Matrix properties
number of rows	2,380,515
number of columns	2,380,515
nonzeros	26,515,867
structural full rank?	yes
structural rank	2,380,515
# of blocks from dmperm	1
# strongly connected comp.	1
explicit zero entries	0
nonzero pattern symmetry	symmetric
numeric value symmetry	symmetric
type	real
structure	symmetric
Cholesky candidate?	yes
positive definite?	no

author	A. Bodendiek
editor	T. Davis
date	2012
kind	model reduction problem
2D/3D problem?	yes

Notes:

Curl-Curl operator of 2nd order Maxwell's equations, A. Bodendiek    
                                                                     
From Andre Bodendiek, Institut Computational Mathematics,            
TU Braunschweig                                                      
                                                                     
The following matrix collection consists of the curl-curl-operator   
of a second-order Maxwell's equations with PEC boundary conditions,  
i.e. E x n = 0, where E and n denote the electric field strength     
and the unit outer normal of the computational domain. The           
curl-curl-operator has been discretized using the Finite Element     
Method with first-order Nedelec elements resulting in the weak       
formulation                                                          
                                                                     
   1/mu0 ( curl E, curl v ),                                         
                                                                     
where v resembles a test function of H(curl) and                     
mu0 = 1.25 1e-9 H / mm denotes the magnetic permeability of vacuum,  
see [Hipt02].                                                        
                                                                     
In general, the underlying model problem of Maxwell's equations      
results from a Coplanar Waveguide, which will be considered for      
the analysis of parasitic effects in the development of new          
semiconductors. Since the corresponding dynamical systems are often  
high-dimensional, model order reduction techniques have become an    
appealing approach for the efficient simulation and accurate analysis
of the parasitic effects. However, different kinds of model order    
techniques require the repeated solution of high-dimensional linear  
systems of the original model problem, see [Bai02,An09]. Therefore,  
the development of efficient solvers resembles an important task     
in model order reduction.                                            
                                                                     
Each matrix CurlCurl_ consists of a different number of degrees  
of freedom, given in the following table:                            
                                                                     
 = 0:   11083                                                    
 = 1:  226451                                                    
 = 2:  806529                                                    
 = 3: 1219574                                                    
 = 4: 2380515                                                    
                                                                     
References.                                                          
                                                                     
@ARTICLE{Bai02,                                                      
  author = {Z. Bai},                                                 
  title = {Krylov subspace techniques for reduced-order modeling     
    of large-scale dynamical systems},                               
  journal = {Applied Numerical Mathematics},                         
  year = {2002},                                                     
  volume = {43},                                                     
  pages = {9--44},                                                   
  number = {1--2}                                                    
}                                                                    
                                                                     
@ARTICLE{Hipt02,                                                     
  author = {R. Hiptmair},                                            
  title = {Finite elements in computational electromagnetism},       
  journal = {Acta Numerica, Cambridge University Press},             
  year = {2002},                                                     
  pages = {237-339}                                                  
}                                                                    
                                                                     
@BOOK{An09,                                                          
  title = {Approximation of {L}arge-{S}cale {D}ynamical {S}ystems},  
  publisher = {Society for Industrial Mathematics},                  
  year = {2009},                                                     
  author = {Athanasios C. Antoulas}                                  
}                                                                    

Ordering statistics:	result
nnz(chol(P*(A+A'+s*I)*P')) with AMD	6,380,637,405
Cholesky flop count	1.2e+14
nnz(L+U), no partial pivoting, with AMD	12,758,894,295
nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD	17,110,956,357
nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD	31,714,179,756

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.