diff --git a/docs/html/node9.html b/docs/html/node9.html
index cfa4fc22..497319a0 100644
--- a/docs/html/node9.html
+++ b/docs/html/node9.html
@@ -60,7 +60,7 @@ Example and test programs
The package contains the examples
and tests
directories;
both of them are further divided into fileread
and
-pargen
subdirectories. Their purpose is as follows:
+pdegen
subdirectories. Their purpose is as follows:
- examples
- contains a set of simple example programs with a
diff --git a/docs/src/building.tex b/docs/src/building.tex
index 7c1d699e..405dc853 100644
--- a/docs/src/building.tex
+++ b/docs/src/building.tex
@@ -225,7 +225,7 @@ make install
\subsection{Example and test programs\label{sec:ex_and_test}}
The package contains the \verb|examples| and \verb|tests| directories;
both of them are further divided into \verb|fileread| and
-\verb|pargen| subdirectories. Their purpose is as follows:
+\verb|pdegen| subdirectories. Their purpose is as follows:
\begin{description}
\item[\tt examples] contains a set of simple example programs with a
predefined choice of preconditioners, selectable via integer
diff --git a/docs/userguide.pdf b/docs/userguide.pdf
index dd11bc10..5ba94b6f 100644
--- a/docs/userguide.pdf
+++ b/docs/userguide.pdf
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-/F15 10.9091 Tf 93.6002 606.3811 Td[(where)]TJ/F24 10.9091 Tf 31.3179 0 Td[(A)]TJ/F15 10.9091 Tf 11.5907 0 Td[(is)-313(a)-312(s)-1(q)1(uare,)-316(re)-1(al)-312(or)-312(com)-1(p)1(le)-1(x)1(,)-317(spar)1(s)-1(e)-313(matri)1(x)-313(with)-312(a)-312(s)-1(y)1(m)-1(metric)-313(spar)1(s)-1(it)28(y)-312(pat)1(te)-1(r)1(n.)]TJ -42.9086 -13.5492 Td[(Thes)-1(e)-333(prec)-1(on)1(di)1(tioners)-334(h)1(a)28(v)28(e)-334(the)-333(foll)1(o)27(win)1(g)-334(general)-333(f)1(e)-1(atu)1(res)-1(:)]TJ
-ET
-1 0 0 1 93.6002 571.4462 cm
+ -398.512 -23.152 Td [(where)]TJ/F22 10.9091 Tf 31.318 0 Td [(A)]TJ/F15 10.9091 Tf 11.591 0 Td [(is)-312(a)-313(square,)-317(real)-312(or)-312(c)-1(ompl)1(e)-1(x,)-316(sparse)-313(matrix)-312(with)-313(a)-312(symmetric)-313(sparsit)28(y)-312(pattern.)]TJ -42.909 -13.549 Td [(These)-333(preconditioners)-334(ha)28(v)28(e)-333(the)-334(follo)28(wing)-333(general)-334(f)1(e)-1(atu)1(res)-1(:)]TJ
0 g 0 G
-1 0 0 1 -93.6002 -571.4462 cm
-BT
-/F27 10.9091 Tf 109.9639 571.4462 Td[(\017)]TJ
-ET
-1 0 0 1 115.4185 571.4462 cm
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-BT
