Vector inverse and mean

This post is based on research papers by Anderson and Trapp, Berlinet, and the post on Reciprocal arithmetic.

The vector inverse x⁻¹ is defined as

$\mathbf{x}^{-1}=\frac{\mathbf{x}}{\|x\|^{2}} =\frac{\hat{\mathbf{x}}}{\|x\|} =\frac{\mathbf{x}}{\mathbf{x}\cdot \mathbf{x}}$

with positive norm. For a non-zero scalar k,

$(k\mathbf{x})^{-1}=\frac{k\mathbf{x}}{\|k\mathbf{x}\|^{2}}=\frac{\mathbf{\mathbf{x}}}{k\|\mathbf{x}\|^{2}}=\frac{1}{k}\mathbf{x}^{-1}$

$\|\mathbf{x}^{-1}\| = \|\mathbf{x}||^{-1}$

The reciprocal (or harmonic or parallel) sum is symbolized in various ways, but I prefer a “boxplus” to maintain its relation with addition. The reciprocal sum of vectors x and y is defined as

$\mathbf{x}\boxplus \mathbf{y}=\left [ \mathbf{x}^{-1}+\mathbf{y}^{-1} \right ]^{-1}$

if x ≠ 0, y ≠ 0, and x + y ≠ 0; otherwise the sum equals 0.

The reciprocal sum is commutative and associative, among other properties. Given two vectors with x + y ≠ 0 and non-zero scalar k,

$\newline \mathbf{x} \boxplus \mathbf{y} = \mathbf{y} \boxplus \mathbf{x} \newline \newline \mathbf{x} \boxplus (\mathbf{y} \boxplus \mathbf{z}) = (\mathbf{x} \boxplus \mathbf{y}) \boxplus \mathbf{z} \newline \newline \mathbf{x} \boxplus \mathbf{x} = \mathbf{x}/2 \newline \newline (k\mathbf{x}) \boxplus (k\mathbf{y})=k(\mathbf{x} \boxplus \mathbf{y}) \newline \newline \mathbf{x} \boxplus \mathbf{y} = \frac{\mathbf{x} \|\mathbf{y}\|^{2}+ \mathbf{y}\|\mathbf{x}\|^{2}}{\|\mathbf{x}+\mathbf{y}\|}$

The arithmetic mean for vectors x and y is

$A(\mathbf{x},\mathbf{y}) = (\mathbf{x}+\mathbf{y})/2$

The reciprocal difference of vectors x and y such that x + y ≠ 0 is defined as

$\mathbf{x} \boxminus \mathbf{y} = \mathbf{x} \boxplus (-\mathbf{y})$

The harmonic mean for vectors x and y is

$H(\mathbf{x}, \mathbf{y}) = 2(\mathbf{x} \oplus \mathbf{y})$

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