• SJSU Singular Matrix Database
• Matrix group: Regtools
• Click here for a description of the Regtools group.
• Click here for a list of all matrices
• Click here for a list of all matrix groups

• Matrix: Regtools/tomo_4900
• Description: TOMO 4900x4900: Create a 2D tomography test problem.
• download as a MATLAB mat-file, file size: 4 MB. Use SJget(268) or SJget('Regtools/tomo_4900') in MATLAB.
• download in Matrix Market format, file size: 5 MB.
• download in Rutherford/Boeing format, file size: 5 MB.

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.