), &(do), A ) is the (Krylov) MGMRES: A Generalization of GMRES for Solving Large Sparse Nomymetric Linear Systems

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1 I. MGMRES: A Generalzaton of GMRES for Solvng Large Sparse Nomymetrc Lnear Systems Davd M.Young and Jen Yuan Chen Center for Numercal Analyas The Unversty of Texaa at Aurrtn Austn, Texas We are concerned wth the soluton of the lnear system (1): Au = b, where A s a real square nonsngular matrx whch a large, sparee and nonsymmetrc. We coneder the u8e of Krylov subspace methods. We fnt choose an ntal approxmaton do)to the soluton 5 = A'lb of (1). We also choose an auxlary mbtrx 2 whch s nonsngular. For n = 1,2, we determne such that t ~ ( ~ ~) ( ~ ) & ~ ( ra) { ~where ), &(do), A ) s the (Krylov) eubspace spanned by the Krylov vectors do), Ado),.,*,An"(") and where do)= b Ado). If Z A s SPD we ale0 requre that (dn)t, ZA(u(") a)) be mnmzed. I, on the other hand, ZA s not SPD,then we requre that the... Galerkn condton, (Zr("),v)= 0, be aatefed for all vek,(r(o),a),where b Au("). Wth the GMRES method, whch was developed by Saad and Schultz [1986],snd whch has for many years been used extenewly for solvng large sparse nonsymmetrc systems one lets Z = AT. One generates set of mutually orthogonal vectors do), tu('),...tu(") such that do) = do)and such that Sp(w(O),..., = Kk(r('),Af for k = D,1,2,...,n, To do ths, for each s we let wtl) be a lnear combnaton of Awfbl),d k l ),...,do). Eext, for each n we choose cfl, c?),. BO that dn) = u(o) c?)w(*) C ~ ~ W ( ~ and ' ) so that (~("1, d"))s mnmzed, The cy) are determned by solvng a related system of lnear equaton8 n the lea& squares sense, Ths s done n a etable mnner usng Gvens rotatons. In ths paper we consder L generalzaton of GMRES. Ths generalzed,("i method, whch we refer to a8 "MGMRES", s very smiar to GMRES cxcept that we let 2 = ATY where Y s a nansngular matrx whch s symmetrc be mutually orthogonal but not necessarly SPD. We requre that the

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4 wth respect to Y. Of course f Y e not SPD t s pasahle that the process of generatng the ut(9 may break down. Also, unlesa Y s SPD we must replace the mnmzaton condton on (dn),dn)) by a Galerkn condton whch q u r e a that (2rtn),d)) = 0 for = O, l,..,, n 1, The d e t d n a t o n of the coeffcents cp) can be c d e d out usng Gvem ro tatons, as n the case of GMRES, though the overall p m d u e s somewhat more complcated. It can, however be shown that, for gven tag and do), one can unquely determne u(')), ~ ( " ' 1 provded that the pro~ 6 of8 computng w@),&),,,tu[n*ol) does not break down and provded that for n = 1,2,...,IZ* there actually exst8 a unque vector dn)such that ~ ( n) d0)trkn(r(0), A) and such that the Galerkn condton a satsfed, * The MGMRES algorthm s consderably eprplfed f YA ae well aa Y s symmetrc, Under ths assumpton one can determne to(") n terms of..,,do) aa would be w(n3) and w(n3) nstead of n ternzs of requred n the general m e. The determnaton of the coeecente cy' whch are nvolved n the Galerkn condton s also coneder&blysmplfed. An exampie o a c a e where Y and YA are symmetrc e the "double system" whch correaponda to the Lanczos method for solvng (1). Thus gven the lnear system (1) one can conader the double system { A ) ( u } = {a} where for some 6 and 2 we have.. I We also choose {*I=( O I I 0) (Y)and (Y)(A)hte symmetrc. The applcaton of MGMRES wth Y = (Y}to the double system yelds the 'LANGMRES" method gven Evdently by Young and Chen [1994], An mportant feature of the GMRES algorthm Sa that one can determne (dn),dn)) or a gven n and test for convergence wthout actually carryng out the complete GMRES procese to determne t("). Thus, one can compute (dn), d"))for each teraton and only actually compute dn)when (dn),dn)) s smaller than a prescrbed tolerance level,?ve descrbe a smlar procedure for MGMRES, For each n we frst compute the resdual F(") for ORTHORES

5 (Y)by determnng the scalng factor cn such that F(") = GW("), The resdual dn)for MGMRES can be determned from?(n) by a short aeres of elmentary vectm operatons. No matrxvector operatons or nner products are requred tu get rnl. References (11 S a d, Youcef and Schultz, Martn R , "GMRES,A Generalzed Mnmum Resdual AIgarthm far Solvng Non Symmetrc Lnear Sptern=, SIAM J. Sc. Stat. Comput, 7, [2] Young, Davd M,and Chen, Jen Yuan [1994], "LANGMRES, An Alternatve t o the Bconjugate Gradent Algorthm for Solvng Large Sparse Konqmmetrc Lnear Systems", to appear n the proceedngs of a con ference held at h'orth Carolna State Unversty n December 1993 n honor of Cornelus Lanczos.