Pivot element

Pivot element in computational mathematics, particularly in algorithms that solve systems of linear equations and in those that find the determinant of a matrix, is a chosen element of the matrix at a certain stage in the calculation. The choice of a pivot element and the pivoting strategy can significantly affect the efficiency and accuracy of numerical algorithms such as the Gaussian elimination and Simplex algorithm.

Definition

A pivot element is selected from the elements of the current working matrix in algorithms that manipulate matrices. In the context of Gaussian elimination, a pivot is typically chosen to be a non-zero element from the diagonal of the matrix, around which we perform row reductions to transform the matrix into its row echelon form. The process of selecting a pivot and rearranging the rows of the matrix is known as pivoting. Pivoting strategies are crucial for reducing round-off errors and ensuring numerical stability in computations.

Pivoting Strategies

There are several strategies for selecting pivot elements, each with its advantages and disadvantages:

• Partial Pivoting: The pivot is chosen as the largest absolute value from the column of the matrix that is currently being considered. This method helps in minimizing numerical errors and is widely used in practical applications.

• Complete Pivoting: The pivot is the largest absolute value in the entire matrix that has not been fixed yet. While this method offers the highest stability, it is more computationally intensive than partial pivoting.

• Scaled Partial Pivoting: This strategy involves scaling the rows before selecting the pivot to account for the possibility that the matrix rows may have widely varying magnitudes. It strikes a balance between stability and computational efficiency.

Applications

Pivot elements play a crucial role in various numerical algorithms:

• In Gaussian elimination, pivots are used to eliminate variables and solve systems of linear equations.
• In the Simplex algorithm, used for solving linear programming problems, pivot operations are used to move from one vertex of the feasible region to another, improving the objective function at each step.
• Determining the determinant of a matrix often involves selecting pivot elements to simplify the matrix to a form where the determinant can be easily calculated.

Importance

The choice of pivot element affects the numerical stability of the algorithm. Poor pivoting strategies can lead to significant round-off errors, making the solution obtained unreliable. Moreover, efficient pivoting can reduce the computational complexity of matrix operations, making it possible to solve larger systems of equations more quickly.

Challenges

While pivoting is essential for numerical stability and efficiency, it introduces additional computational steps and complexity into matrix algorithms. Choosing the most appropriate pivoting strategy requires a balance between minimizing round-off errors and computational overhead.