-/F15 10.9091 Tf 120.873 571.4462 Td[(b)-27(oth)]TJ/F20 10.9091 Tf 24.7943 0 Td[(additive)-276(a)-1(nd)-276(hybri)1(d)-277(mu)1(ltil)1(evel)]TJ/F15 10.9091 Tf 141.7891 0 Td[(v)56(arian)29(ts)-246(ar)1(e)-246(impl)1(e)-1(me)-1(n)29(te)-1(d)1(,)-263(i.)1(e)-1(.)-245(v)56(ar)1(ian)28(ts)-245(that)-245(ar)1(e)]TJ -166.5834 -13.5492 Td[(add)1(itiv)28(e)-226(among)-225(the)-226(lev)28(e)-1(l)1(s)-227(an)1(d)-225(inside)-226(eac)27(h)-225(lev)28(e)-1(l,)-247(an)1(d)-225(v)55(ar)1(ian)28(ts)-226(th)1(at)-226(are)-226(m)28(ulti)1(pli)1(c)-1(ativ)28(e)]TJ 0 -13.5492 Td[(among)-334(the)-334(lev)28(e)-1(ls)-334(and)-334(ad)1(di)1(tiv)28(e)-335(in)1(s)-1(i)1(de)-335(eac)27(h)-333(le)-1(v)28(el;)-334(the)-334(basic)-335(Ad)1(diti)1(v)28(e)-335(S)1(c)27(h)28(w)28(arz)-335(\050AS)1(\051)]TJ 0 -13.5492 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-334(are)-334(ob)1(tain)1(e)-1(d)-333(b)29(y)-334(consid)1(e)-1(r)1(ing)-333(onl)1(y)-334(on)1(e)-334(lev)28(e)-1(l)1(;)]TJ
-ET
-1 0 0 1 93.6002 508.7349 cm
+/F15 10.9091 Tf 10.909 0 Td [(b)-28(oth)]TJ/F18 10.9091 Tf 24.794 0 Td [(additive)-277(and)-276(hybrid)-277(mu)1(ltilevel)]TJ/F15 10.9091 Tf 141.789 0 Td [(v)56(arian)27(ts)-245(are)-245(implemen)28(ted,)-263(i.e.)-245(v)56(arian)28(ts)-245(that)-245(are)]TJ -166.583 -13.549 Td [(additiv)28(e)-226(among)-226(th)1(e)-226(lev)28(e)-1(ls)-225(and)-226(inside)-226(eac)28(h)-226(lev)28(el,)-247(and)-226(v)56(arian)28(ts)-226(that)-226(are)-225(m)27(ultipl)1(ic)-1(ati)1(v)27(e)]TJ 0 -13.549 Td [(among)-334(the)-334(lev)27(els)-334(and)-334(additiv)28(e)-334(inside)-334(eac)27(h)-334(lev)28(el;)-335(the)-334(basic)-334(Additiv)28(e)-334(Sc)27(h)28(w)28(arz)-334(\050AS\051)]TJ 0 -13.549 Td [(preconditioners)-333(are)-334(obtai)1(ned)-334(b)28(y)-333(considering)-334(on)1(ly)-334(one)-333(lev)28(el;)]TJ
0 g 0 G
-1 0 0 1 -93.6002 -508.7349 cm
-BT
-/F27 10.9091 Tf 109.9639 508.7349 Td[(\017)]TJ
-ET
-1 0 0 1 115.4185 508.7349 cm
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0 g 0 G
-1 0 0 1 -115.4185 -508.7349 cm
-BT
-/F15 10.9091 Tf 120.873 508.7349 Td[(a)]TJ/F20 10.9091 Tf 10.3888 0 Td[(pur)52(ely)-467(al)1(gebr)51(aic)]TJ/F15 10.9091 Tf 80.6232 0 Td[(app)1(roac)28(h)-452(is)-453(u)1(s)-1(ed)-452(to)-452(ge)-1(n)1(e)-1(r)1(ate)-453(a)-452(s)-1(equence)-453(of)-452(coarse)-1(-lev)28(e)-1(l)-452(cor-)]TJ -91.012 -13.5492 Td[(rec)-1(t)1(ions)-415(to)-415(a)-415(b)1(as)-1(ic)-415(AS)-414(pr)1(e)-1(cond)1(ition)1(e)-1(r,)-435(wit)1(hout)-414(e)-1(xp)1(licitly)-414(using)-414(an)28(y)-415(in)1(for)1(m)-1(ati)1(on)]TJ 0 -13.5492 Td[(on)-373(the)-374(ge)-1(ometry)-374(of)-373(the)-374(ori)1(ginal)-373(prob)1(lem)-375(\050e.g.)-373(the)-374(disc)-1(r)1(e)-1(ti)1(z)-1(ati)1(on)-374(of)-373(a)-374(PDE)1(\051.)-374(The)]TJ/F20 10.9091 Tf 0 -13.5492 Td[(smo)51(othe)51(d)-313(aggr)51(e)52(ga)-1(ti)1(o)-1(n)]TJ/F15 10.9091 Tf 103.9531 0 Td[(tec)27(hn)1(iqu)1(e)-285(is)-285(ap)1(plied)-284(as)-285(algebrai)1(c)-285(c)-1(oar)1(s)-1(enin)1(g)-285(strategy)-284([)]TJ
