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In article <BtAx00.Lq9@news.cso.uiuc.edu> tcsg3046@uxa.cso.uiuc.edu (Timothy C. Sozen) writes:
>
>I would like to know if there is a way to find max and min eigenvalues 
>of a system of equations. That is, is there a way to find them without
>finding every eigenvalue and testing for max and min?
>
>Thanks in advance
>
>Tim
>
>PS My matrix is symmetric positive definite and 7-diagonal.

The standard procedure for this problem is to combine Sturm sequences,
which count the number of eigenvalues above a point on the line, with
interval bisection (perhaps accelerated by a root finder applied to
the determinant).  The Sturm sequence can be calculated by what
amounts to Gaussian elimination, which is cheap for band matrices.
After the eigenvalues have been found, you can calculate the
eigenvectors by the inverse power method.  I am appending some
references.

Pete Stewart

\begin{thebibliography}{1}

\bibitem{gupk:72}
K.~K. Gupta.
\newblock Solution of eigenvalue problems by {Sturm} sequence method.
\newblock {\em International Journal for Numerical Methods in Engineering},
  4:379--404, 1972.

\bibitem{gupk:74}
K.~K. Gupta.
\newblock Eigenproblem solution by a combined {Sturm} sequence and inverse
  iteration technique.
\newblock {\em Int. J. Numer. Meth. Eng.}, 8:877--911, 1974.

\bibitem{bamw:67}
R.~S.~Martin W.~Barth and J.~H. Wilkinson.
\newblock Calculation of the eigenvalues of a symmetric tridiagonal matrix by
  the method of bisection.
\newblock {\em Numerische Mathematik}, 9:386--393, 1967.
\newblock Also in \cite[pp.249--256]{wire:71}.

\bibitem{wire:71}
J.~H. Wilkinson and C.~Reinsch.
\newblock {\em Handbook for Automatic Computation. Vol.~II Linear Algebra}.
\newblock Springer, New York, 1971.

\end{thebibliography}