VectorLess
✖
VectorLess
Details
- VectorLess gives a partial ordering of vectors, matrices and arrays that is compatible with vector space operations, so that and imply for all .
- VectorLess is typically used to specify vector inequalities for constrained optimization, inequality solving and integration.
- When x and y are -vectors, xy is equivalent to . That is, each part of x is less than the corresponding part of y for the relation to be true.
- When x and y are dimension arrays, xy is equivalent to . That is, each part of x is less than the corresponding part of y for the relation to be true.
- xy remains unevaluated if x or y has non-numeric elements, typically gives True or False otherwise.
- When x is an n-vector and y is a scalar, xy yields True if xi<y for all components .
- By using the character , entered as v< or \[VectorLess], with subscripts vector inequalities can be entered as follows:
-
VectorLess[{x,y}] the standard vector inequality VectorLess[{x,y},κ] vector inequality defined by a cone κ - In general, one can use a proper convex cone κ to specify a vector inequality. The set is the same as κ.
- Possible cone specifications κ in for vectors x include:
-
{"NonNegativeCone", n} such that {"NormCone", n} such that Norm[{x1,…,xn-1}]<xn "ExponentialCone" such that "DualExponentialCone" such that {"PowerCone",α} such that {"DualPowerCone",α} such that - Possible cone specifications κ in for matrices x include:
-
"NonNegativeCone" such that {"SemidefiniteCone", n} symmetric positive definite matrices - Possible cone specifications κ in for arrays x include:
-
"NonNegativeCone" such that - For exact numeric quantities, VectorLess internally uses numerical approximations to establish numerical ordering. This process can be affected by the setting of the global variable $MaxExtraPrecision.
Examples
open allclose allBasic Examples (3)Summary of the most common use cases
x<y yields True when xi < yi is True for all i=1,…,n:
https://wolfram.com/xid/0tz9k9o2-bm4a60
x<y yields False when xi ≥ yi for any i=1,…,n:
https://wolfram.com/xid/0tz9k9o2-bx19bo
Represent a vector inequality:
https://wolfram.com/xid/0tz9k9o2-j6uv96
When v is replaced by numerical vector space elements, the inequality gives True or False:
https://wolfram.com/xid/0tz9k9o2-esdvy6
https://wolfram.com/xid/0tz9k9o2-g1838z
https://wolfram.com/xid/0tz9k9o2-lqink
https://wolfram.com/xid/0tz9k9o2-dog27u
https://wolfram.com/xid/0tz9k9o2-dlrjpq
Scope (7)Survey of the scope of standard use cases
Determine if all of the elements in a vector are strictly positive:
https://wolfram.com/xid/0tz9k9o2-egz77b
https://wolfram.com/xid/0tz9k9o2-dsvgs7
Determine if all components are strictly less than 1:
https://wolfram.com/xid/0tz9k9o2-f21cxf
https://wolfram.com/xid/0tz9k9o2-cwq9cq
https://wolfram.com/xid/0tz9k9o2-oso6qe
https://wolfram.com/xid/0tz9k9o2-cu0xkj
For each component, !xi<yi does imply xi≥yi:
https://wolfram.com/xid/0tz9k9o2-d7s40k
Compare the components of two matrices:
https://wolfram.com/xid/0tz9k9o2-bsdwb5
https://wolfram.com/xid/0tz9k9o2-do2gvh
https://wolfram.com/xid/0tz9k9o2-g9emso
https://wolfram.com/xid/0tz9k9o2-g63km1
https://wolfram.com/xid/0tz9k9o2-y63je
https://wolfram.com/xid/0tz9k9o2-kgkn7p
https://wolfram.com/xid/0tz9k9o2-j308dt
Represent the condition that Norm[{x,y}]<1:
https://wolfram.com/xid/0tz9k9o2-vvt94
https://wolfram.com/xid/0tz9k9o2-b9zhpw
Represent the condition that :
https://wolfram.com/xid/0tz9k9o2-bvbxar
https://wolfram.com/xid/0tz9k9o2-mhpo9n
Show the boundary where for non-negative x,y with α between 0 and 1:
https://wolfram.com/xid/0tz9k9o2-h8dfui
Applications (1)Sample problems that can be solved with this function
VectorLess is a fast way to compare many elements:
https://wolfram.com/xid/0tz9k9o2-bb90pl
https://wolfram.com/xid/0tz9k9o2-kpxflt
https://wolfram.com/xid/0tz9k9o2-fbt5y9
Properties & Relations (3)Properties of the function, and connections to other functions
VectorLess is compatible with a vector space operation:
https://wolfram.com/xid/0tz9k9o2-djby60
Adding vectors to both sides of for any vector :
https://wolfram.com/xid/0tz9k9o2-cbatcv
https://wolfram.com/xid/0tz9k9o2-c76en
https://wolfram.com/xid/0tz9k9o2-fidirk
Multiplying by positive constants for any :
https://wolfram.com/xid/0tz9k9o2-8csn5
https://wolfram.com/xid/0tz9k9o2-gx7g3f
https://wolfram.com/xid/0tz9k9o2-coe8bo
xy is a (strict) partial order, i.e. irreflexive, asymmetric and transitive:
https://wolfram.com/xid/0tz9k9o2-dxz3jr
Irreflexive, i.e. for all elements so no element is related to itself:
https://wolfram.com/xid/0tz9k9o2-mrtvfb
https://wolfram.com/xid/0tz9k9o2-dtoi0n
https://wolfram.com/xid/0tz9k9o2-buqqxn
https://wolfram.com/xid/0tz9k9o2-6r4hd
Transitive, i.e. if and then :
https://wolfram.com/xid/0tz9k9o2-fegzw
https://wolfram.com/xid/0tz9k9o2-l1sy7g
xκy are partial orders but not total orders, so there are incomparable elements:
https://wolfram.com/xid/0tz9k9o2-ixvup8
Neither nor is true, because and are incomparable elements:
https://wolfram.com/xid/0tz9k9o2-bttlje
The set of vectors and . These are the comparable elements to :
https://wolfram.com/xid/0tz9k9o2-d5uj7q
Wolfram Research (2019), VectorLess, Wolfram Language function, https://reference.wolfram.com/language/ref/VectorLess.html.
Text
Wolfram Research (2019), VectorLess, Wolfram Language function, https://reference.wolfram.com/language/ref/VectorLess.html.
Wolfram Research (2019), VectorLess, Wolfram Language function, https://reference.wolfram.com/language/ref/VectorLess.html.
CMS
Wolfram Language. 2019. "VectorLess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/VectorLess.html.
Wolfram Language. 2019. "VectorLess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/VectorLess.html.
APA
Wolfram Language. (2019). VectorLess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/VectorLess.html
Wolfram Language. (2019). VectorLess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/VectorLess.html
BibTeX
@misc{reference.wolfram_2024_vectorless, author="Wolfram Research", title="{VectorLess}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/VectorLess.html}", note=[Accessed: 09-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_vectorless, organization={Wolfram Research}, title={VectorLess}, year={2019}, url={https://reference.wolfram.com/language/ref/VectorLess.html}, note=[Accessed: 09-January-2025
]}