Verified strong solution of interval linear inequalities.
For a rectangular interval matrix A and a matching interval vector b, this function either computes a strong solution x to
A * x ≤ b
(i. e., a real vector x verified to satisfy Ao * x ≤ bo for each Ao in A and bo in b), or verifies nonexistence of such a solution, or yields no verified result:
x is a verified strong solution of A * x ≤ b, and As is an interval matrix of empty intervals,
As is a very right (“almost thin”) interval matrix verified to contain a real matrix Ao such that the system Ao * x ≤ b.inf has no solution (which proves that no strong solution exists), and x is a vector of NaNs,
no verified output.
A theoretical result [1] asserts that if each system Ao * x ≤ bo, where Ao in A and bo in b, has a solution (depending generally on Ao and bo), then there exists a vector x satisfying Ao * x ≤ bo for each Ao in A and bo in b. Such a vector x is called a strong solution of the system A * x ≤ b.
[1] J. Rohn and J. Kreslova, Linear Interval Inequalities, LAMA 38 (1994), 79–82.
Based on Section 2.13 in M. Fiedler, J. Nedoma, J. Ramik, J. Rohn and K. Zimmermann, Linear Optimization Problems with Inexact Data, Springer-Verlag, New York 2006.
This work was supported by the Czech Republic National Research Program “Information Society”, project 1ET400300415.
See also: .
Package: interval