-ET
-1 0 0 1 477.4929 468.0873 cm
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1 0 0 rg 1 0 0 RG
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-BT
-/F15 10.9091 Tf 477.4929 468.0873 Td[(1)]TJ
-ET
-1 0 0 1 482.9474 468.0873 cm
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0 g 0 G
-1 0 0 1 -482.9474 -468.0873 cm
-BT
-/F15 10.9091 Tf 482.9474 468.0873 Td[(,)]TJ
-ET
-1 0 0 1 489.0816 468.0873 cm
+ [(,)]TJ
1 0 0 rg 1 0 0 RG
-1 0 0 1 -489.0816 -468.0873 cm
-BT
-/F15 10.9091 Tf 489.0816 468.0873 Td[(24)]TJ
-ET
-1 0 0 1 499.9907 468.0873 cm
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0 g 0 G
-1 0 0 1 -499.9907 -468.0873 cm
-BT
-/F15 10.9091 Tf 499.9907 468.0873 Td[(].)]TJ -389.454 -21.3857 Td[(The)-287(p)1(ac)27(k)56(age)-288(i)1(s)-287(w)-1(r)1(itten)-287(i)1(n)]TJ/F20 10.9091 Tf 122.3962 0 Td[(F)77(ortr)51(an)-315(95)]TJ/F15 10.9091 Tf 50.4226 0 Td[(,)-296(f)1(ollo)28(wing)-286(an)]TJ/F20 10.9091 Tf 66.4546 0 Td[(obje)51(ct-or)1(ient)-1(e)52(d)-315(ap)-1(pr)52(o)51(ach)]TJ/F15 10.9091 Tf 118.6351 0 Td[(th)1(rough)]TJ -374.845 -13.5492 Td[(th)1(e)-331(e)-1(xp)1(loitat)1(ion)-330(of)-330(featur)1(e)-1(s)-331(suc)28(h)-330(as)-331(abstract)-331(d)1(ata)-331(t)28(yp)-27(e)-331(creation,)-330(fun)1(c)-1(t)1(ional)-330(o)28(v)28(e)-1(r)1(loadi)1(ng)]TJ 0 -13.5492 Td[(and)-359(dy)1(namic)-361(mem)-1(or)1(y)-360(m)-1(an)1(age)-1(men)28(t.)-524(The)-360(paral)1(le)-1(l)-359(im)-1(p)1(lem)-1(en)28(tati)1(on)-360(is)-360(base)-1(d)-359(on)-360(a)-360(S)1(ingle)]TJ 0 -13.5492 Td[(Pr)1(ogram)-231(Mu)1(ltip)1(le)-231(Data)-230(\050SP)1(MD\051)-230(parad)1(igm)-231(f)1(or)-231(d)1(is)-1(t)1(rib)1(uted-m)-1(emory)-230(arc)27(h)1(itec)-1(tu)1(res)-1(.)-410(S)1(ingle)]TJ 0 -13.5492 Td[(and)-376(d)1(oubl)1(e)-378(p)1(rec)-1(ision)-376(impleme)-1(n)28(t)1(ations)-377(of)-376(MLD2P)1(4)-377(are)-377(a)28(v)55(ai)1(lable)-377(f)1(or)-377(b)-27(oth)-376(the)-377(real)-377(an)1(d)]TJ 0 -13.5492 Td[(th)1(e)-334(c)-1(ompl)1(e)-1(x)-333(cas)-1(e,)-333(that)-333(can)-333(b)-28(e)-333(use)-1(d)-333(th)1(rou)1(gh)-333(a)-334(sin)1(gle)-334(in)28(t)1(e)-1(rf)1(ac)-1(e.)]TJ 16.9365 -13.5492 Td[(MLD2P)1(4)-230(h)1(as)-230(b)-27(e)-1(en)-229(des)-1(i)1(gned)-229(to)-230(imp)1(le)-1(men)28(t)-230(scalable)-229(and)-229(eas)-1(y-to-use)-230(m)27(u)1(ltil)1(e)-1(v)28(el)-230(p)1(rec)-1(on)1(-)]TJ -16.9365 -13.5492 Td[(di)1(tioners)-348(in)-348(th)1(e)-349(c)-1(on)29(te)-1(x)1(t)-349(of)-347(the)]TJ/F20 10.9091 Tf 147.4316 0 Td[(PS)-1(B)1(L)-1(AS)-372(\050Par)51(al)-50(lel)-371(Sp)51(arse)-371(BLA)-1(S\051)-371(c)51(omputationa)-1(l)-371(f)1(r)51(ame-)]TJ -147.4316 -13.5492 Td[(wo)-1(r)1(k)]TJ/F15 10.9091 Tf 27.2865 0 Td[([)]TJ
