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SquaredEuclideanDistance

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

SquaredEuclideanDistance[u, v]

Gives Sum Abs[u_i - v_i]^2, the squared Euclidean distance. Rational for rational input, and monotone in EuclideanDistance, so ranking on it orders points identically without taking a root.

Examples (6)

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (6)

In[1]:= EuclideanDistance[{1, 2}, {4, 6}]
Out[1]= 5

In[2]:= SquaredEuclideanDistance[{1/3, 0}, {0, 1/7}]
Out[2]= 58/441

In[3]:= ManhattanDistance[{1, 2}, {4, 6}]
Out[3]= 7

In[4]:= EuclideanDistance[{0, 0}, {1, 1}]
Out[4]= Sqrt[2]

In[5]:= CosineDistance[{1, 0}, {0, 1}]
Out[5]= 1

In[6]:= CosineDistance[{1, 0}, {-1, 0}]
Out[6]= 2

Implementation notes

  • Protected. Not Listable: threading over a List argument is exactly what these must not do, because the list is the point.
  • Exact input gives an exact result where the value is rational. SquaredEuclideanDistance[{1, 2}, {4, 6}] is 25, not 25., and SquaredEuclideanDistance[{1/3, 0}, {0, 1/7}] is 58/441. Squared Euclidean is monotone in Euclidean, so ranking on it orders points identically without introducing a root -- which is how FindClusters stays exact in n dimensions.
  • Complex components contribute their modulus, because the definition takes Abs before squaring rather than squaring the difference. This matters only for complex input, where the two orders differ, and follows Mathematica.
  • Symbolic input survives rather than being rejected: ManhattanDistance[{a}, {b}] is Abs[a - b], as in Mathematica.
  • CosineDistance ranges over [0, 2] -- 0 parallel, 1 orthogonal, 2 antiparallel -- and ignores magnitude. It is not a metric (it violates the triangle inequality) and has no squared form that ranks identically, so it is used directly. A zero vector on either side gives 0, following Mathematica; that is a convention, not a derivation, since the quotient is 0/0.
  • Mismatched lengths, or an argument that is a matrix, leave the call unevaluated.

Attributes: Protected.

References

See also: EuclideanDistance, ManhattanDistance, CosineDistance, List, FindClusters, Abs