Matrix properties
number of rows	7,441
number of columns	7,431
nonzeros	82,270
structural full rank?	no
structural rank	7,110
# of blocks from dmperm	893
# strongly connected comp.	282
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 7432-by-1
smooth_number	full 7441-by-27
solution	full 2983-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 = 73363722971930954428433124842779099222294372095286387 (176-bits)     
passes primality test, n is composite, continuing...                     
1) Initial bound: 180000, pi(180000) estimate: 14875,                    
    largest found: 162359 (actual bound)                                 
2) Number of quadratic residues estimate: 9918, actual number found: 7431
3) Modular square roots found: 14862(2x residues)                        
4) Constructing smooth number list [sieving] (can take a while)...       
Sieving for: 7441                                                        
5. Constructing a matrix of size: 7441x7432                              
Set a total of 82270 exponents, with 3725 negatives                      
Matrix solution found with: 2983 combinations                            
Divisor: 236037985789994529800050193 (probably prime)                    
Divisor: 310813205452462332837525059 (probably prime)                    

Ordering statistics:	result
nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD	6,717,048
nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD	6,250,729

SVD-based statistics:
norm(A)	96.1924
min(svd(A))	1.31948e-47
cond(A)	7.29019e+48
rank(A)	7,110
sprank(A)-rank(A)	0
null space dimension	321
full numerical rank?	no
singular value gap	1.95793e+13

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.