-ET
-1 0 0 1 123.917 338.308 cm
+ [(].)]TJ -114.289 -21.385 Td [(The)-287(pac)28(k)56(age)-287(is)-287(written)-286(in)]TJ/F18 10.9091 Tf 122.396 0 Td [(F)77(ortr)51(an)-315(95)]TJ/F15 10.9091 Tf 50.422 0 Td [(,)-296(follo)28(wing)-287(an)]TJ/F18 10.9091 Tf 66.455 0 Td [(obje)51(ct-oriente)51(d)-315(appr)52(o)51(ach)]TJ/F15 10.9091 Tf 118.635 0 Td [(through)]TJ -374.845 -13.55 Td [(the)-330(e)-1(xp)1(loitation)-331(of)-330(features)-331(suc)28(h)-330(as)-331(abstract)-330(data)-331(t)28(yp)-28(e)-330(creation,)-331(functional)-331(o)28(v)28(erloading)]TJ 0 -13.549 Td [(and)-360(dynamic)-360(memory)-360(managemen)28(t.)-525(The)-360(parallel)-360(implemen)28(tation)-360(is)-360(based)-360(on)-360(a)-360(Single)]TJ 0 -13.549 Td [(Program)-231(Mul)1(tiple)-231(Data)-231(\050SPMD\051)-230(paradigm)-231(for)-230(distributed-memory)-231(arc)28(hitectures.)-410(Single)]TJ 0 -13.549 Td [(and)-377(double)-376(precis)-1(i)1(on)-377(implemen)27(tat)1(ions)-377(of)-377(MLD2P4)-377(are)-377(a)28(v)56(ailable)-377(for)-377(b)-28(oth)-376(the)-377(real)-377(and)]TJ 0 -13.549 Td [(the)-333(complex)-334(case,)-333(that)-333(can)-334(b)-27(e)-334(used)-333(through)-333(a)-334(single)-333(in)28(terface.)]TJ 16.937 -13.549 Td [(MLD2P4)-229(has)-230(b)-27(e)-1(en)-229(designed)-229(to)-230(implemen)28(t)-230(scalable)-229(and)-229(easy-to-use)-230(m)28(ultilev)28(el)-230(precon-)]TJ -16.937 -13.55 Td [(ditioners)-348(in)-348(the)-349(con)28(text)-348(of)-348(the)]TJ/F18 10.9091 Tf 147.432 0 Td [(PSBLAS)-372(\050Par)52(al)-52(l)1(el)-372(Sp)51(arse)-371(BLAS\051)-372(c)51(omputational)-371(fr)51(ame-)]TJ -147.432 -13.549 Td [(work)]TJ/F15 10.9091 Tf 27.287 0 Td [([)]TJ
1 0 0 rg 1 0 0 RG
-1 0 0 1 -123.917 -338.308 cm
-BT
-/F15 10.9091 Tf 123.917 338.308 Td[(15)]TJ
-ET
-1 0 0 1 134.8261 338.308 cm
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0 g 0 G
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-BT
-/F15 10.9091 Tf 134.8261 338.308 Td[(].)-453(PS)1(B)-1(LAS)-336(i)1(s)-337(a)-337(l)1(ibr)1(ary)-336(origin)1(ally)-336(d)1(e)-1(v)28(elop)-27(e)-1(d)-336(to)-336(add)1(res)-1(s)-337(th)1(e)-337(par)1(allel)-336(im)-1(p)1(lem)-1(en-)]TJ -41.2259 -13.5492 Td[(tation)-441(of)-442(i)1(te)-1(r)1(ativ)28(e)-442(s)-1(olv)28(ers)-442(for)-441(s)-1(p)1(arse)-443(l)1(inear)-442(syste)-1(m,)-469(b)28(y)-441(pro)28(vi)1(din)1(g)-442(basic)-442(linear)-442(al)1(ge)-1(b)1(ra)]TJ 0 -13.5492 Td[(op)-27(e)-1(r)1(ators)-443(an)1(d)-442(dat)1(a)-443(manageme)-1(n)28(t)-442(f)1(ac)-1(il)1(ities)-443(f)1(or)-442(distri)1(but)1(e)-1(d)-442(spar)1(s)-1(e)-443(matr)1(ic)-1(es)-1(;)-496(it)-442(also)-442(in-)]TJ 0 -13.5492 Td[(clud)1(e)-1(s)-337(par)1(allel)-337(Kr)1(ylo)28(v)-337(solv)28(e)-1(r)1(s)-1(,)-337(bu)1(ilt)-337(on)-336(the)-337(top)-336(of)-337(th)1(e)-338(b)1(as)-1(i)1(c)-338(P)1(SBLAS)-337(k)28(ern)1(e)-1(ls.)