• SJSU Singular Matrix Database
• Matrix group: Regtools

• Matrix: Regtools/heat_1000
• Description: HEAT 1000x1000 Test problem: inverse heat equation.
• download as a MATLAB mat-file, file size: 4 MB. Use SJget(244) or SJget('Regtools/heat_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? yes structural rank 1,000 numerical rank 588 dimension of the numerical null space 412 numerical rank / min(size(A)) 0.588 Euclidean norm of A 0.35515 calculated singular value # 588 5.592e-014 numerical rank defined using a tolerance max(size(A))*eps(norm(A)) = 5.5511e-014 calculated singular value # 589 5.4486e-014 gap in the singular values at the numerical rank: singular value # 588 / singular value # 589 1.0263 calculated condition number 2.6144e+233 condest Inf nonzeros 500,500 # of blocks from dmperm 1,000 # strongly connected comp. 1,000 entries not in dmperm blocks 499,500 explicit zero entries 0 nonzero pattern symmetry 0% numeric value symmetry 0% type real structure unsymmetric Cholesky candidate? no positive definite? no

 author Carasso, Elden editor Per Christian Hansen date 1982 kind ill-posed problem 2D/3D problem? no SJid 244 UFid -

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

Notes:

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

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

[A,b,x] = heat(n,kappa)

A first kind Volterra integral equation with [0,1] as
integration interval.  The kernel is K(s,t) = k(s-t) with
k(t) = t^(-3/2)/(2*kappa*sqrt(pi))*exp(-1/(4*kappa^2*t)) .
Here, kappa controls the ill-conditioning of the matrix:
kappa = 5 gives a well-conditioned problem
kappa = 1 gives an ill-conditioned problem.
The default is kappa = 1.

An exact soltuion is constructed, and then the right-hand side
b is produced as b = A*x.
```

 Ordering statistics: AMD METIS DMPERM+ nnz(chol(P*(A+A'+s*I)*P')) 500,500 500,500 1,000 Cholesky flop count 3.3e+008 3.3e+008 1.0e+003 nnz(L+U), no partial pivoting 1,000,000 1,000,000 500,500 nnz(V) for QR, upper bound nnz(L) for LU 284,041 500,500 1,000 nnz(R) for QR, upper bound nnz(U) for LU 500,500 500,500 500,500

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.