6.5-08 Comparing numerical methods

Comparing numerical methods

Numerical QR methods can agree mathematically while differing in rounding behavior. These questions compare classical and modified Gram–Schmidt with Householder QR, separating conditioning of the problem from stability of the algorithm.

What you will practise

• Interpret exact versus floating-point equivalence.
• Recognize the cost of forming a Gram matrix.
• Assess both orthogonality and reconstruction errors.
• Explain why stable methods do not remove intrinsic sensitivity.

Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail.

Before you start

Review QR, normal equations and numerical near-dependence. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports choosing and evaluating numerical linear-algebra methods within Module 6 of the Complete Linear Algebra Course.