• 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/tomo_900
  • Description: TOMO 900x900: Create a 2D tomography test problem.
  • download as a MATLAB mat-file, file size: 307 KB. Use SJget(266) or SJget('Regtools/tomo_900') in MATLAB.
  • download in Matrix Market format, file size: 395 KB.
  • download in Rutherford/Boeing format, file size: 362 KB.

    Regtools/tomo_900

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

    Regtools/tomo_900

    Matrix properties (click for a legend)  
    number of rows900
    number of columns900
    structural full rank?yes
    structural rank900
    numerical rank 893
    dimension of the numerical null space7
    numerical rank / min(size(A))0.99222
    Euclidean norm of A 32.88
    calculated singular value # 8930.0091677
    numerical rank defined using a tolerance
    max(size(A))*eps(norm(A)) =
    6.3949e-012
    calculated singular value # 8941.4718e-015
    gap in the singular values at the numerical rank:
    singular value # 893 / singular value # 894
    6.2291e+012
    calculated condition number5.2441e+017
    condestInf
    nonzeros35,598
    # of blocks from dmperm1
    # strongly connected comp.1
    entries not in dmperm blocks0
    explicit zero entries0
    nonzero pattern symmetry 4%
    numeric value symmetry 0%
    typereal
    structureunsymmetric
    Cholesky candidate?no
    positive definite?no

    authorHansen
    editorPer Christian Hansen
    date2007
    kindill-posed problem
    2D/3D problem?no
    SJid266
    UFid-

    Additional fieldssize and type
    bfull 900-by-1
    xfull 900-by-1

    Notes:

        Constructed by the call [A,b,x]= tomo(30)                            
                                                                             
      where tomo is from Regularization Tools. The description of            
      tomo from http://www2.imm.dtu.dk/~pch/Regutools/ is:                   
                                                                             
                    [A,b,x] = tomo(N,f);                                     
                                                                             
      This function creates a simple two-dimensional tomography test problem.
      A 2D domain [0,N] x [0,N] is divided into N^2 cells of unit size, and a
      total of round(f*N^2) rays in random directions penetrate this domain. 
      The default value is f = 1.                                            
                                                                             
      Each cell is assigned a value (stored in the vector x), and for each   
      ray the corresponding element in the right-hand side b is the line     
      integral along the ray, i.e.                                           
         sum_{cells in ray}  x_{cell j} * length_{cell j}                    
      where length_{cell j} is the length of the ray in the j-th cell.       
                                                                             
      The matrix A is sparse, and each row (corresponding to a ray) holds    
      the value length_{cell j} in the j-th position.  Hence:                
         b = A*x .                                                           
      Once a solution x_reg has been computed, it can be visualized by means 
      of imagesc(reshape(x_reg,N,N)).                                        
                                                                             
      The exact solution, reshape(x,N,N), is identical to the exact image in 
      the function blur.                                                     
                                                                             
      Note that the code for tomo uses random numbers and repeated calls to  
      tomo will produce different matrices.                                  
    

    Ordering statistics:AMD METIS
    nnz(chol(P*(A+A'+s*I)*P'))324,883 319,609
    Cholesky flop count1.7e+008 1.6e+008
    nnz(L+U), no partial pivoting648,866 638,318
    nnz(V) for QR, upper bound nnz(L) for LU358,038 357,583
    nnz(R) for QR, upper bound nnz(U) for LU398,899 399,620

    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.