonlinear equations, ordinary and partial differential equations, and many other tasks. 

We now show that with some additional book-keeping, Gaussian elimination without partial pivoting returns just such an LU factorization, A = LU. We now perform Gaussian... 

Gaussian elimination (without partial pivoting) for this system was demonstrated earlier, and from (1.70)-(1.89) we obtain the factorization... 

We have seen that to make Gaussian elimination robust, we must include partial pivoting so that all kkj are finite. When the factorization is performed using Gaussian elimination with partial pivoting, the book-keeping is a bit more complex, but the result is similar We obtain lower and upper triangular matrices L and U, and a permutation matrix P, such that... 

I.A.I. Solve the following linear system by hand, using Gaussian elimination with partial pivoting, followed by backward substitution. 

It turns out the number of independent equations can also be found from the rank of the stoichiometric matrix, V/. Recall from linear algebra that the rank of a matrix is defined by the number of linearly independent rows in the matrix. It can be found using Gaussian elimination with partial pivoting or simply by using the rank(...) function in MATLAB. Once the rank is determined, we need to specify that number of independent... 

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