CS223 · Annotated SlidesChapter 6 — Least-Squares Problems
Chapter 6 · Annotated slides

Least-Squares Problems

Least-squares solutions of inconsistent systems, the normal equations AᵀAx̂ = Aᵀb, uniqueness when the columns are independent, the least-squares error, and solving with QR.

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    1. Least-squares solution x̂ minimizes ‖b − Ax‖. Solve the normal equations AᵀAx̂ = Aᵀb. Unique ⇔ A’s columns are independent ⇔ AᵀA invertible: x̂ = (AᵀA)⁻¹Aᵀb.
    TipLeast-squares error = ‖b − Ax̂‖.

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    MCQ

    The least-squares solutions of Ax = b are the solutions of:

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          1. With A = QR: Rx̂ = Qᵀb (more stable numerically).
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