• 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/gravity_100
  • Description: GRAVITY 100x100 Test problem: 1-D gravity surveying model problem
  • download as a MATLAB mat-file, file size: 16 KB. Use SJget(237) or SJget('Regtools/gravity_100') in MATLAB.
  • download in Matrix Market format, file size: 26 KB.
  • download in Rutherford/Boeing format, file size: 9 KB.


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


    Matrix properties (click for a legend)  
    number of rows100
    number of columns100
    structural full rank?yes
    structural rank100
    numerical rank 48
    dimension of the numerical null space52
    numerical rank / min(size(A))0.48
    Euclidean norm of A 6.4593
    calculated singular value # 489.4596e-014
    numerical rank defined using a tolerance
    max(size(A))*eps(norm(A)) =
    calculated singular value # 494.5878e-014
    gap in the singular values at the numerical rank:
    singular value # 48 / singular value # 49
    calculated condition number5.4668e+018
    # of blocks from dmperm1
    # strongly connected comp.1
    entries not in dmperm blocks0
    explicit zero entries0
    nonzero pattern symmetrysymmetric
    numeric value symmetrysymmetric
    Cholesky candidate?yes
    positive definite?no

    editorPer Christian Hansen
    kindill-posed problem
    2D/3D problem?no

    Additional fieldssize and type
    bfull 100-by-1
    xfull 100-by-1


        Constructed by the call [A,b,x]= gravity(100)                        
      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'))5,050 5,050
    Cholesky flop count3.4e+005 3.4e+005
    nnz(L+U), no partial pivoting10,000 10,000
    nnz(V) for QR, upper bound nnz(L) for LU5,050 5,050
    nnz(R) for QR, upper bound nnz(U) for LU5,050 5,050

    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.