How do you use pivoting and scaling techniques to improve the accuracy and stability of Gaussian elimination?

1 What is pivoting?

Pivoting is a technique that involves swapping rows or columns of a matrix to avoid dividing by a small or zero pivot element. A pivot element is the diagonal entry of a matrix that is used to eliminate the entries below it. If the pivot element is too small, the elimination process can introduce large errors or even fail. Pivoting can also help to reduce the growth of round-off errors and improve the conditioning of the matrix.

2 What is scaling?

Scaling is a technique that involves multiplying each row or column of a matrix by a factor to make the entries more balanced and comparable. Scaling can help to avoid overflow or underflow of numbers and reduce the sensitivity of the matrix to perturbations. Scaling can be done before or after pivoting, depending on the criteria and the algorithm.

3 How to perform partial pivoting?

Partial pivoting is a type of pivoting that swaps rows of a matrix to make the pivot element the largest in magnitude in its column. This can help avoid division by zero and minimize error propagation. To perform partial pivoting, you need to locate the row with the largest absolute value in each column below the diagonal. If the largest value is not on the diagonal, you must swap the rows to put it there. Then, perform Gaussian elimination on the pivoted matrix.

4 How to perform complete pivoting?

Complete pivoting is a type of pivoting that swaps both rows and columns of a matrix to make the pivot element the largest in magnitude in the whole matrix. This technique can reduce the error growth and improve the stability of the solution. To perform complete pivoting, you must first search for the row and column with the largest absolute value in the submatrix below and to the right of the diagonal. If it’s not on the diagonal, swap the rows and columns to put it there. Then, carry out Gaussian elimination on the pivoted matrix, keeping track of any row or column permutations.

5 How to perform scaling by equilibration?

Scaling by equilibration is a type of scaling that multiplies each row and column of a matrix by a factor to make the maximum absolute value in each row and column equal to one. This technique can help improve the conditioning of the matrix and reduce the effect of large or small entries. To perform scaling by equilibration, you must first find the maximum absolute value in each row and column. Then, divide each row and column by its maximum value, resulting in a scaled matrix with entries between -1 and 1. Finally, you can perform the desired pivoting and Gaussian elimination on the scaled matrix while keeping track of the scaling factors.

6 How to perform scaling by norms?

Scaling by norms is a type of scaling that multiplies each row and column of a matrix by a factor to make the norm of each row and column equal to one. This norm is a measure of the size or length of a vector, and there are different types, such as the 1-norm, the 2-norm, or the infinity-norm. Scaling by norms can help to balance the relative importance of different rows and columns and reduce error amplification. To do this, you must first find the norm of each row or column using the same type of norm for both rows and columns. Then divide each row and column by its norm, resulting in a scaled matrix with unit norms for each row and column. Finally, perform the desired pivoting and Gaussian elimination on the scaled matrix while keeping track of the scaling factors.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?