• SJSU Singular Matrix Database
  • Matrix group: JGD_Margulies
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  • Matrix: JGD_Margulies/flower_5_1
  • Description: Combinatorial optimization as polynomial eqns, Susan Margulies, UC Davis
  • download as a MATLAB mat-file, file size: 5 KB. Use SJget(588) or SJget('JGD_Margulies/flower_5_1') in MATLAB.
  • download in Matrix Market format, file size: 3 KB.
  • download in Rutherford/Boeing format, file size: 3 KB.

    JGD_Margulies/flower_5_1

    Routine svd from Matlab 7.6.0.324 (R2008a) used to calculate the singular values.

    JGD_Margulies/flower_5_1

    dmperm of JGD_Margulies/flower_5_1

    Matrix properties (click for a legend)  
    number of rows211
    number of columns201
    structural full rank?no
    structural rank191
    numerical rank 179
    dimension of the numerical null space22
    numerical rank / min(size(A))0.89055
    Euclidean norm of A 3.5144
    calculated singular value # 1790.25661
    numerical rank defined using a tolerance
    max(size(A))*eps(norm(A)) =
    9.3703e-014
    calculated singular value # 1801.5163e-015
    gap in the singular values at the numerical rank:
    singular value # 179 / singular value # 180
    1.6924e+014
    calculated condition number1.8993e+016
    condest-2
    nonzeros602
    # of blocks from dmperm13
    # strongly connected comp.1
    explicit zero entries0
    nonzero pattern symmetry 0%
    numeric value symmetry 0%
    typebinary
    structurerectangular
    Cholesky candidate?no
    positive definite?no

    authorS. Margulies
    editorJ.-G. Dumas
    date2008
    kindcombinatorial problem
    2D/3D problem?no
    SJid588
    UFid2,157

    Notes:

    Combinatorial optimization as polynomial eqns, Susan Margulies, UC Davis
    From Jean-Guillaume Dumas' Sparse Integer Matrix Collection,            
    http://ljk.imag.fr/membres/Jean-Guillaume.Dumas/simc.html               
                                                                            
    http://arxiv.org/abs/0706.0578                                          
                                                                            
    Expressing Combinatorial Optimization Problems by Systems of Polynomial 
    Equations and the Nullstellensatz                                       
                                                                            
    Authors: J.A. De Loera, J. Lee, Susan Margulies, S. Onn                 
                                                                            
    (Submitted on 5 Jun 2007)                                               
                                                                            
    Abstract: Systems of polynomial equations over the complex or real      
    numbers can be used to model combinatorial problems. In this way, a     
    combinatorial problem is feasible (e.g. a graph is 3-colorable,         
    hamiltonian, etc.) if and only if a related system of polynomial        
    equations has a solution. In the first part of this paper, we construct 
    new polynomial encodings for the problems of finding in a graph its     
    longest cycle, the largest planar subgraph, the edge-chromatic number,  
    or the largest k-colorable subgraph.  For an infeasible polynomial      
    system, the (complex) Hilbert Nullstellensatz gives a certificate that  
    the associated combinatorial problem is infeasible. Thus, unless P =    
    NP, there must exist an infinite sequence of infeasible instances of    
    each hard combinatorial problem for which the minimum degree of a       
    Hilbert Nullstellensatz certificate of the associated polynomial system 
    grows.  We show that the minimum-degree of a Nullstellensatz            
    certificate for the non-existence of a stable set of size greater than  
    the stability number of the graph is the stability number of the graph. 
    Moreover, such a certificate contains at least one term per stable set  
    of G. In contrast, for non-3- colorability, we found only graphs with   
    Nullstellensatz certificates of degree four.                            
                                                                            
    Filename in JGD collection: Margulies/flower_5_1.sms                    
    

    Ordering statistics:AMD METIS
    nnz(V) for QR, upper bound nnz(L) for LU2,639 2,429
    nnz(R) for QR, upper bound nnz(U) for LU3,494 3,408

    Maintained by Leslie Foster, last updated 24-Apr-2009.

    Entries 5 through 14 in the table of matrix properties and the singular
    value plot were created using SJsingular code. The other plots
    and statistics are produced using utilities from the SuiteSparse package.
    Matrix color plot pictures by cspy, a MATLAB function in the CSparse package.