Gaussian elimination with partial pivoting

The solution of linear algebraic equations by this method is based on the following steps  

202 FINITE E1.EMENT SOFTWARE - MAIN COMPONENTS Step 5 - the last equation is solved to give 

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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. 

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... 

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... 

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... 

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

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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