for Linear Systems With Strictly Diagonally Dominant Matrix

Transcription

1 MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 152 OCTOBER 1980, PAGES On an Accelerated Overrelaxaton Iteratve Method for Lnear Systems Wth Strctly Dagonally Domnant Matrx By M. Madalena Martns* Abstract. We consder a lnear system Ax = b of n smultaneous equatons, where A s a strctly dagonally domnant matrx. We get bounds for the spectral radus of the matrx., r, uj',, whch s accocated wth the Accelerated Overrelaxaton teratve method (AOR). Suffcent condtons for the convergence of that method wll be gven, whch mprove the results of Theorem 3, Secton 4 of [2], appled to ths type of matrces. 1. Introducton. We want the soluton x of a lnear system (1.1) Ax = b, where A s an (n, r) real-matrx and x and b are n-real-vectors. We assume A s strctly dagonally domnant. There are several mportant teratve methods for the approxmaton of (1.1). We wll take the (AOR) of [1]. For that, the matrx s expressed as the matrx sum (1.2) A=I-E-F, where / s the dentty and E and F are respectvely strctly lower and upper trangular n, r) matrces. From the (AOR) method we can wrte the followng equatons: (1.3) X(+D = (/ _ rey [(l - coy + (co - r)^ + cof]x(,) + co(/ - retxb, 1 = 0, 1,2,- So, Lr w wll be the pont-(aor)-matrx assocated wth the matrx A, (1.4) Lr>w = (/ - re)~x [(1 - wx + (w - r)e + cof], and PÍLr u) the correspondng spectral radus. 2. Bounds for plr ^). As we assume A strctly dagonally domnant, A s nonsngular and verfes n (2.1) \\> Z Ityl =l'2' >" Receved October 18, 1979; revsed March 18, Mathematcs Subject Classfcaton. Prmary 65F10. *Ths work was done when the author was "equparado a bolsero pelo Insttuto Naconal de Investgaçâb Centífca" Amercan Mathematcal Socety /80/ /$02.25

2 1270 M- MADALENA MARTINS Bounds for PÍLrw) are obtaned from Theorem 1. If A o/(l.l) s a strctly dagonally domnant matrx, then PÍLr u) satsfes the followng: (2.2) 11 - col - Ico - r\e - co /;. Ico - r\e + \w\ft col T-ttm^-< p(l-) < T-ThTí^-' = 1, 2,..., n, where \r\ < 1 /e and e, f are respectvely the -row sums of the modul of the entres of E and F, respectvely. Proof. Snce the egenvalues of Lr ^ are gven from (2-3) det(l,w-a/) = 0 after some manpulatons, t s easy to verfy that to solve (2.3) s equvalent to solvng (2.4) det ß = 0, where Q s r(x-l) + co _^, ^ X-l+co X-l+co If we take the parameter r, co, X, n order that Q be strctly dagonally domnant, we get (2.5) IX-1 +co > co+ (X-l)r e; + Icol/;., /=1,..., t- Then, the values of X satsfyng (2.5) wll not satsfy (2.4) and cannot be egenvalues of (2.3). From (2.5), as X can take any value, we have or X col > Ico - r\e, + Xr e,. + w /f, = 1.n, X (1 - r e,) > Ico - re,. + Icol/; col, í = 1,..., n. Assumng \r\ < l/e, we have 1 - \r\et If X satsfes ths nequalty, t cannot be an egenvalue of Lr LJ, and then Ico-rle,.+ co /;- + l-col pa^xmax-j-^- forlrk-. In order to get the lower bound, from (2.5) we wrte 11 - col - IXI > Ico - r\e + Xr e(- + Icol/;., 1=1,2,...,«, and

3 AN ACCELERATED OVERRELAXATION ITERATIVE METHOD co - Ico -r\e - Icol/; X < mn- -. í 1 + Me,- Snce the values of X satsfyng ths nequalty are not egenvalues of Lr u, then 1 -co - co-r e, - \œ\f p(l^)> mn- -^-, =l,...,n. Q.E.D. As t s well known, for convenent values of r and co, the (AOR) method becomes the well-known teratve methods: r = 0, co = 1 Jacob Method, r = 1, co = 1 Gauss-Sedel Method, r = 0, co Smultaneous Overrelaxaton Method, r = co Successve Overrelaxaton Method. Takng these values, we get from (2.2) the known results (2-6a) P(L0l) <max(e,-+/;.), (2.6b) pl1,)<max ~-, l - e (2.6c) p(l0>j < max M(e, +/,) co, M/z + U-wl (2-6d) P(-co,c)<max ', ' -. - Me,- 3. Convergence of the (AOR) Method. Theorem 2. If A o/(l.l) s a strctly dagonally domnant matrx and co > r > 0, then a suffcent condton for the convergence of the {AOR) method s Proof. 0<co< max(e,. + f )' From [1, Secton 3], wth A strctly dagonally domnant, the (AOR) method s convergent f0<co<l,0<r<l. be less than 1 f From (2.2) we see that p(l,.jcj) wll (3.1) co-rk,. + Icol/;-+ II-col + r e,. < 1, z = l,...,n. Wth co > r > 0 these condtons wll be fulflled. Takng /(Ô) = (5 - r)e,. + 6/j. + (1-5) + re,., we see that /(5) s a nonncreasng functon of ô f 0 < 5 < 1 and /(0) = 1, /(l) = e + f{ < 1. Wth 5 > 1, /(Ô) s an ncreasng functon wth /(ô) = 1 for 5=2/(14- e,. + /.). Then p{l ) < 1 f max (e,. + f ) and the (AOR) method wll be convergent. Q.E.D.

