Without pivoting, Gaussian elimination is applied to a system where one step produces a very small pivot ε ≈ 0.0001. The multiplier for the next row is approximately 10,000. What is the danger?
AThe algorithm will fail to find a solution because the system is ill-conditioned
BFloating-point rounding errors in that row are amplified by 10,000 before being subtracted, potentially ruining the result
CThe pivot ε causes division by zero, halting the computation
Small pivots don't cause division by zero — they produce huge multipliers. When that large multiplier (10,000) is applied to a row with even tiny floating-point errors, those errors are scaled up by a factor of 10,000. The cumulative effect can render the numerical answer completely wrong even for a well-conditioned system. Option A is wrong because the system's condition number is a property of the matrix, independent of pivot order — a well-conditioned system can still be destroyed by poor pivot choice.
Question 2 Multiple Choice
Partial pivoting guarantees that all multipliers |mᵢₖ| satisfy what condition, and why does this matter?
A|mᵢₖ| ≤ 1, which ensures rounding errors are not amplified as they propagate through subsequent row operations
B|mᵢₖ| ≥ 1, which ensures the pivot rows dominate and the algorithm converges
C|mᵢₖ| = 1 exactly, which balances all rows and eliminates rounding error
D|mᵢₖ| < n, where n is the matrix size — keeping multipliers below the matrix dimension
Partial pivoting selects the row with the largest absolute value in the current column as the pivot. By placing the maximum value in the denominator of each multiplier mᵢₖ = aᵢₖ / aₖₖ, it guarantees |mᵢₖ| ≤ 1 for all entries below the pivot. When multipliers are at most 1, rounding errors are not amplified — they can only stay the same or shrink as they propagate. This is the mechanism by which partial pivoting achieves numerical stability.
Question 3 True / False
A system of linear equations is well-conditioned (small condition number), but Gaussian elimination without pivoting produces a wildly inaccurate answer. This can happen.
TTrue
FFalse
Answer: True
This is the crucial point of the topic. The condition number measures sensitivity of the solution to perturbations in the data — a well-conditioned system has a unique, stable solution. But Gaussian elimination without pivoting can still fail numerically by encountering a small pivot, amplifying floating-point rounding errors catastrophically even though the mathematical problem is perfectly well-posed. Condition number and pivot behavior are separate issues. Pivoting addresses the numerical algorithm's stability, not the problem's inherent sensitivity.
Question 4 True / False
Complete pivoting is typically preferred over partial pivoting in practice because it provides a stronger stability guarantee.
TTrue
FFalse
Answer: False
Despite its stronger theoretical guarantee, complete pivoting is rarely used in practice. It requires searching the entire remaining submatrix (O(n²) entries per step) rather than just one column (O(n) entries), and it also permutes column order, complicating the recovery of the solution. For virtually all problems in scientific computing, partial pivoting provides sufficient stability at much lower overhead. The better guarantee of complete pivoting does not justify its cost in practice — partial pivoting is the standard choice.
Question 5 Short Answer
Why does reordering the rows of a linear system (as partial pivoting does) produce the same mathematical solution but better numerical results?
Think about your answer, then reveal below.
Model answer: Row swapping is a valid elementary row operation that doesn't change the solution set of the system — the equations are the same, just reordered. Numerically, however, order determines which value becomes the pivot at each step. By choosing the largest available entry as the pivot, partial pivoting ensures multipliers stay ≤ 1, preventing rounding errors from being amplified. The mathematics is unchanged; only the numerical behavior improves.
The key insight is that pivoting is a strategy for the algorithm's numerical behavior, not a change to the mathematical problem. Swapping row 3 and row 7 doesn't alter the set of equations or their solution — it merely changes the sequence of operations. Since floating-point arithmetic is the source of error, and since the magnitude of multipliers determines how errors propagate, choosing large pivots controls error growth without touching the underlying mathematics.