• 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_100
  • Description: TOMO 100x100: Create a 2D tomography test problem.
  • download as a MATLAB mat-file, file size: 13 KB. Use SJget(265) or SJget('Regtools/tomo_100') in MATLAB.
  • download in Matrix Market format, file size: 16 KB.
  • download in Rutherford/Boeing format, file size: 16 KB.

    Regtools/tomo_100

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

    Regtools/tomo_100

    Matrix properties (click for a legend)  
    number of rows100
    number of columns100
    structural full rank?yes
    structural rank100
    numerical rank 98
    dimension of the numerical null space2
    numerical rank / min(size(A))0.98
    Euclidean norm of A 11.657
    calculated singular value # 980.010966
    numerical rank defined using a tolerance
    max(size(A))*eps(norm(A)) =
    1.7764e-013
    calculated singular value # 992.7158e-016
    gap in the singular values at the numerical rank:
    singular value # 98 / singular value # 99
    4.0379e+013
    calculated condition number9.8564e+016
    condest1.6924e+019
    nonzeros1,342
    # of blocks from dmperm1
    # strongly connected comp.1
    entries not in dmperm blocks0
    explicit zero entries0
    nonzero pattern symmetry 13%
    numeric value symmetry 0%
    typereal
    structureunsymmetric
    Cholesky candidate?no
    positive definite?no

    authorHansen
    editorPer Christian Hansen
    date2007
    kindill-posed problem
    2D/3D problem?no
    SJid265
    UFid-

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

    Notes:

        Constructed by the call [A,b,x]= tomo(10)                            
                                                                             
      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'))3,556 3,686
    Cholesky flop count1.7e+005 1.8e+005
    nnz(L+U), no partial pivoting7,012 7,272
    nnz(V) for QR, upper bound nnz(L) for LU2,876 3,373
    nnz(R) for QR, upper bound nnz(U) for LU4,577 4,587

    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.