Algorithm for Non-Hermitian

An Implementation of the Look-Ahead Lanczos Algorithm for Non-Hermitian Matrices. 

Freund, R. W, and Malhotra, M. (1997) A block QMR algorithm for non-Hermitian linear S5rstems with multiple right-hand sides. Linear Algebra Appl, 254,119-157. 

The formal properties of operator L eq 2.18 (known as the symplectic structure ) allow the introduction of a variational principle eq D3, " a scalar product (eq Bl), and ultimately to reduce the original non-Hermitian eigenvalue problem (eq 2.18) to the equivalent Hermitian problem which may be solved using standard numerical algorithms (Appendices B—E). For example, F is a Hermitian operator. Lowdin s symmetric orthogonalization procedure " " leads to the Hermitian eigenvalue problem as well (eq E5), which may be subsequently solved by Davidson s algorithm (Appendix E). The spectral transform Lanczos method developed by Ruhe and Ericsson is another example of such transformation. 

In practice, this approach is numerically unstable for electron structure problems because the weights fluctuate strongly due to the Coulomb potential. Numerically stable and efficient algorithms are obtained with the importance sampling transformation. Let (f>(x) be an approximation to the ground state wave function ipoix) that has been obtained from other methods. (j> is assumed to be real like all other wave functions in this review. The operator H = (p x)H(j) xY —Eref is non-Hermitian and has the eigenfunctions fi x) = i/ fc(x) (x) and the shifted eigenvalues Ek — Ere/- The function y(t, x) = (j) x) l/ t, x) solves the equation... 

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