Lanczos Method
  Z. Bai and D. Day

The non-Hermitian Lanczos method is an oblique projection method (see §3.2) for solving the NHEP,

$$Ax=\lambda x \quad \mbox{and} \quad y^{\ast} A = \lambda y^{\ast},$$ (138)

where $A$ is a non-Hermitian matrix. For complex symmetric matrices ($A=A^T$ but $A \ne A^{\ast}$), a special Lanczos method is presented §7.11.

With two starting vectors $q_1$ and $p_1$, the Lanczos method builds a pair of biorthogonal bases for the Krylov subspaces ${\cal K}^j(A,q_1)$ and ${\cal K}^j(A^{\ast},p_1)$, provided that the matrix-vector multiplications $Az$ and $A^{\ast} z$ for an arbitrary vector $z$ are available. The inner loop uses two three-term recurrences. These recurrences use less memory and fewer memory references than the corresponding recurrences in the Arnoldi method discussed in §7.5. The Lanczos method provides approximations for both right and left eigenvectors. When estimating errors and condition numbers of the computed eigenpairs, it is crucial that both the left and right eigenvectors be available. However, there are risks of breakdown and numerical instability with the method, since it does not work with orthogonal transformations. This section defines the basic Lanczos method and its main properties, and presents algorithmic techniques that enhance the method's numerical stability and accuracy.