-455(The)-337(pr)1(e)-1(-)]TJ 0 -13.5492 Td[(condi)1(tioners)-380(a)28(v)56(ailab)1(le)-380(in)-379(MLD)1(2P4)-380(can)-379(b)-27(e)-380(use)-1(d)-379(with)-379(th)1(e)-1(se)-380(Kry)1(lo)28(v)-380(solv)28(ers)-1(.)-582(T)-1(h)1(e)-380(c)27(h)1(oic)-1(e)]TJ 0 -13.5492 Td[(of)-361(PS)1(B)-1(LAS)-361(h)1(as)-363(b)-27(ee)-1(n)-361(mainl)1(y)-362(motiv)56(ate)-1(d)-361(b)28(y)-361(th)1(e)-363(n)1(e)-1(ed)-361(of)-362(h)1(a)28(vin)1(g)-362(a)-362(p)-27(ortab)1(le)-362(and)-361(e\016)-1(cien)28(t)]TJ 0 -13.5492 Td[(soft)28(w)27(ar)1(e)-313(in)1(frastru)1(c)-1(t)1(ure)-312(im)-1(p)1(lem)-1(en)28(tin)1(g)-312(\134de)-313(f)1(ac)-1(to")-312(stand)1(ard)-312(p)1(arallel)-312(s)-1(p)1(arse)-313(lin)1(e)-1(ar)-312(al)1(ge)-1(b)1(ra)]TJ 0 -13.5492 Td[(k)28(ernels,)-304(to)-297(p)1(ursue)-297(goal)1(s)-298(suc)28(h)-297(as)-297(p)-27(erfor)1(m)-1(an)1(c)-1(e,)-304(p)-27(ortab)1(ilit)28(y)84(,)-304(mo)-28(du)1(larit)28(y)-296(e)-1(d)-296(extensibil)1(it)28(y)-297(i)1(n)]TJ 0 -13.5492 Td[(th)1(e)-273(dev)28(e)-1(lop)1(me)-1(n)28(t)-272(of)-272(th)1(e)-273(precondi)1(tioner)-272(pac)28(k)55(age.)-424(On)-272(th)1(e)-273(other)-272(han)1(d,)-284(the)-272(im)-1(p)1(lem)-1(en)28(tati)1(on)]TJ 0 -13.5492 Td[(of)-378(MLD2P)1(4)-379(has)-379(led)-379(to)-378(s)-1(ome)-379(revis)-1(i)1(ons)-379(and)-378(exte)-1(n)29(tions)-379(of)-378(the)-379(P)1(SBLAS)-379(k)28(ern)1(e)-1(ls,)-390(leadi)1(ng)]TJ 0 -13.5492 Td[(to)-439(th)1(e)-440(r)1(e)-1(ce)-1(n)29(t)-439(PS)1(B)-1(LAS)-438(2.0)-438(v)28(e)-1(rsion)-438([)]TJ
-ET
-1 0 0 1 268.6586 202.8161 cm
+ [(].)-454(PSBLAS)-336(is)-336(a)-337(library)-336(originally)-336(dev)28(elop)-28(ed)-337(to)-336(address)-336(the)-337(parallel)-336(implemen-)]TJ -27.287 -13.549 Td [(tation)-442(of)-442(iterativ)28(e)-442(solv)28(ers)-442(for)-442(sparse)-442(linear)-442(system,)-469(b)28(y)-442(pro)28(viding)-442(basic)-442(linear)-442(algebra)]TJ 0 -13.549 Td [(op)-28(erators)-442(and)-442(data)-442(managemen)27(t)-442(facilities)-442(for)-442(distributed)-442(sparse)-443(matrices;)-496(it)-443(also)-442(in-)]TJ 0 -13.549 Td [(cludes)-337(parallel)-337(Krylo)28(v)-337(solv)28(ers,)-338(built)-337(on)-336(the)-337(top)-337(of)-337(the)-337(basic)-337(PSBLAS)-337(k)28(ernels.)-455(The)-337(pre-)]TJ 0 -13.55 Td [(conditioners)-379(a)27(v)56(ailable)-380(in)-379(MLD2P4)-380(can)-379(b)-28(e)-379(used)-380(with)-379(these)-380(Krylo)28(v)-380(solv)28(ers.)