BIZNIQUE / LINEAR ALGEBRA / 29
Compute distances without losing their meaning
Compute distances without losing their meaning
Use an algebraic identity to organise repeated Euclidean comparisons.
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THE IDEA
Read the example.
Read the example.
The identity expresses squared distance through norms and a dot product. For many pairs, these terms can be organised as array operations and norms can be reused. The example compares a point with the origin.
‖x−y‖₂² = ‖x‖₂² + ‖y‖₂² − 2xᵀy
Use it well
Use it well
Roundoff can produce tiny negative values when subtracting nearly equal floating-point quantities. A numerical safeguard is different from accepting a materially invalid distance.
TRY IT YOURSELF
A quick check.
A quick check.
What is the distance between (0, 0) and (3, 4)?
Check your reasoning
Check your reasoning
The squared distance is 25, so the Euclidean distance is 5.
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