Problem 1
Leading digits cancel, magnifying relative error in the remaining difference.
Faculty use
Floating-Point Arithmetic and Error Propagation
Reward correct mathematics, explicit dimensions/hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
The computed result is the exact solution to a nearby problem, with the perturbation controlled.
Problem 3
Definitions should identify the vector/matrix/space/transformation, dimensions or domain/codomain, and the hypotheses behind distinguishing representation error, rounding, truncation, and instability. Use the formulas fl(x op y)=(x op y)(1+δ); relative error=‖x−x̂‖/‖x‖.
Problem 4
A strong error analysis identifies the first step that violates a definition, dimension, theorem hypothesis, or matrix-operation rule; then repairs that step and re-verifies the final result.
Problem 5
Application responses vary. Require defined variables, a compatible matrix/vector model connected to large systems, reproducible computation, research communication, ethical data modelling, and advanced mathematics, a complete computation, interpretation, and limitations.
Problem 6
Accept two independent checks appropriate to the task: reconstruction, residual, RREF comparison, determinant/rank test, orthogonality, substitution, eigenpair check Av=λv, factorization product, or a second method.
Scoring guidance
Six problems × 10 points: 5 mathematical accuracy, 2 structure/justification, 2 verification/interpretation, 1 notation and communication.