-583(The)-380(c)28(hoice)]TJ 0 -13.549 Td [(of)-362(PSBLAS)-361(has)-362(b)-28(een)-361(mainly)-362(motiv)56(ated)-362(b)28(y)-362(the)-362(need)-361(of)-362(ha)28(ving)-362(a)-361(p)-28(ortable)-362(and)-361(e\016cien)27(t)]TJ 0 -13.549 Td [(soft)28(w)28(are)-313(infrastructure)-312(implemen)28(ting)-312(\134de)-313(facto")-312(standard)-312(parallel)-312(sparse)-313(linear)-312(algebra)]TJ 0 -13.549 Td [(k)28(ernels,)-304(to)-297(pursue)-297(goals)-296(suc)27(h)-296(as)-297(p)-28(erformance,)-304(p)-28(or)1(tabilit)27(y)84(,)-304(mo)-28(dularit)28(y)-297(ed)-296(e)-1(xt)1(e)-1(nsib)1(ilit)27(y)-296(in)]TJ 0 -13.549 Td [(the)-272(dev)27(elopmen)28(t)-272(of)-273(the)-272(preconditioner)-272(pac)27(k)56(age.)-424(On)-273(the)-272(other)-272(hand,)-285(the)-272(implemen)27(tation)]TJ 0 -13.55 Td [(of)-379(MLD2P4)-378(has)-379(led)-379(to)-379(some)-379(revisions)-379(and)-378(exten)27(tion)1(s)-379(of)-379(the)-379(PSBLAS)-379(k)28(ernels,)-390(leading)]TJ 0 -13.549 Td [(to)-439(the)-439(recen)28(t)-439(PSBLAS)-439(2.0)-438(v)27(ersion)-438([)]TJ
1 0 0 rg 1 0 0 RG
-1 0 0 1 -268.6586 -202.8161 cm
-BT
-/F15 10.9091 Tf 268.6586 202.8161 Td[(14)]TJ
-ET
-1 0 0 1 279.5677 202.8161 cm
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0 g 0 G
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-/F15 10.9091 Tf 279.5677 202.8161 Td[(].)-761(Th)1(e)-440(i)1(n)28(te)-1(r)1(-)-1(p)1(ro)-28(ces)-1(s)-439(c)-1(om)28(un)1(ic)-1(at)1(ion)-439(r)1(e)-1(qu)1(ired)-438(b)28(y)]TJ -185.9675 -13.5492 Td[(MLD2P)1(4)-355(is)-355(encapsul)1(ate)-1(d)-354(in)29(to)-355(th)1(e)-355(PSBLAS)-354(rou)1(tines,)-360(e)-1(x)1(c)-1(ept)-354(few)-355(c)-1(ases)-356(wh)1(e)-1(r)1(e)-356(M)1(PI)-354([)]TJ
-ET
-1 0 0 1 492.1119 189.2669 cm
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1 0 0 rg 1 0 0 RG
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-ET
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-/F15 10.9091 Tf 216.7217 529.3831 Td[(]\051.)-530(Th)1(e)-363(main)-361(m)-1(oti)1(v)55(at)1(ion)-362(f)1(or)-362(c)28(ho)-28(osing)-362(Ad)1(di)1(tiv)28(e)-362(Sc)27(h)29(w)27(arz)-362(p)1(re)-1(-)]TJ -123.1215 -13.5492 Td[(condi)1(tioners)-361(is)-362(t)1(heir)-361(in)29(trin)1(s)-1(ic)-361(par)1(allelism)-1(.)-527(A)-361(dr)1(a)27(wb)1(ac)27(k)-361(of)-361(th)1(e)-1(se)-362(p)1(rec)-1(on)1(dit)1(ioners)-361(is)-362(th)1(at)]TJ 0 -13.5492 Td[(th)1(e)-389(n)28(u)1(m)27(b)-27(e)-1(r)-388(of)-387(iteration)1(s)-389(of)-388(th)1(e)-389(p)1(rec)-1(on)1(diti)1(oned)-388(s)-1(ol)1(v)28(e)-1(r)1(s)-389(generally)-388(gr)1(o)27(ws)-388(w)-1(i)1(th)-388(th)1(e)-389(n)28(u)1(m)-1(-)]TJ 0 -13.5492 Td[(b)-27(e)-1(r)-370(of)-369(s)-1(u)1(bmatri)1(c)-1(es)-1(.)