Name of the matrix:

Exponential matrix

Pattern:

 

$$a_{ij} = a_{i-j} = \frac{1+\sigma\vert i-j\vert}{\sigma} \; {\mathrm{e}}^{-\sigma\vert i-j\vert}, \qquad i,j=0,1,\ldots$$

Properties:

symmetric, positive definite, the asymptotic condition number is

$${\mathrm{cond}}(A) = \frac{p(\sigma)+2q(\sigma)}{p(\sigma)-2q(\sigma)} \; \left(\frac{1+{\mathrm{e}}^{-\sigma}}{1-{\mathrm{e}}^{-\sigma}}\right)^4$$

where

$$\begin{cases} p(\sigma) = \frac{1}{\sigma} \left( 1-{\mathrm... ...1 - \frac{1}{\sigma} \right) {\mathrm{e}}^{-\sigma} \end{cases}$$

Source/utilization:

test for the solution of linear systems

References:

 
C.V.M. van der Mee, Z.M. Nashed and S. Seatzu, Sampling expansions in unitarily translation invariant reproducing kernel Hilbert spaces, Adv. Comput. Math. 19(4), 355-372 (2003).
C.V.M. van der Mee and S. Seatzu, A method for generating infinite positive self-adjoint test matrices and Riesz bases, SIAM J. Matr. Anal. Appl., to appear.

Matlab:

t=tptest(2,n,sigma)