Pointwise Conditioning

Geometry of the image ellipse

Formulation

\kappa(x) = \frac{||A|| \cdot ||x||}{||Ax||}

||A|| = \sigma_1 = 2

||x|| = 1

The unit circle in the domain $\mathbb{R}^2$ maps to an ellipse lying inside the plane Span(A) in $\mathbb{R}^3$ .

The worst-case perturbation Aδx always aligns with the long axis of the ellipse.
When Ax is near the short axis, ||Ax|| is small, so κ(x) is large.
Pointwise conditioning depends on both the input location x and the worst-case perturbation amplification.

φ (phi)45°

Position of x on the unit circle in the domain.

Domain (R²)

x = (0.71, 0.71)

||Ax||

1.458

κ(x)

1.372

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