Routine svd from Matlab 7.6.0.324 (R2008a) used to calculate the singular values.

Matrix properties (click for a legend) | |

number of rows | 6,071 |

number of columns | 12,230 |

structural full rank? | yes |

structural rank | 6,071 |

numerical rank | 6,058 |

dimension of the numerical null space | 6,172 |

numerical rank / min(size(A)) | 0.99786 |

Euclidean norm of A | 15.917 |

calculated singular value # 6058 | 0.045648 |

numerical rank defined using a tolerance max(size(A))*eps(norm(A)) = | 2.1725e-011 |

calculated singular value # 6059 | 5.322e-015 |

gap in the singular values at the numerical rank: singular value # 6058 / singular value # 6059 | 8.5771e+012 |

calculated condition number | 4.162e+015 |

condest | -2 |

nonzeros | 35,632 |

# of blocks from dmperm | 1 |

# strongly connected comp. | 1 |

entries not in dmperm blocks | 0 |

explicit zero entries | 0 |

nonzero pattern symmetry | 0% |

numeric value symmetry | 0% |

type | real |

structure | rectangular |

Cholesky candidate? | no |

positive definite? | no |

author | M. Meketon |

editor | D. Gay |

date | 1990 |

kind | linear programming problem |

2D/3D problem? | no |

SJid | 512 |

UFid | 619 |

Additional fields | size and type |

b | full 6071-by-1 |

c | full 12230-by-1 |

lo | full 12230-by-1 |

hi | full 12230-by-1 |

z0 | full 1-by-1 |

Notes:

A Netlib LP problem, in lp/data. For more information send email to netlib@ornl.gov with the message: send index from lp send readme from lp/data The following are relevant excerpts from lp/data/readme (by David M. Gay): The column and nonzero counts in the PROBLEM SUMMARY TABLE below exclude slack and surplus columns and the right-hand side vector, but include the cost row. We have omitted other free rows and all but the first right-hand side vector, as noted below. The byte count is for the MPS compressed file; it includes a newline character at the end of each line. These files start with a blank initial line intended to prevent mail programs from discarding any of the data. The BR column indicates whether a problem has bounds or ranges: B stands for "has bounds", R for "has ranges". The BOUND-TYPE TABLE below shows the bound types present in those problems that have bounds. PROBLEM SUMMARY TABLE Name Rows Cols Nonzeros Bytes BR Optimal Value DFL001 6072 12230 41873 353192 B 1.12664E+07 ** BOUND-TYPE TABLE DFL001 UP Submitted by Marc Meketon. DFL001, says Marc Meketon, "is a 'real-world' airline schedule planning (fleet assignment) problem. This LP was preprocessed by a modified version of the KORBX(r) System preprocessor. The problem reduced in size (rows, columns, non-zeros) significantly. The row and columns were randomly sorted and renamed, and a fixed adjustment to the objective function was eliminated. The name of the problem is derived from the initials of the person who created it." Bob Bixby reports that the CPLEX solver (running on a Sparc station) finds slightly different optimal values for some of the problems. On a MIPS processor, MINOS version 5.3 (with crash and scaling of December 1989) also finds different optimal values for some of the problems. The following table shows the values that differ from those shown above. (Whether CPLEX finds different values on the recently added problems remains to be seen.) Problem CPLEX(Sparc) MINOS(MIPS) DFL001 1.1266396047E+07 ** David Gay reports: ** On an IEEE-arithmetic machine (an SGI 4D/380S), I (dmg) succeeded in getting MINOS 5.3 to report optimal objective values, 1.1261702419E+07 and 1.1249281428E+07, for DFL001 only by starting with LOAD files derived from the solution obtained on the same machine by Bob Vanderbei's ALPO (an interior-point code); starting from one of the resulting "optimal" bases, MINOS ran 23914 iterations on a VAX before reporting an optimal value of 1.1253287141E+07. When started from the same LOAD file used on the SGI machine, MINOS on the VAX reported an optimal value of 1.1255107696E+07. Changing the FEASIBILITY TOLERANCE to 1.E-10 (from its default of 1.E-6) led MINOS on the SGI machine to report "optimal" values of 1.1266408461E+07 and 1.1266402835E+07. This clearly is a problem where the FEASIBILITY TOLERANCE, initial basis, and floating-point arithmetic strongly affect the "optimal" solution that MINOS reports. On the SGI machine, ALPO with SPLIT 3 found primal: obj value = 1.126639607e+07 FEASIBLE ( 2.79e-09 ) dual: obj value = 1.126639604e+07 FEASIBLE ( 1.39e-16 ) Bob Bixby reports the following about his experience solving DFL001 with CPLEX: First, the value for the objective function that I get running defaults is 1.1266396047e+07, with the following residuals: Max. unscaled (scaled) bound infeas.: 4.61853e-14 (2.30926e-14) Max. unscaled (scaled) reduced-cost infeas.: 6.40748e-08 (6.40748e-08) Max. unscaled (scaled) Ax-b resid.: 4.28546e-14 (4.28546e-14) Max. unscaled (scaled) c_B-B'pi resid.: 8.00937e-08 (8.00937e-08) The L_infinity condition number of the (scaled) optimal basis is 213737. I got exactly the same objective value solving the problem in several different ways. I played a bit trying to get a better reduced-cost infeasibility, but that seems hopeless (if not pointless) given the c-Bpi residuals. Just as an aside, this problem exhibits very interesting behavior when solved using a simplex method. I ran reduced-cost pricing on it in phase I, with the result that it took 465810 iterations to get feasible. Running the default CPLEX pricing scheme, the entire problem solved in 94337 iterations (33059 in phase I) on a Sparcstation. Steepest-edge pricing (and a different scaling) took 25803 iterations. This is a nasty problem. Added to Netlib on 11 Oct. 1990 9 Jan. 1991: Bixby's remarks about DFL001 added to lp/data/readme.

Ordering statistics: | AMD |
METIS |

nnz(V) for QR, upper bound nnz(L) for LU | 9,529,914 | 7,249,767 |

nnz(R) for QR, upper bound nnz(U) for LU | 1,544,706 | 1,383,665 |

*Maintained by Leslie Foster, last updated 24-Apr-2009.*

Entries 5 through 14 in the table of matrix properties and the singular

value plot were created using SJsingular code. The other plots

and statistics are produced using utilities from
the SuiteSparse package.

Matrix color plot pictures by cspy, a MATLAB function in the CSparse package.