Lanczos reduction

T. J. Park and J.C. Light Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85 (1986) 5870-5876... 

Yang-Mills fields, 249-250, 255-257 Lagrangian multiplier, conical intersection location, 488-489, 565 Laguerre polynomials, Renner-Teller effect, triatomic molecules, 589—598 Lanczos reduction ... 

In fact, the Lanczos reduction was originally proposed as a tridiagonalization scheme, predating the Givens and Householder methods. Unlike the latter methods, however, the Lanczos method is recursive. This means that the dimensionality of the T matrix is determined by the number of steps of the Lanczos recursion (K), which is usually much smaller than the dimensionality of the Hamiltonian matrix (N) in real calculations. 

This reduction (called also the QT decomposition) can be accomplished by the Lanczos algorithm (E.29) ... 

The Lanczos method is based on generating the orthonormal basis in Krylov space Ki =span c, Ac, A c by applying the Gram-Schmidt orthogonaliza-tion process, described in Appendix A. In matrix notations this approach is associated with the reduction of the symmetric matrix A to a tridiagonal matrix and also with the special properties of T/,. This reduction (called also QT decomposition) is described by the formula... 

In contrast, partial least squares (PLS) [73] reduces the calculation time by a different type of reduction of the E-matrix by use of the Lanczos bidiagonalisation. 

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