• 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/heat_500
• Description: HEAT 500x500 Test problem: inverse heat equation.
• download as a MATLAB mat-file, file size: 487 KB. Use SJget(243) or SJget('Regtools/heat_500') in MATLAB.
• download in Matrix Market format, file size: 504 KB.
• download in Rutherford/Boeing format, file size: 169 KB.

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

 Matrix properties (click for a legend) number of rows 500 number of columns 500 structural full rank? yes structural rank 500 numerical rank 492 dimension of the numerical null space 8 numerical rank / min(size(A)) 0.984 Euclidean norm of A 0.35525 calculated singular value # 492 3.3503e-013 numerical rank defined using a tolerance max(size(A))*eps(norm(A)) = 2.7756e-014 calculated singular value # 493 3.7152e-023 gap in the singular values at the numerical rank: singular value # 492 / singular value # 493 9.0181e+009 calculated condition number 1.8493e+124 condest 5.4564e+270 nonzeros 125,250 # of blocks from dmperm 500 # strongly connected comp. 500 entries not in dmperm blocks 124,750 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 243 UFid -

 Additional fields size and type b full 500-by-1 x full 500-by-1

Notes:

```    Constructed by the call [A,b,x]= heat(500)

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')) 125,250 125,250 500 Cholesky flop count 4.2e+007 4.2e+007 5.0e+002 nnz(L+U), no partial pivoting 250,000 250,000 125,250 nnz(V) for QR, upper bound nnz(L) for LU 63,257 125,250 500 nnz(R) for QR, upper bound nnz(U) for LU 125,250 125,250 125,250

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.