normal

$$(\omega S-H)G = 1$$ (159)

$$H = \left[ \begin{array}{ccccc} H_{00} & H_{01} & 0 & 0 & 0... ...& \ddots \\ 0 & 0 & 0 & \ddots & \ddots \end{array} \right]$$ (160)

$$S = \left[ \begin{array}{ccccc} S_{00} & S_{01} & 0 & 0 & 0... ...& \ddots \\ 0 & 0 & 0 & \ddots & \ddots \end{array} \right]$$ (161)

$$G = \left[ \begin{array}{ccccc} G_{00} & G_{01} & G_{02} & ... ... & & & G_{33} \\ \vdots & & & & \ddots \end{array} \right]$$ (162)

Use $G_{i0}$ column,

$$(\omega S_{00}-H_{00} )G_{00} + (\omega S_{01}-H_{01}) G_{10} = I$$ (163)

$$(\omega S_{10} -H_{10} )G_{00} + (\omega S_{11}-H_{11}) G_{10} +(\omega S_{12}-H_{12}) G_{20} = 0$$ (164)

$$(\omega S_{21}-H_{21}) G_{10} + (\omega S_{22} -H_{22}) G_{20} +(\omega S_{23} -H_{23}) G_{30} = 0$$ (165)

$$\vdots$$ (166)

The last two eqs are generalized as

$$(\omega S_{nn}-H_{nn}) G_{n0} = -(\omega S_{n,n-1}-H_{n,n-1}) G_{n-1,0} - (\omega S_{n,n+1}- H_{n,n+1}) G_{n+1,0}$$ (167)

rewrite the relation between the neighboring site.

$$g_{s0}^{-1} G_{00} = I + \alpha _0 G_{10}$$ (168)

$$g_{0}^{-1} G_{n0} = \beta _0 G_{n-1,0} + \alpha _0 G_{n+1,0}$$ (169)

The relationship between the second neighboring is

$$g_{s0}^{-1} G_{00} = I + \alpha _0 g_0 ( \beta _0 G_{0,0} + \alpha _0 G_{2,0} )$$

$$( g_{s0}^{-1} - \alpha _0 g_0 \beta _0) ) G_{00} = I + \alpha _0 g_0 \alpha _0 G_{2,0}$$

$$g_{s1}^{-1} G_{00} = I + \alpha _1 G_{20}$$ (170)

And

$$g_0^{-1} G_{n0} = \beta _0 g_0 ( \beta _0 G_{n-2,0} + \alpha _0 G_{n,0} )$$

$$+ \alpha _0 g_0 ( \beta _0 G_{n,0} + \alpha _0 G_{n+2,0} )$$

$$\{ g_0^{-1} - \beta _0 g_0 \alpha _0 - \alpha _0 g_0 \beta _0 \} G_{n0} = \beta _0 g_0 \beta _0 G_{n-2,0} + \alpha _0 g_0 \alpha _0 G_{n+2,0}$$ (171)

$$g_1^{-1} G_{n0} = b_1 G_{n-2,0} + \alpha _1 G_{n+2,0}$$ (172)

the relation between $2^i$th neighboring layer is

$$g_{si}^{-1} G_{00} = I + \alpha _i G_{2^i,0}$$ (173)

$$g_{i}^{-1} G_{n0} = \beta _i G_{n-2^i,0} + \alpha _i G_{n+2^i,0}$$ (174)

$g_{si},g_{i}, \alpha _i, \beta _i$ can be evaluated iteratively

$$g_{s,i+1}^{-1} = g_{s,i}^{-1} - \alpha _i g_i \beta _i$$ (175)

$$g_{i+1}^{-1} = g_{i}^{-1} - \beta _i g_i \alpha _i - \alpha _i g_i \beta _i$$ (176)

$$a_{i+1} = \alpha _i g_i \alpha _i$$ (177)

$$b_{i+1} = \beta _i g_i \beta _i$$ (178)

especially

$$G_{00} = g_{s,i} \vert_{i\rightarrow \infty}$$ (179)