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key term - Pivot

Citation:

Definition

In the context of solving systems with Gaussian elimination, a pivot is the leading, non-zero entry in a row that is used as the basis for the elimination process. The pivot is the key element that allows the elimination method to systematically reduce the system of equations to an equivalent system in row echelon form.

5 Must Know Facts For Your Next Test

  1. The pivot element must be non-zero in order to perform the elimination process effectively.
  2. Choosing the appropriate pivot element is crucial for ensuring numerical stability and avoiding division by zero during the elimination steps.
  3. Pivoting, or the strategic selection of pivot elements, is often necessary when the initial system does not have a leading 1 in each row.
  4. Partial pivoting, where the row with the largest absolute value in the pivot column is chosen, is a common strategy to improve the stability of the elimination process.
  5. Successful pivoting allows the system to be transformed into row echelon form, facilitating the back-substitution process to find the solution.

Review Questions

  • Explain the role of the pivot element in the Gaussian elimination method for solving systems of linear equations.
    • The pivot element is the leading, non-zero entry in a row that is used as the basis for the elimination process in Gaussian elimination. The pivot is the key element that allows the elimination method to systematically reduce the system of equations to an equivalent system in row echelon form, where the solution can be easily obtained. The pivot element must be non-zero in order to perform the elimination process effectively, and choosing the appropriate pivot element is crucial for ensuring numerical stability and avoiding division by zero during the elimination steps.
  • Describe the concept of partial pivoting and its importance in the Gaussian elimination method.
    • Partial pivoting is a strategy used in Gaussian elimination where the row with the largest absolute value in the pivot column is chosen as the pivot row. This is done to improve the numerical stability of the elimination process and avoid potential issues with division by zero. Partial pivoting helps ensure that the pivot element is sufficiently large, which is important for maintaining the accuracy of the solution and preventing the accumulation of rounding errors during the elimination steps. By strategically selecting the pivot elements, the system can be transformed into row echelon form more reliably, facilitating the back-substitution process to find the final solution.
  • Analyze the relationship between the pivot element, row echelon form, and the solution process in Gaussian elimination.
    • The pivot element is the key to transforming the system of equations into row echelon form, which is a crucial step in the Gaussian elimination method. Successful pivoting allows the system to be reduced to an equivalent system in row echelon form, where the leading 1 in each row represents the pivot element. This row echelon form simplifies the back-substitution process used to find the solution, as the variables can be solved for one by one in a systematic manner. The pivot elements play a vital role in this transformation, as they must be non-zero and strategically chosen to ensure numerical stability and the accurate determination of the solution. The relationship between the pivot, row echelon form, and the solution process is essential for understanding and effectively applying the Gaussian elimination method.

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