Which of the following statements is false regarding partial pivoting?

Question

BlackTom AI · Accepted Answer

Partial pivoting is a strategy used during Gaussian elimination to select a suitable pivot in each column by swapping rows so the pivot element is large (in magnitude) and nonzero.

1) Relevant knowledge points and concepts
- Pivot: the diagonal element used to eliminate entries below it in a given column during Gaussian elimination.  
- Partial pivoting: at elimination step k, find the row i ≥ k with the largest |a_i,k| and swap that row with row k before eliminating.  
- Purpose: avoid division by zero and reduce round-off error. Partial pivoting changes the order of rows (equations), but it is not an iterative method and does not require an initial guess.

2) Step-by-step reasoning
- At step 1, inspect column 1. If a11 = 0 (or very small), find a row below with a larger |a_i1|.  
- Swap that row into the first position (this is row swapping and changes equation order).  
- Use the new pivot to eliminate entries below it by multiplying and subtracting rows.  
- Repeat for subsequent columns.

3) Simple illustrative calculation
Matrix: [[0, 2], [1, 3]]. a11 = 0 so partial pivoting swaps row1 and row2 → [[1,3],[0,2]]. Now eliminate (no division by zero), proceed to back substitution.

4) Why option (a) is false
Option (a) claims partial pivoting “uses an initial guess for the solution,” which is incorrect: partial pivoting is a direct-elimination technique applied to the coefficient matrix and does not require or use any initial guess. Initial guesses are associated with iterative solvers (e.g., Jacobi, Gauss–Seidel, Newton).

5) Why the other choices are incorrect as false statements (i.e., why they are actually true)
- (b) True: partial pivoting involves swapping rows so the pivot is nonzero (and preferably large).  
- (c) True: by swapping to a nonzero pivot, it prevents division by zero on the diagonal.  
- (d) True: swapping rows changes the order of equations (row order), though the solution set is unchanged.

Conclusion: The false statement is (a). This matches the provided correct answer.

Which of the following statements is false regarding partial pivoting?单项选择题

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