1420 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



8.2 Matrix description of the successive approximations 



From the developments of Section IV, we know i,o{t), i„o(t) and ino'it) 

 or what is equivalent, the vectors Iro , Ino and I„o'. The first approxi- 

 mation takes into account the effect of i„o{t) and t„u'(0 on ir{t). [See 

 eciuation (l.c) and (l.d)]. The time functions i„o(0 and z„o'(0 affect the 

 system R only during the interval (0, r) . Therefore we must consider the 

 vector G(I„o + /„o') which corresponds to the excitation of the resonant 

 circuit. Since the opening of the switch after a closure time r forcibly 

 brings ir{t) to zero we have 



In = GFG{InO + Ino'), (29) 



where the matrix G has been defined above and the matrix F is a diago- 

 nal matrix whose diagonal elements Fa (/.' = ••• — 1, 0, +1 •••) are 

 defined by Fk = F(jco, + /27rA-). Note that (28) implies that | F^ | ^ 1 

 for all A-'s. It should be kept in mind that Iro + In is the first approxima- 

 tion to the exact Ir{p)- 



The next iteration is obtained by first taking into account the effect 

 of In on the rest of the network: 



/„i = E In , 



(30) 



Inl = E In , 



where E is a diagonal matrix whose elements Ek {k = • • ■ , — 1, 0, +1, 

 • • •) are defined by Ek = E{jus + 27rAy), and then the effects of /„2 

 and In2 on /, , that is, 



1,2 = GFG{I,a + /„i'), (31) 



combining (30) and (31), hi = 2G F G E In ■ A repetition of the same 

 procedure would lead to la — 2 G F G E Ir2 , and in general Im + i = 



2GFGE Irn. 



Since the nth. approximation to Ir(p) is given by the sum ^ILoIrk , 

 the successive approximation scheme will be conA'ergent only if the series 



converges. This will be the case if and only if the series 



[\ -\-2GFGE -^ ■■■ + (2 G F G F)" + • • •]/,! (32) 



converges. 



