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

Matrix properties (click for a legend) | |

number of rows | 500 |

number of columns | 500 |

structural full rank? | yes |

structural rank | 500 |

numerical rank | 492 |

dimension of the numerical null space | 8 |

numerical rank / min(size(A)) | 0.984 |

Euclidean norm of A | 0.35525 |

calculated singular value # 492 | 3.3503e-013 |

numerical rank defined using a tolerance max(size(A))*eps(norm(A)) = | 2.7756e-014 |

calculated singular value # 493 | 3.7152e-023 |

gap in the singular values at the numerical rank: singular value # 492 / singular value # 493 | 9.0181e+009 |

calculated condition number | 1.8493e+124 |

condest | 5.4564e+270 |

nonzeros | 125,250 |

# of blocks from dmperm | 500 |

# strongly connected comp. | 500 |

entries not in dmperm blocks | 124,750 |

explicit zero entries | 0 |

nonzero pattern symmetry | 0% |

numeric value symmetry | 0% |

type | real |

structure | unsymmetric |

Cholesky candidate? | no |

positive definite? | no |

author | Carasso, Elden |

editor | Per Christian Hansen |

date | 1982 |

kind | ill-posed problem |

2D/3D problem? | no |

SJid | 243 |

UFid | - |

Additional fields | size and type |

b | full 500-by-1 |

x | full 500-by-1 |

Notes:

Constructed by the call [A,b,x]= heat(500) where heat is from Regularization Tools. The description of heat from http://www2.imm.dtu.dk/~pch/Regutools/ is: [A,b,x] = heat(n,kappa) A first kind Volterra integral equation with [0,1] as integration interval. The kernel is K(s,t) = k(s-t) with k(t) = t^(-3/2)/(2*kappa*sqrt(pi))*exp(-1/(4*kappa^2*t)) . Here, kappa controls the ill-conditioning of the matrix: kappa = 5 gives a well-conditioned problem kappa = 1 gives an ill-conditioned problem. The default is kappa = 1. An exact soltuion is constructed, and then the right-hand side b is produced as b = A*x.

Ordering statistics: | AMD |
METIS |
DMPERM+ |

nnz(chol(P*(A+A'+s*I)*P')) | 125,250 | 125,250 | 500 |

Cholesky flop count | 4.2e+007 | 4.2e+007 | 5.0e+002 |

nnz(L+U), no partial pivoting | 250,000 | 250,000 | 125,250 |

nnz(V) for QR, upper bound nnz(L) for LU | 63,257 | 125,250 | 500 |

nnz(R) for QR, upper bound nnz(U) for LU | 125,250 | 125,250 | 125,250 |

*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.