The conjugate-gradient m ethods (H estenes, 1980), (Sarkar et al, 1988) are iterative m ethods of solving m atrix-equations w hose convergence are in theory sure. There are m any d ifferen t co njugate-gradient m eth o d s to choose from . Some conjugate-gradient m ethods req u ire th e m atrix in the equation to be positive-definite. The m atrices in electrom agnetic scattering problem s are n o t positive-definite, a n d for the non-positive-definite case a suitable conjugate-gradient m ethod to use is given in (H estenes, 1980, eqn. 12.7(a) - (d), p. 297). W e w ill refer to this m eth o d as the least-square- conjugate-gradient (LSCG) m ethod.
In sp ite of the th eo re tic al a ssu ra n ce of co n v erg en ce, it is n o t uncom m on to find in the literatu re references to the iteratio n diverging (Peterson an d M ittra, 1984), (Peterson and M ittra, 1985), (Sarkar et al, 1988). We have ourselves been a p p ly in g the LSCG m eth o d to the problem of scattering from ro u g h surfaces, an d have fo u n d th a t for larg e surfaces convergence is n o t sure. The LSCG m ethod proceeds by generating at each iteration a conjugate-vector th at satisfies som e orthogonality properties in theory. The convergence is sure by virtue of these properties. H ow ever, the conjugate-vectors are g e n erate d recu rsiv ely , an d as a consequence of ro u n d in g errors, m ay fail to satisfy their theoretical p ro p erties (Scott and Peterson, 1988). In this section w e use Gram -Schm idt orthogonalization to enforce the o rth o g o n ality p ro p erties at each iteration. In fact, a G ram - S chm idt co n ju g ate-g rad ien t m eth o d for the p o sitiv e-d efin ite case w as giv en so m etim e ago by H esten es (1980), an d w e hav e a d a p te d this p rocedure for the non-positive-definite case. We call this m odified LSCG m eth o d the G ram -Schm idt, least-sq u are, co n ju g ate-g rad ien t (GS-LSCG) m ethod. We w ill show that in the absence of ro u n d in g errors, the GS-LSCG m eth o d a n d the LSCG m e th o d w ill d eterm in e th e sam e sequence of conjugate-vectors. In this resp ec t the GS-LSCG m e th o d is n o t a new conjugate-gradient m ethod. H ow ever, in the presence of ro u n d in g errors w e have fo u n d the GS-LSCG m eth o d to be very m uch less susceptible to ro u n d in g errors th an the LSCG m ethod.
The LSCG an d the GS-LSCG m ethods are applied to solving the m atrix- eq u atio n
Lu = f. (312)
In this stu d y w e shall only consider the case w here the o p erator L is an N by N , n o n -sin g u lar m atrix. The conjugate-gradient m eth o d s are iterative m eth o d s of solving the m atrix -eq u atio n (312). A t the iteration, the m ethods determ ine a conjugate-vector pj^ in the dom ain of L, an d a vector
Lpi^ in the range of L. The estim ate uj^ to the so lu tio n of the m atrix- equation is determ ined as an expansion of the vectors pj, j = 0 , k-1.
k-1
Z Pj (313)
j=0
The coefficients aj, j = 0, k-1, of the expansion (3-13) are calculated to force the error
r]^ = f - Luk, (314)
betw een f an d L u^ orthogonal to the vectors Lpj, j = 0 , k-1, i.e.
< r]^, Lpj > = 0, for j = 0,..., k-1. (3-15)
This is the n atu ral criterion to choose for determ ining the coefficients aj, j = 0, ..., k-1, for the follow ing reason. A ny set of N , linearly in d ep en d en t vectors in the range R(L) of L are a basis spanning R(L) (Kreysig, 1978). At the N th iteratio n of the conjugate-gradient m ethod, the N vectors Lpj, j = 0, ..., N-1, in the range of L w ill have been determ ined. M oreover, as w e will show later these vectors are linearly independent, and, therefore, sp an R(L). A t the N^^ iteration, the estim ate ujq to the solution of the m atrix- e q u atio n (312), the difference betw een the vectors f an d L u ^ is the error r%q. W ith the coefficients of the expansion determ ined according to (3-15), the error rjq is either orthogonal to the space sp an n ed by Lpj, j = 0,..., N-1, else it is zero. H ow ever, since the vectors Lpj, j = 0 ,..., N-1, sp an R(L), the only vector in R(L) th at can satisfy (3 15) is the zero vector. W ith zero on the left-hand-side (LHS) of (314), u N solves the m atrix -eq u atio n (312) u n iq u ely fo r n o n -sin g u la r L. In th is m an n e r, th e co n ju g a te -g rad ie n t m eth o d determ ines the exact solution of the m atrix-equation in at m ost N iterations.
The condition (315) can be w ritten in term s of the coefficients aj, i = 0, k-1, by substituting the right-hand-side (RHS) of (314) into the LHS of (315),
<rj^, Lpj> = <f, Lpj> - <Luj^, Lpj>
k-1
<f, Lpj> - % aj <Lpi, Lpj>.= 0, for j = 0 , k-1. (316) i=0
The second line of (316) is o b tain ed from the first line b y u sin g the expansion (313) for the solution uk- As (316) stands, the coefficients aj, j = 0, ..., k-1, are them selves the so lu tio n of a system of lin ear equations. H ow ever, the vectors pj, j = 0 ,..., k-1, determ ined by the conjugate-gradient m e th o d are te rm e d " c o n ju g a te -v e c to rs " , b e ca u se th e y sa tisfy the orthogonality p roperty
< Lpi, Lpj > = 0, for i j. (317)
This p ro p erty diagonalizes (316). It also guarantees th at the vectors Lpj, j = 0 ,..., k-1, are linearly independent, as w e h ad required earlier. A pplying the p ro p erty (3-17) to the RHS of (316), the coefficients aj, j = 0, ..., k-1, are determ ined according to
< r^, Lpj > = <f, Lpj> - aj I I Lpj I I ^ = 0, for j < k. (318)
The solution
3j = < f, Lpj > / I I Lpj I 12, for j < k, (3-19)
solves (318), as m ay be v erified b y su b stitu tio n . A n im p o rta n t fact to recognize from (319) is th a t only the vector p]^ is used to com pute aj.. Therefore, if w e have already generated the sequence of vectors pj, j = 0,...,
k-2, a n d by some m eans generate a new vector we need only deduce the coefficient to augm ent the solution (313) according to
u k = u k -l+ ak_i Pk-l- (3 20)
Thus w e have an iterative m ethod of solving (3-12). Similarly, the error r^ is determ ined recursively