• SJSU Singular Matrix Database
• Matrix group: Regtools

• Matrix: Regtools/i_laplace_1000
• Description: I_LAPLACE 1000x1000 Test problem: inverse Laplace transformation.
• download as a MATLAB mat-file, file size: 2 MB. Use SJget(248) or SJget('Regtools/i_laplace_1000') 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 1,000 number of columns 1,000 structural full rank? no structural rank 581 numerical rank 43 dimension of the numerical null space 957 numerical rank / min(size(A)) 0.043 Euclidean norm of A 7.5262 calculated singular value # 43 1.2196e-012 numerical rank defined using a tolerance max(size(A))*eps(norm(A)) = 8.8818e-013 calculated singular value # 44 5.3894e-013 gap in the singular values at the numerical rank: singular value # 43 / singular value # 44 2.2629 calculated condition number Inf condest Inf nonzeros 301,876 # of blocks from dmperm 22 # strongly connected comp. 276 explicit zero entries 0 nonzero pattern symmetry 67% numeric value symmetry 0% type real structure unsymmetric Cholesky candidate? no positive definite? no

 author Varah editor Per Christian Hansen date 1983 kind ill-posed problem 2D/3D problem? no SJid 248 UFid -

 Additional fields size and type b full 1000-by-1 x full 1000-by-1

Notes:

```    Constructed by the call [A,b,x]= i_laplace(1000)

where i_laplace is from Regularization Tools. The description of
i_laplace from http://www2.imm.dtu.dk/~pch/Regutools/ is:

[A,b,x,t] = i_laplace(n,example)

Discretization of the inverse Laplace transformation by means of
Gauss-Laguerre quadrature.  The kernel K is given by
K(s,t) = exp(-s*t) ,
and both integration intervals are [0,inf).

The following examples are implemented, where f denotes
the solution, and g denotes the right-hand side:
1: f(t) = exp(-t/2),        g(s) = 1/(s + 0.5)
2: f(t) = 1 - exp(-t/2),    g(s) = 1/s - 1/(s + 0.5)
3: f(t) = t^2*exp(-t/2),    g(s) = 2/(s + 0.5)^3
4: f(t) = | 0 , t <= 2,     g(s) = exp(-2*s)/s.
| 1 , t >  2

The quadrature points are returned in the vector t.
```

 Ordering statistics: AMD METIS nnz(chol(P*(A+A'+s*I)*P')) 201,905 201,889 Cholesky flop count 4.5e+007 4.5e+007 nnz(L+U), no partial pivoting 402,810 402,778 nnz(V) for QR, upper bound nnz(L) for LU 213,086 170,188 nnz(R) for QR, upper bound nnz(U) for LU 263,450 263,450

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.