• SJSU Singular Matrix Database
• Matrix group: Regtools

• Matrix: Regtools/heat_200
• Description: HEAT 200x200 Test problem: inverse heat equation.
• download as a MATLAB mat-file, file size: 40 KB. Use SJget(242) or SJget('Regtools/heat_200') 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 200 number of columns 200 structural full rank? yes structural rank 200 numerical rank 195 dimension of the numerical null space 5 numerical rank / min(size(A)) 0.975 Euclidean norm of A 0.35555 calculated singular value # 195 3.6249e-008 numerical rank defined using a tolerance max(size(A))*eps(norm(A)) = 1.1102e-014 calculated singular value # 196 3.449e-021 gap in the singular values at the numerical rank: singular value # 195 / singular value # 196 1.051e+013 calculated condition number 2.7654e+058 condest 3.5615e+152 nonzeros 20,100 # of blocks from dmperm 200 # strongly connected comp. 200 entries not in dmperm blocks 19,900 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 242 UFid -

 Additional fields size and type b full 200-by-1 x full 200-by-1

Notes:

```    Constructed by the call [A,b,x]= heat(200)

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')) 20,100 20,100 200 Cholesky flop count 2.7e+006 2.7e+006 2.0e+002 nnz(L+U), no partial pivoting 40,000 40,000 20,100 nnz(V) for QR, upper bound nnz(L) for LU 11,641 20,100 200 nnz(R) for QR, upper bound nnz(U) for LU 20,100 20,100 20,100

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.