• SJSU Singular Matrix Database
  • Matrix group: Regtools
  • Click here for a description of the Regtools group.
  • Click here for a list of all matrices
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  • Matrix: Regtools/gravity_500
  • Description: GRAVITY 500x500 Test problem: 1-D gravity surveying model problem
  • download as a MATLAB mat-file, file size: 813 KB. Use SJget(239) or SJget('Regtools/gravity_500') in MATLAB.
  • download in Matrix Market format, file size: 491 KB.
  • download in Rutherford/Boeing format, file size: 122 KB.

    Regtools/gravity_500

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

    Regtools/gravity_500

    Matrix properties (click for a legend)  
    number of rows500
    number of columns500
    structural full rank?yes
    structural rank500
    numerical rank 46
    dimension of the numerical null space454
    numerical rank / min(size(A))0.092
    Euclidean norm of A 6.4592
    calculated singular value # 465.4977e-013
    numerical rank defined using a tolerance
    max(size(A))*eps(norm(A)) =
    4.4409e-013
    calculated singular value # 472.7363e-013
    gap in the singular values at the numerical rank:
    singular value # 46 / singular value # 47
    2.0092
    calculated condition number2.5879e+020
    condest1.5215e+020
    nonzeros250,000
    # of blocks from dmperm1
    # strongly connected comp.1
    entries not in dmperm blocks0
    explicit zero entries0
    nonzero pattern symmetrysymmetric
    numeric value symmetrysymmetric
    typereal
    structuresymmetric
    Cholesky candidate?yes
    positive definite?no

    authorHansen
    editorPer Christian Hansen
    date2002
    kindill-posed problem
    2D/3D problem?no
    SJid239
    UFid-

    Additional fieldssize and type
    bfull 500-by-1
    xfull 500-by-1

    Notes:

        Constructed by the call [A,b,x]= gravity(500)                        
                                                                             
      where gravity is from Regularization Tools. The description of         
      gravity from http://www2.imm.dtu.dk/~pch/Regutools/ is:                
                                                                             
                 [A,b,x] = gravity(n,example,a,b,d)                          
                                                                             
      Discretization of a 1-D model problem in gravity surveying, in which   
      a mass distribution f(t) is located at depth d, while the vertical     
      component of the gravity field g(s) is measured at the surface.        
                                                                             
      The resulting problem is a first-kind Fredholm integral equation       
      with kernel                                                            
         K(s,t) = d*(d^2 + (s-t)^2)^(-3/2) .                                 
      The following three examples are implemented (example = 1 is default): 
         1: f(t) = sin(pi*t) + 0.5*sin(2*pi*t),                              
         2: f(t) = piecewise linear function,                                
         3: f(t) = piecewise constant function.                              
      The problem is discretized by means of the midpoint quadrature rule    
      with n points, leading to the matrix A and the vector x.  Then the     
      right-hand side is computed as b = A*x.                                
                                                                             
      The t integration interval is fixed to [0,1], while the s integration  
      interval [a,b] can be specified by the user. The default interval is   
      [0,1], leading to a symmetric Toeplitz matrix.                         
                                                                             
      The parameter d is the depth at which the magnetic deposit is located, 
      and the default value is d = 0.25. The larger the d, the faster the    
      decay of the singular values.                                          
    

    Ordering statistics:AMD METIS
    nnz(chol(P*(A+A'+s*I)*P'))125,250 125,250
    Cholesky flop count4.2e+007 4.2e+007
    nnz(L+U), no partial pivoting250,000 250,000
    nnz(V) for QR, upper bound nnz(L) for LU125,250 125,250
    nnz(R) for QR, upper bound nnz(U) for LU125,250 125,250

    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.