biochemical network; left nullspace is required
Name N_reactome
Group Schulthess
Matrix ID 2554
Num Rows 10,204
Num Cols 16,559
Nonzeros 43,816
Pattern Entries 43,816
Kind Biochemical Network
Symmetric No
Date 2012
Author P. Schulthess
Editor T. Davis
Structural Rank 9,308
Structural Rank Full false
Num Dmperm Blocks 925
Strongly Connect Components 418
Num Explicit Zeros 0
Pattern Symmetry 0%
Numeric Symmetry 0%
Cholesky Candidate no
Positive Definite no
Type integer
SVD Statistics
Matrix Norm 6.068857e+01
Minimum Singular Value 0
Condition Number Inf
Rank 9,025
sprank(A)-rank(A) 283
Null Space Dimension 1,179
Full Numerical Rank? no
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Matrices from Pascal Schulthess, Institute for Pathology,            
Chariteplatz 1, 10117 Berlin, Germany.                               
Three large biochemical networks (N_biocarta, N_pid, and N_reactome).
These are stoichiometric matrices extracted from three biochemical   
databases (BioCarta, PID, and REACTOME) describing cell signaling    
pathways and protein-protein interaction networks.  The goal is to   
find the left nullspace of the matrix; in MATLAB notation:           
N = null (Problem.A') ;                                              
The matrix (Problem.A')*N will thus be essentially zero.             
This can be done much more efficiently with the spqr_rank toolbox by 
Leslie Foster and Tim Davis, as:                                     
N = spqr_null (Problem.A') ;                                         
The matrix A is transposed, then N = null (A) or N = spqr_null (A)   
is computed.  The size statistic is the memory taken by N.           
spqr_null can compute either an explicit matrix N, or an implicit    
Householder-based representation.  The latter takes less memory.     
Matrix: N_biocarta  size: 1996 by 1922 (transposed)                  
spqr_null stats:                                                     
                       flag: 0                                       
                       rank: 1023                                    
                        tol: 3.5456e-12                              
      est_sval_upper_bounds: [0.1689 3.4534e-15]                     
      est_sval_lower_bounds: [0.1203 0]                              
    sval_numbers_for_bounds: [1023 1024]                             
         est_norm_A_times_N: 2.4349e-15                              
spqr_null, implicit:  0.03 sec, norm(A*N)  9e-15 size:  0.08 MB      
spqr_null, explicit:  0.10 sec, norm(A*N)  9e-15 size:  0.11 MB      
MATLAB null:          3.31 sec, norm(A*N)  2e-13 size: 13.82 MB      
all report dim(N) of 899.                                            
Matrix: N_pid  size: 3923 by 3625 (transposed)                       
spqr_null stats:                                                     
                       flag: 0                                       
                       rank: 2048                                    
                        tol: 1.3937e-11                              
      est_sval_upper_bounds: [0.0922 5.1310e-15]                     
      est_sval_lower_bounds: [0.0585 0]                              
    sval_numbers_for_bounds: [2048 2049]                             
         est_norm_A_times_N: 1.6751e-15                              
spqr_null, implicit:  0.05 sec, norm(A*N)  4e-14 size:  0.21 MB      
spqr_null, explicit:  0.34 sec, norm(A*N)  4e-14 size:  1.32 MB      
MATLAB null:         24.86 sec, norm(A*N)  9e-13 size: 45.73 MB      
all report dim(N) of 1577                                            
Matrix: N_reactome  size: 16559 by 10204 (transposed)                
spqr_null stats:                                                     
                       flag: 0                                       
                       rank: 9025                                    
                        tol: 1.1766e-10                              
      est_sval_upper_bounds: [0.6722 1.3042e-14]                     
      est_sval_lower_bounds: [0.0106 0]                              
    sval_numbers_for_bounds: [9025 9026]                             
         est_norm_A_times_N: 9.4695e-15                              
spqr_null, implicit: 0.95 sec, norm(A*N)  2e-13 size:  7.5 MB        
spqr_null, explicit: 3.53 sec, norm(A*N)  2e-13 size: 25.2 MB        
MATLAB null:       904.54 sec, norm(A*N)  2e-10 size: 96.2 MB        
all report dim(N) of 1179.