Matrix properties
number of rows	18,368
number of columns	18,368
nonzeros	223,801
structural full rank?	yes
structural rank	18,368
# 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?	no
positive definite?	no

author	B. Senses, A. Rao
editor	T. Davis
date	2015
kind	optimal control problem
2D/3D problem?	no

Additional fields	size and type
b	full 18368-by-1
rowname	full 18368-by-82
mapping	full 18368-by-1

Notes:

Optimal control problem, Vehicle Dynamics & Optimization Lab, UF       
Anil Rao and Begum Senses, University of Florida                       
http://vdol.mae.ufl.edu                                                
                                                                       
This matrix arises from an optimal control problem described below.    
Each optimal control problem gives rise to a sequence of matrices of   
different sizes when they are being solved inside GPOPS, an optimal    
control solver created by Anil Rao, Begum Senses, and others at in VDOL
lab at the University of Florida.  This is one of the matrices in one  
of these problems.  The matrix is symmetric indefinite.                
                                                                       
Rao, Senses, and Davis have created a graph coarsening strategy        
that matches pairs of nodes.  The mapping is given for this matrix,    
where map(i)=k means that node i in the original graph is mapped to    
node k in the smaller graph.  map(i)=map(j)=k means that both nodes    
i and j are mapped to the same node k, and thus nodes i and j have     
been merged.                                                           
                                                                       
This matrix consists of a set of nodes (rows/columns) and the          
names of these rows/cols are given                                     
                                                                       
Anil Rao, Begum Sense, and Tim Davis, 2015.                            
                                                                       
VDOL/lowThrust                                                         
                                                                       
Low-thrust orbit transfer optimal control problem is taken from        
Ref.~\cite{betts2010practical}. The goal of the low-thrust orbit       
transfer problem is to determine the state and the control that        
minimize the fuel consumption during the orbit transfer of a           
spacecraft that starts from a low-earth orbit and terminates at the    
geostationary orbit via low-thrust propulsion systems.  The highly     
nonlinear dynamics of the low-thrust orbit transfer problem is given   
in modified equinoctial elements (state of the system) and the thrust  
direction (control of the system).  Furthermore, the low-thrust        
optimal control problem is a badly scaled problem because of the       
small thrust-to-initial-mass ratio, that is typically on the order of  
$O(10^{-4})$, and the long orbit transfer duration. Badly scaling of   
the problem leads to a lot of delayed pivots. The specified accuracy   
tolerance of $10^{-8}$ were satisfied after thirteen mesh iterations.  
As the mesh refinement proceeds, the size of the KKT matrices          
increases from 584 to 18476.                                           
                                                                       
@book{betts2010practical,                                              
  title={Practical Methods for Optimal Control and Estimation Using    
     Nonlinear Programming},                                           
  author={Betts, John T},                                              
  volume={19},                                                         
  year={2010},                                                         
  publisher={SIAM Press},                                              
  address = {Philadelphia, Pennsylvania},                              
}                                                                      

Ordering statistics:	result
nnz(chol(P*(A+A'+s*I)*P')) with AMD	359,077
Cholesky flop count	1.1e+07
nnz(L+U), no partial pivoting, with AMD	699,786
nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD	17,277,774
nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD	38,180,696

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

Maintained by Tim Davis, last updated 04-Jun-2015.
Matrix pictures by cspy, a MATLAB function in the CSparse package.
Matrix graphs by Yifan Hu, AT&T Labs Visualization Group.