4 1272 M. MADALENA MARTINS Let us consder a frst-degree lnear statonary teratve method (3-2) x(,+ 1) = G*(0-M/=o,,2,...> wth G a known n x n matrx, d a known n-vector, and x^0' an arbtrary ntal approxmaton for the soluton x. The followng method (3.3) x(''+1) = [(1 -coy + cog]*(,) +corf,. = 012_, wll be called the extrapolated method of (3.2). We recall now, the Theorem of Extrapolaton [2, p. 2], as we need t n the sequel. Theorem 3 (Theorem of Extrapolaton). The suffcent condtons for the convergence of (3.3) are: (1) The orgnal (3.2) s convergent, (2) 0 < co < 2/(1 + p{g)). Settng r = 0 n (1.3), we obtan *(< +!) = [( - cox + co{e + F)]x + w =0(1(2>.. Ths s the extrapolated Jacob method, where co s the extrapolaton parameter. After some computaton, t s easy to verfy that (1.3) s the extrapolated SOR method, when r = 0 and ts extrapolaton parameter s co/r. Theorem 4. If A from (1.1) s strctly dagonally domnant, then p{l0jj) < 1 provded 0 < co < 2/(1 + p(l01)). Proof. Bearng n mnd that A strctly dagonally domnant mples p{l01) < 1 [4, p. 73], t s an mmedate consequence of Theorem 3. Theorem 5. If A o/(l.l) s strctly dagonally domnant, then p{lr, co) < 1 wth 0 < r < 1 provded 0 < co < 2r/(l + p{lrr)). Proof. From (2.6d) we easly deduce that the SOR method wll converge wth 0 < r < 1 (A s strctly dagonally domnant). Then, by the Theorem of Extrapolaton, the AOR method wll converge for 0<r<landO<co< 2r/(l + p(lr>,)). Theorem 6. The AOR method s convergent,.e. p{lr^) < 1, for: () 0 < r < co and 0 < co < 2/(1 + max,-(e,- + f )) f 2r < 2 1 +PÍLr r) " 1 + maxe,. + f ) ' (n) 0 < r < 1 and 0 < co < 2r/(l + p{lr r)) or 1 < r < co and 0 < co < 2/(1 + max,-(e,. + f )) f 2r 2 1 +PLr>r) 1 +max(e,-+/;.) '

5 Proof. an accelerated overrelaxaton teratve method 1273 These results come from Theorems 2 and 5. They mprove the results of Theorem 3, Secton 4 of [2], when the matrx A of (1.1) s strctly dagonally domnant: ( 2r p{lr U})<l f 0 < r < 1 and 0 < co < max-jl, { +. Acknowledgment. The author wshes to thank Professor Fernanda de Olvera and Professor Gracano de Olvera for ther suggestons and crtcsm durng the wrtng of ths paper. Department of Mathematcs Unversty of Combra Combra, Portugal 1. A. HADJIDIMOS, "Accelerated overrelaxaton method,"math. Comp., v. 32, 1978, pp A. HADJIDIMOS & A. YEYIOS, The Prncple of Extrapolaton n Connecton wth the Accelerated Overrelaxaton (AOR) Method, T. R. No. 16, Department of Mathematcs, Unversty of Ioannna, Ioannna, Greece, G. AVDELAS, A. HADJIDIMOS & A. YEYIOS, Some Theoretcal and Computatonal Results Concernng the Accelerated Overrelaxaton (AOR) Method, T. R. No. 8, Department of Mathematcs, Unversty of Ioannna, Ioannna, Greece, R. S. VARGA, Matrx Iteratve Analyss, Prentce-Hall, Englewood Clffs, N. J., D. M. YOUNG, Iteratve Soluton of Large Lnear Systems, Academc Press, New York and London, 1971.