Approximately 1200%, because amplification applies when both and contain errors

Approximately 24%, because condition number amplifies the 0.1% measurement error by a factor of 240

Approximately 0.04%, because reduces the error

Approximately 2.4%, since for square-root scaling

Approximately 240%, since , making the computed dimensions highly unreliable

Approximately 0.1%, matching the input tolerance, because solving linear systems preserves error magnitudes

No, because Gaussian elimination is numerically unstable for large

Yes, modern CPUs have enough cache to handle matrices

Yes, Gaussian elimination is the fastest direct method and handles any size

No, because Gaussian elimination cannot process sparse matrices at all

Yes, partial pivoting reduces memory from to for large systems

No, dense storage needs TB and FLOPs would take years on one CPU

Setting achieves exactly for any matrix via perfect conditioning

Equilibration cannot reduce because condition number is invariant under diagonal scaling transformations

Equilibration increases because added diagonal matrices introduce parameters that amplify errors

Normalizing row norms via and column norms via balances matrix entries, reducing the condition number effectively

Choosing based on eigenvalues aligns the system with principal components to minimize

Setting yields with trivially

Method 2 is approximately 5 times faster since half the elimination work is reused across loads.

Method 2 is slower because storing L and U matrices requires additional memory allocation overhead.

Method 2 and Method 1 have similar performance because both require operations per solve.

Method 2 is approximately 100 times faster due to reduced floating-point operations in forward substitution.

Method 2 is approximately 1000 times faster due to the matrix dimension scaling with complexity.

Method 2 is approximately 10 times faster because LU factorization is done once for all cases.