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

Matrix properties (click for a legend) | |

number of rows | 4,900 |

number of columns | 4,900 |

structural full rank? | yes |

structural rank | 4,900 |

numerical rank | 4,897 |

dimension of the numerical null space | 3 |

numerical rank / min(size(A)) | 0.99939 |

Euclidean norm of A | 77.135 |

calculated singular value # 4897 | 0.00013581 |

numerical rank defined using a tolerance max(size(A))*eps(norm(A)) = | 6.9633e-011 |

calculated singular value # 4898 | 6.4091e-016 |

gap in the singular values at the numerical rank: singular value # 4897 / singular value # 4898 | 2.119e+011 |

calculated condition number | 5.1232e+017 |

condest | 7.6299e+016 |

nonzeros | 457,007 |

# of blocks from dmperm | 1 |

# strongly connected comp. | 1 |

entries not in dmperm blocks | 0 |

explicit zero entries | 0 |

nonzero pattern symmetry | 2% |

numeric value symmetry | 0% |

type | real |

structure | unsymmetric |

Cholesky candidate? | no |

positive definite? | no |

author | Hansen |

editor | Per Christian Hansen |

date | 2007 |

kind | ill-posed problem |

2D/3D problem? | no |

SJid | 268 |

UFid | - |

Additional fields | size and type |

b | full 4900-by-1 |

x | full 4900-by-1 |

Notes:

Constructed by the call [A,b,x]= tomo(70) where tomo is from Regularization Tools. The description of tomo from http://www2.imm.dtu.dk/~pch/Regutools/ is: [A,b,x] = tomo(N,f); This function creates a simple two-dimensional tomography test problem. A 2D domain [0,N] x [0,N] is divided into N^2 cells of unit size, and a total of round(f*N^2) rays in random directions penetrate this domain. The default value is f = 1. Each cell is assigned a value (stored in the vector x), and for each ray the corresponding element in the right-hand side b is the line integral along the ray, i.e. sum_{cells in ray} x_{cell j} * length_{cell j} where length_{cell j} is the length of the ray in the j-th cell. The matrix A is sparse, and each row (corresponding to a ray) holds the value length_{cell j} in the j-th position. Hence: b = A*x . Once a solution x_reg has been computed, it can be visualized by means of imagesc(reshape(x_reg,N,N)). The exact solution, reshape(x,N,N), is identical to the exact image in the function blur. Note that the code for tomo uses random numbers and repeated calls to tomo will produce different matrices.

Ordering statistics: | AMD |
METIS |

nnz(chol(P*(A+A'+s*I)*P')) | 10,405,178 | 10,048,489 |

Cholesky flop count | 3.1e+010 | 2.9e+010 |

nnz(L+U), no partial pivoting | 20,805,456 | 20,092,078 |

nnz(V) for QR, upper bound nnz(L) for LU | 11,242,706 | 7,780,797 |

nnz(R) for QR, upper bound nnz(U) for LU | 11,950,713 | 11,997,846 |

*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.