• 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_1000
  • Description: GRAVITY 1000x1000 Test problem: 1-D gravity surveying model problem
  • download as a MATLAB mat-file, file size: 6 MB. Use SJget(240) or SJget('Regtools/gravity_1000') in MATLAB.
  • download in Matrix Market format, file size: 2 MB.
  • download in Rutherford/Boeing format, file size: 873 KB.


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


    Matrix properties (click for a legend)  
    number of rows1,000
    number of columns1,000
    structural full rank?yes
    structural rank1,000
    numerical rank 45
    dimension of the numerical null space955
    numerical rank / min(size(A))0.045
    Euclidean norm of A 6.4592
    calculated singular value # 451.1139e-012
    numerical rank defined using a tolerance
    max(size(A))*eps(norm(A)) =
    calculated singular value # 465.5492e-013
    gap in the singular values at the numerical rank:
    singular value # 45 / singular value # 46
    calculated condition number4.8668e+020
    # 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 1000-by-1
    xfull 1000-by-1


        Constructed by the call [A,b,x]= gravity(1000)                       
      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'))500,500 500,500
    Cholesky flop count3.3e+008 3.3e+008
    nnz(L+U), no partial pivoting1,000,000 1,000,000
    nnz(V) for QR, upper bound nnz(L) for LU500,500 500,500
    nnz(R) for QR, upper bound nnz(U) for LU500,500 500,500

    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.