• SJSU Singular Matrix Database
• Matrix group: Regtools

• Matrix: Regtools/tomo_2500
• Description: TOMO 2500x2500: Create a 2D tomography test problem.
• download as a MATLAB mat-file, file size: 1 MB. Use SJget(267) or SJget('Regtools/tomo_2500') in MATLAB.

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

 Matrix properties (click for a legend) number of rows 2,500 number of columns 2,500 structural full rank? yes structural rank 2,500 numerical rank 2,496 dimension of the numerical null space 4 numerical rank / min(size(A)) 0.9984 Euclidean norm of A 55.317 calculated singular value # 2496 0.0016308 numerical rank defined using a tolerance max(size(A))*eps(norm(A)) = 1.7764e-011 calculated singular value # 2497 1.21e-015 gap in the singular values at the numerical rank: singular value # 2496 / singular value # 2497 1.3477e+012 calculated condition number 2.6355e+017 condest Inf nonzeros 166,782 # of blocks from dmperm 1 # strongly connected comp. 1 entries not in dmperm blocks 0 explicit zero entries 0 nonzero pattern symmetry 3% 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 267 UFid -

 Additional fields size and type b full 2500-by-1 x full 2500-by-1

Notes:

```    Constructed by the call [A,b,x]= tomo(50)

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')) 2,655,352 2,628,222 Cholesky flop count 4.0e+009 3.9e+009 nnz(L+U), no partial pivoting 5,308,204 5,253,944 nnz(V) for QR, upper bound nnz(L) for LU 2,917,949 2,901,781 nnz(R) for QR, upper bound nnz(U) for LU 3,106,075 3,102,952

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.