Question

Can you explain in detail what Gaussian Elimination with pivoting is? and how is it different from Gaussian Elimination without pivoting?

Answer 1

A system of linear equations can be placed into matrix form. Each equation becomes a row and each variable becomes a column. An additional column is added for the right hand side.

The goal when solving a system of equations is to place the augmented matrix into reduced row-echelon form, if possible.

There are three elementary row operations that you may use to accomplish placing a matrix into reduced row-echelon form.

Each of the requirements of a reduced row-echelon matrix can satisfied using the elementary row operations.

• If there is a row of all zeros, then it is at the bottom of the matrix.
  Interchange two rows of a matrix to move the row of all zeros to the bottom.
• The first non-zero element of any row is a one. That element is called the leading one also called pivot element.
  Multiply (divide) the row by a non-zero constant to make the first non-zero element into a one.
• The leading one of any row is to the right of the leading one of the previous row.
  Multiply a row by a non-zero constant and add it to another row, replacing that row. The point of this elementary row operation is to make numbers into zeros. By making the numbers under the leading ones into zero, it forces the first non-zero element of any row to be to the right of the leading one of the previous row.

All elements above and below a leading one are zero.

• Multiply a row by a non-zero constant and add it to another row, replacing that row. The point of this elementary row operation is to make numbers into zero. The difference here is that you're clearing (making zero) the elements above the leading one instead of just below the leading one.

This is Gaussian Elimination with pivoting.

Gaussian elimination without pivoting is similar to Gaussian elimination with pivoting.

In this method the leading one or pivot element is not made 1 and the elements only below the pivots are made zero not above it as in the case of Gaussian elimination with pivoting and rest of the things are same as in the above method.