19.
Explanation
Pivoting in the solution of simultaneous linear equations is a numerical technique used to improve the numerical stability and accuracy of solving a system of linear equations, particularly when using techniques like Gaussian elimination. It involves selecting a pivot element (typically the largest absolute value in a column) as the leading element in a row to minimize the effect of rounding errors and avoid division by very small numbers, which can lead to numerical instability.
Here's an example to illustrate pivoting:
Consider the following system of linear equations:
2x+y−z=8
−3x−y+2z=−11
−2x+y+2z=−3
If we attempt to solve this system using Gaussian elimination without pivoting, the process might look like this:
Step 1: Subtract 2×2× Row 1 from Row 2 and subtract −1×−1× Row 1 from Row 3 to make the elements below the first pivot (2) become zeros. This leads to potential rounding errors and loss of precision.
Step 2: Continue Gaussian elimination to obtain the solution. However, the intermediate results may accumulate errors, and the final solution may be less accurate.
On the other hand, when we use pivoting, the process is as follows:
Step 1: Select the pivot element as the largest absolute value in the first column, which is 3 (from the second equation).
Step 2: Swap the second equation with the first equation to make the pivot element (3) the leading element in the first row:
3x+y−2z=−11
2x+y−z=8
−2x+y+2z=−3
Step 3: Continue Gaussian elimination with this new order of equations. The pivot element (3) ensures that division by small numbers is avoided, and the solution is more accurate and stable.
Pivoting is a crucial technique in numerical linear algebra to enhance the reliability and accuracy of solving systems of linear equations, especially when dealing with ill-conditioned or badly scaled systems.