// FILE. . . . . d:/hak/hlt/src/hlt/math/matrix/sources/LinearFixPointEquationSystem.java
// EDIT BY . . . Hassan Ait-Kaci
// ON MACHINE. . Hak-Laptop
// STARTED ON. . Sat Nov 16 15:03:29 2019
/**
* @version Last modified on Mon Dec 16 09:52:22 2019 by hak
* @author Hassan Aït-Kaci
* @copyright © by the author
*/
package hlt.math.matrix;
/**
* package hlt.math.matrix
* documentation listing
*/
/**
* This implements a system of linear fix-point equations with double
* coefficients of the form:
*
* A × X + B = X where A is an m
* × n Matrix, B is an
* m × 1 Matrix (i.e., a
* column vector of order m), and X is a column vector
* of order n of variables Xi,
* i=1, ... n. For example, if m = 3, n = 4
* and:
*
* 1.1 1.2 1.3 1.4 1.0
* A = 2.1 2.2 2.3 2.4 B = 2.0
* 3.1 3.2 3.3 3.4 3.0
*
* then:
*
* 1.1 × X1 + 1.2 × X2 + 1.3 × X3 + 1.4 × X4 + 1.0 = X1
* 2.1 × X1 + 2.2 × X2 + 2.3 × X3 + 2.4 × X4 + 2.0 = X2
* 3.1 × X1 + 3.2 × X2 + 3.3 × X3 + 3.4 × X4 + 3.0 = X3
*
* which gives the equivalent non-fix-point formulation (A-I) × X = -B:
*
* 0.1 × X1 + 1.2 × X2 + 1.3 × X3 + 1.4 × X4 = -1.0
* 2.1 × X1 + 1.2 × X2 + 2.3 × X3 + 2.4 × X4 = -2.0
* 3.1 × X1 + 3.2 × X2 + 2.3 × X3 + 3.4 × X4 = -3.0
*
* and conversely, from the more conventional non-fix-point
* formulation of the form A' × X = B', one recovers the
* corresponding equivalent fix-point form as:
* (A'+I) × X + (-B') = X, and then use this class.
* Non square equational systems may only have parametric solutions if
* there are more variables than equations (expressed in terms of the
* missing variables), or redundant solutions if there are more
* equations than variables. For a square matrix, a solution will exist
* iff it has an inverse (i.e., iff it has a non-zero
* determinant).
*
* @see Matrix
*/
public class LinearFixPointEquationSystem
{
}