-554(T)-1(h)1(is)-370(m)-1(a)28(y)-370(b)-27(e)-371(a)-370(se)-1(r)1(ious)-370(limitation)-369(on)-370(p)1(arallel)-370(com)-1(p)1(uters,)-379(s)-1(in)1(c)-1(e)-370(the)]TJ 0 -13.5492 Td[(n)28(u)1(m)27(b)-27(e)-1(r)-403(of)-402(s)-1(u)1(bmatrices)-404(usual)1(ly)-403(m)-1(at)1(c)27(hes)-404(th)1(e)-404(n)28(u)1(m)27(b)-27(e)-1(r)-402(of)-403(a)28(v)55(ai)1(lable)-403(pr)1(o)-28(c)-1(ess)-1(ors.)-654(Op)1(timal)]TJ 0 -13.5492 Td[(con)28(v)28(e)-1(r)1(ge)-1(n)1(c)-1(e)-262(rates,)-276(i.e.)-262(it)1(e)-1(rati)1(on)-262(n)29(um)28(b)-28(ers)-262(in)1(dep)-28(end)1(e)-1(n)28(t)-261(of)-262(t)1(he)-262(n)28(um)28(b)-27(e)-1(r)-261(of)-262(sub)1(m)-1(at)1(rice)-1(s,)-276(can)]TJ 0 -13.5492 Td[(b)-27(e)-370(obtai)1(ned)-369(b)28(y)-369(c)-1(or)1(re)-1(ctin)1(g)-370(th)1(e)-370(pr)1(e)-1(cond)1(ition)1(e)-1(r)-369(th)1(rou)1(gh)-369(a)-370(suit)1(able)-370(ap)1(pr)1(o)27(x)1(im)-1(at)1(ion)-369(of)-369(the)]TJ 0 -13.5492 Td[(ori)1(ginal)-323(l)1(inear)-323(syste)-1(m)-323(in)-322(a)-323(c)-1(oar)1(s)-1(e)-323(s)-1(p)1(ac)-1(e,)-325(whi)1(c)27(h)-323(gl)1(oball)1(y)-323(c)-1(ou)1(ples)-323(the)-323(inf)1(ormation)-322(related)]TJ 0 -13.5492 Td[(to)-333(the)-333(s)-1(in)1(gle)-334(sub)1(m)-1(atr)1(ice)-1(s.)]TJ/F20 10.9091 Tf 16.9365 -14.0113 Td[(Two-l)1(evel)-308(S)-1(c)1(h)-1(warz)]TJ/F15 10.9091 Tf 91.5293 0 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-280(are)-280(obt)1(ained)-279(b)28(y)-280(com)27(b)1(in)1(ing)-279(basic)-280(\050one-le)-1(v)28(el\051)-279(Sc)28(h-)]TJ -108.4658 -13.5492 Td[(w)28(arz)-363(p)1(rec)-1(on)1(diti)1(oners)-363(with)-362(a)-362(coarse)-1(-lev)28(e)-1(l)-362(correction.)-531(In)-362(t)1(his)-362(c)-1(on)28(text,)-369(th)1(e)-363(one-le)-1(v)28(el)-362(pr)1(e)-1(-)]TJ 0 -13.5492 Td[(condi)1(tioner)-344(is)-345(of)1(te)-1(n)-344(called)-344(`sm)-1(o)-27(other'.)-477(Di\013e)-1(r)1(e)-1(n)29(t)-345(t)28(w)28(o-)-1(l)1(e)-1(v)28(el)-345(p)1(rec)-1(on)1(dit)1(ioners)-345(ar)1(e)-345(obtai)1(ned)]TJ 0 -13.5492 Td[(b)28(y)-370(v)55(ar)1(yin)1(g)-371(th)1(e)-372(c)28(hoice)-371(of)-371(t)1(he)-371(s)-1(mo)-28(oth)1(e)-1(r)-370(and)-370(of)-370(the)-371(coarse)-1(-lev)28(e)-1(l)-370(c)-1(or)1(re)-1(ction)1(,)-380(and)-370(th)1(e)-372(w)28(a)28(y)]TJ 0 -13.5492 Td[(th)1(e)-1(y)-373(are)-374(c)-1(om)28(bi)1(ned)-374([)]TJ
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10.9091 Tf 39.5379 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 3.9588 Td[(\016)]TJ -1.5152 -7.0143 Td[(i)]TJ/F27 10.9091 Tf 9.0896 3.0555 Td[(\033)]TJ/F24 10.9091 Tf 11.5151 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 4.5878 Td[(\016)]TJ/F28 7.9701 Tf 4.046 0 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -12.1477 -7.8446 