qit.utils.majorize

qit.utils.majorize(x: array[float], y: array[float], tol: float = 1e-08) → bool

Majorization partial order of real vectors.

Returns True iff x is majorized by y, denoted by $x⪯y$. This is equivalent to

$$\sum _{k=1}^n{x}_k^{\downarrow }\le \sum _{k=1}^n{y}_k^{\downarrow }\text{for all}n\in \{1,2,\dots ,d\},$$

where $x^{\downarrow }$ is the vector x with the elements sorted in nonincreasing order.

$x⪯y$ if and only if x is in the convex hull of all the coordinate permutations of y.

Parameters

• x (array[float]) – real vectors, equal length

• y (array[float]) – real vectors, equal length

• tol (float) – numerical tolerance

Returns

$x⪯y$

Return type

bool