Adding restart for non converging Krylov algorithm

[kbidzhiev]

Summary

This PR updates the Lanczos implementation to use a residual-minimizing restart strategy for DMRG only. After each iteration the algorithm evaluates the residuals of the candidate vectors, selects the vector with the smallest residual, and restarts the search from that vector.

What changed

• Added residual-based selection across the current set of candidate vectors.
• Modified the restart logic so the next Lanczos run begins from the best (lowest-residual) vector instead of continuing from the last generated vector.
• Updated the iteration flow to support “pick-best-and-restart” without altering the external API/inputs.

exhaustive guide to numerical errors etc in Lanczos algo
Lectures on Lanczos with detail study of residual
https://www.stat.uchicago.edu/~lekheng/courses/302/demmel/demmch7.pdf

[a-quelle-pasqal]

Please also add the literature, I'm especially interested in the justification for
resid = (beta * y[-1]).abs() # == norm(op(ritz_vec) - ritz_value * ritz_vec)

[kbidzhiev]

Please also add the literature, I'm especially interested in the justification for resid = (beta * y[-1]).abs() # == norm(op(ritz_vec) - ritz_value * ritz_vec)

see Saad (1), Prop. 6.8 (Ch. 6, p. 131).

(1) Numerical Methods for Large Eigenvalue Problems, Y. Saad.
https://epubs.siam.org/doi/book/10.1137/1.9781611970739

[kbidzhiev]

Please also add the literature, I'm especially interested in the justification for resid = (beta * y[-1]).abs() # == norm(op(ritz_vec) - ritz_value * ritz_vec)

done in the comments in the code and in the head of the file

[elmerhej]

LGTM

[a-quelle-pasqal]

LGTM