Matrix: MathWorks/QRpivot

Description: Limitation of basic solution to x=A\b using qr(A); needs min 2-norm

MathWorks/QRpivot graph
(bipartite graph drawing)

MathWorks/QRpivot dmperm of MathWorks/QRpivot

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  • download as a MATLAB mat-file, file size: 13 KB. Use UFget(1894) or UFget('MathWorks/QRpivot') in MATLAB.
  • download in Matrix Market format, file size: 21 KB.
  • download in Rutherford/Boeing format, file size: 14 KB.

    Matrix properties
    number of rows660
    number of columns749
    structural full rank?yes
    structural rank660
    # of blocks from dmperm9
    # strongly connected comp.1
    explicit zero entries0
    nonzero pattern symmetry 0%
    numeric value symmetry 0%
    Cholesky candidate?no
    positive definite?no

    authorP. Quillen
    editorT. Davis
    kindcounter-example problem
    2D/3D problem?no

    Additional fieldssize and type
    bsparse 660-by-1


    Counter-example problem from The MathWorks, Pat Quillen                 
    This matrix was obtained from a MATLAB user.  It illustrates the        
    limitations inherent in computing a basic solution to an under-         
    determined system without the use of column pivoting.                   
    With column pivoting (which can only be done in MATLAB with full        
    matrices) the problem is solved properly.                               
    When finding the min 2-norm solution (ignoring fill-in):                
        [Q,R] = qr (A') ;                                                   
        x = Q*(R'\b) ;                                                      
    a good solution is found.  To reduce fill-in:                           
        p = colamd (A') ;                                                   
        [Q,R] = qr (A (p,:)') ;                                             
        x = Q*(R'\b(p)) ;                                                   
    which also finds a good solution.                                       
    However, x=A\b computes a basic solution, using this algorithm:         
        q = colamd (A) ;                                                    
        [Q,R] = qr (A (:,q)) ;                                              
        x = R\(Q'*b) ;                                                      
        x (q) = q ;                                                         
    which finds an error with norm(A*x-b) of 1e-9 in MATLAB 7.6.            
    With random permutations, and determining the cond(R1) of the leading   
    trianglar part (R is "squeezed" and the columns can be partitioned into 
    [R1 R2] where R1 is square and upper triangular) leads to the following 
    Note that the error is high when condest(R1) is high.  Note in          
    particular the last trial.                                              
    So this clinches the question.  MATLAB's QR, and my new sparse QR, both 
    use a rank-detection method (by Heath) that does not do column pivoting,
    and which is known to fail for some problems - for which Grimes & Lewis'
    method will likely succeed.                                             
    The advantage of my QR is that I now always return R as upper           
    trapezoidal, so if the user is concerned, he/she can easily check       
    condest(R(:,1:m)) if m < n.                                             
        err 7.71e-07 condest R1 2.18e+12                                    
        err 1.25e-09 condest R1 9.82e+08                                    
        err 2.47e-09 condest R1 2.46e+11                                    
        err 4.00e-09 condest R1 4.03e+09                                    
        err 9.88e-10 condest R1 4.73e+09                                    
        err 2.25e-08 condest R1 5.34e+09                                    
        err 2.00e-08 condest R1 1.04e+09                                    
        err 1.09e-09 condest R1 6.83e+08                                    
        err 6.18e-08 condest R1 8.13e+10                                    
        err 3.13e-10 condest R1 4.23e+09                                    
        err 6.64e-10 condest R1 2.46e+10                                    
        err 5.76e-09 condest R1 4.31e+09                                    
        err 7.61e-07 condest R1 5.08e+10                                    
        err 2.27e-09 condest R1 4.94e+09                                    
        err 3.99e-10 condest R1 2.80e+09                                    
        err 1.37e-09 condest R1 3.13e+09                                    
        err 6.93e-05 condest R1 1.84e+14                                    
        err 1.35e-08 condest R1 7.18e+09                                    
        err 1.09e-08 condest R1 1.79e+11                                    
        err 1.81e-09 condest R1 2.99e+08                                    
        err 1.55e-01 condest R1 2.45e+18                                    
    In summary, this is a "feature" not a "bug".  If you want a reliable    
    solution to an underdetermined system, find the min 2norm solution      
    via a QR factorization of A'.                                           

    Ordering statistics:result
    nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD15,330
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD15,836

    SVD-based statistics:
    null space dimension0
    full numerical rank?yes

    singular values (MAT file):click here
    SVD method used:s = svd (full (A)) ;

    MathWorks/QRpivot svd

    For a description of the statistics displayed above, click here.

    Maintained by Tim Davis, last updated 12-Mar-2014.
    Matrix pictures by cspy, a MATLAB function in the CSparse package.
    Matrix graphs by Yifan Hu, AT&T Labs Visualization Group.