Infinite algorithm method

Consider a linear chain of N sites composed of four blocks, labelled i = 1. 4. At present we consider symmetric systems with open boundary conditions so that blocks 1 and 4, and blocks 2 and 3 are equivalent. This is illustrated in Fig. H.l. A general state of the block i is denoted by TOj) where m is a shorthand label for the quantum numbers of the block (e.g. the conserved quantum numbers, spin and charge, and any other state index). Block i has Ni sites and its Hilbert space is spanned by Mi states. 

Block 1 is augmented by block 2 to form the system block with Ng = N1+N2 sites and a Hilbert space of Mg = Mi M2 states. Similarly, block 4 is augmented by block 3 to form the environment block with Ne = N + A4 sites and a Hilbert space of Me = M3M4 states. 

If this procedure is repeated so that the left- and right- hand blocks are grown by sequentially augmenting them with the middle blocks, the Hilbert space size of these blocks would grow exponentially in size as a function of their physical size. The goal is therefore to truncate the system block Hilbert space so that Ms Ms Ml at each iteration. This goal is achieved by the DMRG method. 

A general system block state, ms)( ), is a direct product of states of blocks 1 and 2, 

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Fig. H.l. The component blocks for the infinite algorithm method of the DMRG technique for open boundary conditions.

Unfortunately, Maxwell s equations can be solved analytically for only a few simple canonical resonator structures, such as spheres (Stratton, 1997) and infinitely long cylinders of circular cross-sections (Jones, 1964). For arbitrary-shape microresonators, numerical solution is required, even in the 2-D formulation. Most 2-D methods and algorithms for the simulation of microresonator properties rely on the Effective Index (El) method to account for the planar microresonator finite thickness (Chin, 1994). The El method enables reducing the original 3-D problem to a pair of 2-D problems for transverse-electric and transverse-magnetic polarized modes and perform numerical calculations in the plane of the resonator. Here, the effective... 

In general case Eqs. (4.60) and (4.61) present infinite sets of the five-term (pentadiagonal) recurrence relations with respect to the index l. In certain special cases (t - 0 or a - 0), they reduce to three-term (tridiagonal) recurrence relations. In this section the sweep procedure for solving such relations is described. This method, also known as the Thomas algorithm, is widely used for recurrence relations entailed by the finite-difference approximation in the solution of differential equations (e.g., see Ref. 61). In our case, however, the recurrence relation follows from the exact expansion (4.60) of the distribution function in the basis of orthogonal spherical functions and free of any seal of proximity, inherent to finite-difference method. Moreover, in our case, as explained below, the sweep method provides the numerical representation of the exact solution of the recurrence relations. 

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