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Pivoting.One of the ways around this problem is to ensure that small values (especially zeros) do not appear on the diagonal and, if they do, to remove them by rearranging the matrix and vectors.

Partial pivoting.If zero element is found in diagonal position i.e., aij for i = j which is called pivot element interchange the corresponding elements of two rows such that new diagonal element i if non-zero and having maximum value in that corresponding column. The process can be explained in following steps. In the first step the largest coefficient of x1 (may be positive or negative) is selected from all the equations. Now we interchange the first

equation with the equation having largest coefficient of xi. In the second step, the numerically largest coefficient of x2 is selected from the remaining equations. In this step we will not consider the first equations now interchange the second equation with the equation having largest coefficient of y. We continue this process till last equation. This procedure is known as partial pivoting. In general, the rearrangement of equation is done even if pivot element is non-zero to improve the accuracy of solution by reducing the round off errors involved in elimination process, by getting a larger determinant, which is done by finding a largest element of the row as the pivotal element.

Complete Pivoting.If the order of elimination of x1, x2, x3, ........ is not important, then we may choose at each stage the largest coefficient of the whole matrix of coefficients. We may search the largest value, not only in rows but also in columns. After searching largest value, we bring at the diagonal position. This method of elimination is known as complete pivoting.

The superiority of this method is that it gives the solution of a system, provided its determinant does not vanish in finite number of steps.

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