
\subsubsection{Steepest descent method.}
The method can be used when only few lowest eigenvalues are needed. It is
based on the variational principle
\begin{equation}
\lambda[\mathbf{x}]\equiv
\frac{\mathbf{x}^TA\mathbf{x}}{\mathbf{x}^{T}\mathbf{x}}\ge\lambda_{0}\;,
\end{equation}
where $\lambda_0$ is the lowest eigenvalue.

We start some initial vector $\mathbf{x}$ and make
an improvement upon it by adding a (small) correction $\mathbf{x}\rightarrow
\mathbf{x}+\alpha\Delta\mathbf{x}$, where
$\Delta\mathbf{x}=(A-\lambda[\mathbf{x}])\mathbf{x}$, and $\alpha$
is a parameter.  For small $\alpha<0$ the new eigenvalue
\begin{equation}
\lambda[\mathbf{x} + \alpha\Delta \mathbf{x}]=
\lambda [\mathbf{x}] + 2\alpha \Delta \mathbf{x}^{T}\Delta \mathbf{x}
+ \alpha^{2}\Delta \mathbf{x}^{T}A\Delta \mathbf{x}
\end{equation}
will always be smaller than the
original unless we are at the solution ($\Delta \mathbf{x}=0$). The parameter $\alpha$
is chosen to minimize the new $\lambda$
\begin{equation}
\alpha = - \frac{\Delta \mathbf{x}^{T}\Delta \mathbf{x}}{\Delta
\mathbf{x}^{T}A\Delta \mathbf{x}}.
\end{equation}
The operation is repeated until the desired accuracy is reached. The
matrix is then deflated and the procedure started again.
%-->
%
%<strong>Problems</strong>
%<ol type=>
%<li> The spring constant (fjederkonstant) is to be determined from an
%experiment where measured were the periods of harmonic oscillations
%of different masses hung on the spring. The masses were m={0.0698,
%0.0743, 0.1194}&plusmn;0.0001 kg, the corresponding periods were
%T={0.78,0.80,1.00}&plusmn;0.01 s. Estimate the spring constant (with
%the error, of course).
%
%<li> (Non-obligatory) Implement QR/QL algorithm for matrix diagonalization.
%<li> Implement inverse iteration algorithm for finding
%eigenvectors/eigenvalues.

