**Matrix: VDOL/goddardRocketProblem_1**

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

(undirected graph drawing) |

Matrix properties | |

number of rows | 831 |

number of columns | 831 |

nonzeros | 8,498 |

structural full rank? | yes |

structural rank | 831 |

# of blocks from dmperm | 45 |

# 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 831-by-1 |

rowname | full 831-by-79 |

mapping | full 831-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/goddardRocketProblem 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. @article{goddard1920method, title={A Method of Reaching Extreme Altitudes.}, author={Goddard, Robert H}, journal={Nature}, volume={105}, pages={809--811}, year={1920} }

Ordering statistics: | result |

nnz(chol(P*(A+A'+s*I)*P')) with AMD | 7,728 |

Cholesky flop count | 7.9e+04 |

nnz(L+U), no partial pivoting, with AMD | 14,625 |

nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD | 59,030 |

nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD | 98,397 |

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.
*