• SJSU Singular Matrix Database
  • Matrix group: Regtools
  • Click here for a description of the Regtools group.
  • Click here for a list of all matrices
  • Click here for a list of all matrix groups


  • Matrix: Regtools/heat_1000
  • Description: HEAT 1000x1000 Test problem: inverse heat equation.
  • download as a MATLAB mat-file, file size: 4 MB. Use SJget(244) or SJget('Regtools/heat_1000') in MATLAB.
  • download in Matrix Market format, file size: 2 MB.
  • download in Rutherford/Boeing format, file size: 1 MB.

    Regtools/heat_1000

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

    Regtools/heat_1000

    dmperm of Regtools/heat_1000

    Matrix properties (click for a legend)  
    number of rows1,000
    number of columns1,000
    structural full rank?yes
    structural rank1,000
    numerical rank 588
    dimension of the numerical null space412
    numerical rank / min(size(A))0.588
    Euclidean norm of A 0.35515
    calculated singular value # 5885.592e-014
    numerical rank defined using a tolerance
    max(size(A))*eps(norm(A)) =
    5.5511e-014
    calculated singular value # 5895.4486e-014
    gap in the singular values at the numerical rank:
    singular value # 588 / singular value # 589
    1.0263
    calculated condition number2.6144e+233
    condestInf
    nonzeros500,500
    # of blocks from dmperm1,000
    # strongly connected comp.1,000
    entries not in dmperm blocks499,500
    explicit zero entries0
    nonzero pattern symmetry 0%
    numeric value symmetry 0%
    typereal
    structureunsymmetric
    Cholesky candidate?no
    positive definite?no

    authorCarasso, Elden
    editorPer Christian Hansen
    date1982
    kindill-posed problem
    2D/3D problem?no
    SJid244
    UFid-

    Additional fieldssize and type
    bfull 1000-by-1
    xfull 1000-by-1

    Notes:

        Constructed by the call [A,b,x]= heat(1000)                          
                                                                             
      where heat is from Regularization Tools. The description of            
      heat from http://www2.imm.dtu.dk/~pch/Regutools/ is:                   
                                                                             
                  [A,b,x] = heat(n,kappa)                                    
                                                                             
      A first kind Volterra integral equation with [0,1] as                  
      integration interval.  The kernel is K(s,t) = k(s-t) with              
         k(t) = t^(-3/2)/(2*kappa*sqrt(pi))*exp(-1/(4*kappa^2*t)) .          
      Here, kappa controls the ill-conditioning of the matrix:               
         kappa = 5 gives a well-conditioned problem                          
         kappa = 1 gives an ill-conditioned problem.                         
      The default is kappa = 1.                                              
                                                                             
      An exact soltuion is constructed, and then the right-hand side         
      b is produced as b = A*x.                                              
    

    Ordering statistics:AMD METIS DMPERM+
    nnz(chol(P*(A+A'+s*I)*P'))500,500 500,500 1,000
    Cholesky flop count3.3e+008 3.3e+008 1.0e+003
    nnz(L+U), no partial pivoting1,000,000 1,000,000 500,500
    nnz(V) for QR, upper bound nnz(L) for LU284,041 500,500 1,000
    nnz(R) for QR, upper bound nnz(U) for LU500,500 500,500 500,500

    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.