A team solves for the nodal displacements of a lightly damped mechanical structure by…
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics
Error, Chance and the Machine, Ch Rounding, backward error and the sensitivity of a computation
Problem
A team solves Ax=b for the nodal displacements of a lightly damped mechanical structure by Gaussian elimination with partial pivoting in binary64 arithmetic (u≈1.1×10−16). The matrix has κ2(A)=4.0×109. The computed solution x^ has residual ∥b−Ax^∥/(∥A∥∥x^∥)=3×10−16, but comparison with a known analytic solution shows the relative error ∥x^−x∥/∥x∥=2.6×10−7. The team's report concludes: "Because 2.6×10−7 is far larger than the unit roundoff 1.1×10−16, the elimination routine must be introducing extra rounding error and should be classed as numerically unstable." Find the error in this reasoning, using the residual and the condition number to say what the numbers actually show.