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

Matrix properties (click for a legend) | |

number of rows | 100 |

number of columns | 100 |

structural full rank? | yes |

structural rank | 100 |

numerical rank | 48 |

dimension of the numerical null space | 52 |

numerical rank / min(size(A)) | 0.48 |

Euclidean norm of A | 6.4593 |

calculated singular value # 48 | 9.4596e-014 |

numerical rank defined using a tolerance max(size(A))*eps(norm(A)) = | 8.8818e-014 |

calculated singular value # 49 | 4.5878e-014 |

gap in the singular values at the numerical rank: singular value # 48 / singular value # 49 | 2.0619 |

calculated condition number | 5.4668e+018 |

condest | 4.6994e+019 |

nonzeros | 10,000 |

# of blocks from dmperm | 1 |

# strongly connected comp. | 1 |

entries not in dmperm blocks | 0 |

explicit zero entries | 0 |

nonzero pattern symmetry | symmetric |

numeric value symmetry | symmetric |

type | real |

structure | symmetric |

Cholesky candidate? | yes |

positive definite? | no |

author | Hansen |

editor | Per Christian Hansen |

date | 2002 |

kind | ill-posed problem |

2D/3D problem? | no |

SJid | 237 |

UFid | - |

Additional fields | size and type |

b | full 100-by-1 |

x | full 100-by-1 |

Notes:

Constructed by the call [A,b,x]= gravity(100) where gravity is from Regularization Tools. The description of gravity from http://www2.imm.dtu.dk/~pch/Regutools/ is: [A,b,x] = gravity(n,example,a,b,d) Discretization of a 1-D model problem in gravity surveying, in which a mass distribution f(t) is located at depth d, while the vertical component of the gravity field g(s) is measured at the surface. The resulting problem is a first-kind Fredholm integral equation with kernel K(s,t) = d*(d^2 + (s-t)^2)^(-3/2) . The following three examples are implemented (example = 1 is default): 1: f(t) = sin(pi*t) + 0.5*sin(2*pi*t), 2: f(t) = piecewise linear function, 3: f(t) = piecewise constant function. The problem is discretized by means of the midpoint quadrature rule with n points, leading to the matrix A and the vector x. Then the right-hand side is computed as b = A*x. The t integration interval is fixed to [0,1], while the s integration interval [a,b] can be specified by the user. The default interval is [0,1], leading to a symmetric Toeplitz matrix. The parameter d is the depth at which the magnetic deposit is located, and the default value is d = 0.25. The larger the d, the faster the decay of the singular values.

Ordering statistics: | AMD |
METIS |

nnz(chol(P*(A+A'+s*I)*P')) | 5,050 | 5,050 |

Cholesky flop count | 3.4e+005 | 3.4e+005 |

nnz(L+U), no partial pivoting | 10,000 | 10,000 |

nnz(V) for QR, upper bound nnz(L) for LU | 5,050 | 5,050 |

nnz(R) for QR, upper bound nnz(U) for LU | 5,050 | 5,050 |

*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.