Matrix: VDOL/goddardRocketProblem_1

Description: goddardRocketProblem optimal control problem (matrix 1 of 2)

VDOL/goddardRocketProblem_1 graph
(undirected graph drawing)

VDOL/goddardRocketProblem_1 dmperm of VDOL/goddardRocketProblem_1

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  • download as a MATLAB mat-file, file size: 90 KB. Use UFget(2689) or UFget('VDOL/goddardRocketProblem_1') in MATLAB.
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    Matrix properties
    number of rows831
    number of columns831
    structural full rank?yes
    structural rank831
    # of blocks from dmperm45
    # strongly connected comp.1
    explicit zero entries0
    nonzero pattern symmetrysymmetric
    numeric value symmetrysymmetric
    Cholesky candidate?no
    positive definite?no

    authorB. Senses, A. Rao
    editorT. Davis
    kindoptimal control problem
    2D/3D problem?no

    Additional fieldssize and type
    bfull 831-by-1
    rownamefull 831-by-79
    mappingfull 831-by-1


    Optimal control problem, Vehicle Dynamics & Optimization Lab, UF       
    Anil Rao and Begum Senses, University of Florida                                                                  
    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.                            
    Goddard rocket maximum ascent optimal control problem is taken from    
    Ref.~\cite{goddard1920method}. The goal of the Goddard rocket maximum  
    ascent problem is to determine the state and the control that          
    maximize the final altitude of an ascending rocket. The state of the   
    system is defined by the altitude, velocity, and the mass of the       
    rocket and the control of the system is the thrust. The Goddard        
    rocket problem contains a singular arc where the continuous-time       
    optimality conditions are indeterminate, thereby the nonlinear         
    programming problem solver will have difficulty determining the        
    optimal control during the singular arc. In order to prevent this      
    difficulty and obtain more accurate solutions the Goddard rocket       
    problem is posed as a three-phase optimal control problem. Phase one   
    and phase three contains the same dynamics and the path constraints    
    as the original problem, while phase two contains an additional path   
    constraint and an event constraint. The specified accuracy tolerance   
    of $10^{-8}$ were satisfied after two mesh iterations. As the mesh     
    refinement proceeds, the size of the KKT matrices increases from 831   
    to 867.                                                                
      title={A Method of Reaching Extreme Altitudes.},                     
      author={Goddard, Robert H},                                          

    Ordering statistics:result
    nnz(chol(P*(A+A'+s*I)*P')) with AMD7,728
    Cholesky flop count7.9e+04
    nnz(L+U), no partial pivoting, with AMD14,625
    nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD59,030
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD98,397

    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.