• SJSU Singular Matrix Database
• Matrix group: Regtools

• Matrix: Regtools/gravity_1000
• Description: GRAVITY 1000x1000 Test problem: 1-D gravity surveying model problem
• download as a MATLAB mat-file, file size: 6 MB. Use SJget(240) or SJget('Regtools/gravity_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 45 dimension of the numerical null space 955 numerical rank / min(size(A)) 0.045 Euclidean norm of A 6.4592 calculated singular value # 45 1.1139e-012 numerical rank defined using a tolerance max(size(A))*eps(norm(A)) = 8.8818e-013 calculated singular value # 46 5.5492e-013 gap in the singular values at the numerical rank: singular value # 45 / singular value # 46 2.0073 calculated condition number 4.8668e+020 condest 1.1904e+021 nonzeros 1,000,000 # of blocks from dmperm 1 # strongly connected comp. 1 entries not in dmperm blocks 0 explicit zero entries 0 nonzero pattern symmetry symmetric numeric value symmetry symmetric type real structure symmetric Cholesky candidate? yes positive definite? no

 author Hansen editor Per Christian Hansen date 2002 kind ill-posed problem 2D/3D problem? no SJid 240 UFid -

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

Notes:

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

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

[A,b,x] = gravity(n,example,a,b,d)

Discretization of a 1-D model problem in gravity surveying, in which
a mass distribution f(t) is located at depth d, while the vertical
component of the gravity field g(s) is measured at the surface.

The resulting problem is a first-kind Fredholm integral equation
with kernel
K(s,t) = d*(d^2 + (s-t)^2)^(-3/2) .
The following three examples are implemented (example = 1 is default):
1: f(t) = sin(pi*t) + 0.5*sin(2*pi*t),
2: f(t) = piecewise linear function,
3: f(t) = piecewise constant function.
The problem is discretized by means of the midpoint quadrature rule
with n points, leading to the matrix A and the vector x.  Then the
right-hand side is computed as b = A*x.

The t integration interval is fixed to [0,1], while the s integration
interval [a,b] can be specified by the user. The default interval is
[0,1], leading to a symmetric Toeplitz matrix.

The parameter d is the depth at which the magnetic deposit is located,
and the default value is d = 0.25. The larger the d, the faster the
decay of the singular values.
```

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