Matrix properties
number of rows	13,691
number of columns	13,682
nonzeros	154,303
structural full rank?	no
structural rank	13,006
# of blocks from dmperm	1,903
# strongly connected comp.	590
explicit zero entries	0
nonzero pattern symmetry	0%
numeric value symmetry	0%
type	binary
structure	rectangular
Cholesky candidate?	no
positive definite?	no

author	D. Priebel
editor	T. Davis
date	2009
kind	combinatorial problem
2D/3D problem?	no

Additional fields	size and type
factor_base	full 13682-by-1
smooth_number	full 13691-by-29
solution	full 5210-by-1

Notes:

Each column in the matrix corresponds to a number in the factor base       
less than some bound B.  Each row corresponds to a smooth number (able     
to be completely factored over the factor base).  Each value in a row      
binary vector corresponds to the exponent of the factor base mod 2.        
For example:                                                               
                                                                           
    factor base: 2 7 23                                                    
    smooth numbers: 46, 28, 322                                            
    2^1       * 23^1 = 46                                                  
    2^2 * 7^1        = 28                                                  
    2^1 * 7^1 * 23^1 = 322                                                 
    Matrix:                                                                
        101                                                                
        010                                                                
        111                                                                
                                                                           
A solution to the matrix is considered to be a set of rows which when      
combined in GF2 produce a null vector. Thus, if you multiply each of       
the smooth numbers which correspond to that particular set of rows you     
will get a number with only even exponents, making it a perfect            
square. In the above example you can see that combining the 3 vectors      
results in a null vector and, indeed, it is a perfect square: 644^2.       
                                                                           
Problem.A: A GF(2) matrix constructed from the exponents of the            
factorization of the smooth numbers over the factor base. A solution of    
this matrix is a kernel (nullspace). Such a solution has a 1/2 chance of   
being a factorization of N.                                                
                                                                           
Problem.aux.factor_base: The factor base used. factor_base(j) corresponds  
to column j of the matrix. Note that a given column may or may not have    
nonzero elements in the matrix.                                            
                                                                           
Problem.aux.smooth_number: The smooth numbers, smooth over the factor      
base.  smooth_number(i) corresponds to row i of the matrix.                
                                                                           
Problem.aux.solution: A sample solution to the matrix. Combine, in GF(2)   
the rows with these indicies to produce a solution to the matrix with the  
additional property that it factors N (a matrix solution only has 1/2      
probability of factoring N).                                               
                                                                           
Problem specific information:                                              
                                                                           
n = 4232562527578032866150921497850842593296760823796443077101 (192-bits)  
passes primality test, n is composite, continuing...                       
1) Initial bound: 350000, pi(350000) estimate: 27417,                      
    largest found: 317729 (actual bound)                                   
2) Number of quadratic residues estimate: 18279, actual number found: 13681
3) Modular square roots found: 27362(2x residues)                          
4) Constructing smooth number list [sieving] (can take a while)...         
Sieving for: 13691                                                         
5. Constructing a matrix of size: 13691x13682                              
Set a total of 154303 exponents, with 6893 negatives                       
Matrix solution found with: 5210 combinations                              
Divisor: 83135929635332984850508004533 (probably prime)                    
Divisor: 50911351399373573113182167897 (probably prime)                    

Ordering statistics:	result
nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD	20,360,732
nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD	18,023,441

SVD-based statistics:
norm(A)	120.05
min(svd(A))	2.04308e-63
cond(A)	5.87592e+64
rank(A)	13,006
sprank(A)-rank(A)	0
null space dimension	676
full numerical rank?	no
singular value gap	1.3401e+11

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

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.