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maxplustslinsol

max-plus algebra solution of twoside linear equation C⊗x=D⊗x

Calling Sequence

[x,x0,r] = maxplustslinsol(C,D)

Parameters

C
: an input matrix of size m by-n.
D
: an input matrix of size m by-n.

Description

Function returns a non unique solution x of equation C⊗x=D⊗x,

a random initial x0 and the number of iterating r. The existence of solution x depend on the initial x0. The solution exists if for every row i-th there is an index j0 such that cij0= dij0. If there exists a row i-th such that ci,j > di,j for all j then the solution does not exist,

and function returns r=1000, x=[].

For detailed, see : Cechlarova, K. (2005), "Eigenvectors of Interval Matrices over Max-Plus Algebra", Discrete Applied Mathematics, vol.150, 2-15.

Examples

c  = [1.    2.    4.    8.; 
      6.    3.    9.    4.; 
      7.    3.    8.    0. ];
d  = [0.    1.    4.    5.; 
      2.    3.    9.    2.;  
      7.    3.    7.    0.] ;

[x,x0,r] = maxplustslinsol(c,d)
 r  =
 
    3.  
 x0  =
 
    6.  
    5.  
    8.  
    8.  
 x  =
 
    8.  
    9.  
    7.  
    3.  
// Check that x is the solution
maxplusotimes(c,x)==maxplusotimes(d,x)
 ans  =
 
  T  
  T  
  T  
// another solution
[x,x0,r] = maxplustslinsol(c,d)
 r  =
 
    5.  
 x0  =
 
    9.  
    0.  
    2.  
    5.  
 x  =
 
    5.  
    4.  
    2.  
  - 2.
// Check that x is the solution
maxplusotimes(c,x)==maxplusotimes(d,x)
 ans  =
 
  T  
  T  
  T  
// The next example gives a solution does not exist.
c  = [10.    6.     13.    16.    6.;   
      17.    15.    9.     10.    9.;   
      2.     12.    18.    11.    11.;  
      4.     8.     5.     7.     8. ;  
      8.     11.    10.    14.    9. ];
d  = [4.    6.    4.    7.    5.;  
      9.    7.    9.    3.    1.;  
      1.    9.    9.    5.    2.;  
      2.    5.    3.    3.    6.;  
      6.    3.    4.    5.    8.];
// Check that c≥d
c >= d
 ans  =
 
  T T T T T  
  T T T T T  
  T T T T T  
  T T T T T  
  T T T T T 
// observe that c(3,:)≠d(3,:)
isequal(c(3,:),d(3,:))
 ans  =
 
  F  
// So, the equation c⊗x=d⊗x does not has a solution
[x,x0,r]=maxplustslinsol(c,d)
 r  =
 
    1000.  
 x0  =
 
    1.   
    7.   
    3.   
    5.   
    10.  
 x  =
 
     [] \\ this means that a solution does not exist.

Author

"Max-Plus Algebra Toolbox", ver. 1.01, April, 2009.

See Also


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