title: How can I get eigenvalues and eigenvectors fast and accurate?
body: <p>I need to compute the eigenvalues and eigenvectors of a big matrix (about 1000*1000 or even more). Matlab works very fast but it does not guaranty accuracy. I need this to be pretty accurate (about 1e-06 error is ok) and within a reasonable time (an hour or two is ok).</p>
  <p>My matrix is symmetric and pretty sparse. The exact values are: ones on the diagonal, and on the diagonal below the main diagonal, and on the diagonal above it. Example:</p>
  <p><a href="https://i.stack.imgur.com/gPFuz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/gPFuz.png" alt="Example Matrix" /></a></p>
  
  <p>How can I do this? C++ is the most convenient to me.</p>