Td[(i)]TJ/F15 10.9091 Tf 20.2247 3.2568 Td[(obt)1(ained)-306(b)28(y)-307(i)1(nclud)1(ing)-306(the)-307(v)28(erti)1(c)-1(es)-307(that)-306(are)-307(adj)1(ac)-1(en)28(t)-306(to)-307(an)28(y)-306(v)28(ertex)]TJ -100.9733 -13.5492 Td[(in)]TJ/F24 10.9091 Tf 12.7273 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 4.5878 Td[(\016)]TJ/F28 7.9701 Tf 4.046 0 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -12.1477 -7.8446 Td[(i)]TJ/F15 10.9091 Tf 16.88 3.2568 Td[(.)]TJ -22.9738 -15.2694 Td[(Let)]TJ/F24 10.9091 Tf 18.79 0 Td[(n)]TJ/F25 7.9701 Tf 6.548 3.9588 Td[(\016)]TJ 0 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 7.425 3.0555 Td[(b)-27(e)-265(th)1(e)-265(size)-265(of)]TJ/F24 10.9091 Tf 63.7053 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 3.9588 Td[(\016)]TJ -1.5152 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 8.9402 3.0555 Td[(and)]TJ/F24 10.9091 Tf 20.4566 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(\016)]TJ -0.0842 -7.0143 Td[(i)]TJ/F27 10.9091 Tf 7.6586 3.0555 Td[(2)-277(<)]TJ/F25 7.9701 Tf 18.1818 3.9588 Td[(n)]TJ/F26 5.9776 Tf 5.1382 2.8128 Td[(\016)]TJ 0 -5.395 Td[(i)]TJ/F28 7.9701 Tf 4.0895 2.5822 Td[(\002)]TJ/F25 7.9701 Tf 6.5865 0 Td[(n)]TJ/F15 10.9091 Tf 8.5172 -3.9588 Td[(th)1(e)-265(res)-1(t)1(riction)-264(op)-27(erator)-264(th)1(at)-264(maps)-265(a)-264(v)28(ec)-1(t)1(or)]TJ/F24 10.9091 Tf -211.5596 -14.9745 Td[(v)]TJ/F27 10.9091 Tf 8.9929 0 Td[(2)-303(<)]TJ/F25 7.9701 Tf 18.4651 3.9588 Td[(n)]TJ/F15 10.9091 Tf 9.4427 -3.9588 Td[(on)28(to)-349(t)1(he)-349(v)28(e)-1(ctor)]TJ/F24 10.9091 Tf 76.601 0 Td[(v)]TJ/F25 7.9701 Tf 5.6793 3.9588 Td[(\016)]TJ -0.3914 -7.0143 Td[(i)]TJ/F27 10.9091 Tf 8.2491 3.0555 Td[(2)-303(<)]TJ/F25 7.9701 Tf 18.4651 3.9588 Td[(n)]TJ/F26 5.9776 Tf 5.1382 2.8128 Td[(\016)]TJ 0 -5.395 Td[(i)]TJ/F15 10.9091 Tf 8.394 -1.3766 Td[(con)28(tain)1(ing)-349(th)1(e)-350(comp)-28(on)1(e)-1(n)29(ts)-350(of)]TJ/F24 10.9091 Tf 144.9832 0 Td[(v)]TJ/F15 10.9091 Tf 9.4857 0 Td[(corres)-1(p)-27(ond)1(in)1(g)-349(to)-349(the)]TJ -313.5049 -14.9745 Td[(v)28(ertice)-1(s)-369(in)]TJ/F24 10.9091 Tf 52.9901 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 3.9588 Td[(\016)]TJ -1.5152 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 6.0593 3.0555 Td[(.)-551(The)-369(transp)-27(os)-1(e)-369(of)]TJ/F24 10.9091 Tf 94.0006 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(\016)]TJ -0.0843 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 8.6538 3.0555 Td[(is)-369(a)-369(pr)1(olongati)1(on)-369(op)-27(e)-1(r)1(ator)-369(fr)1(om)]TJ/F27 10.9091 Tf 155.4601 0 Td[(<)]TJ/F25 7.9701 Tf 7.8788 3.9588 Td[(n)]TJ/F26 5.9776 Tf 5.1382 2.8128 Td[(\016)]TJ 0 -5.395 Td[(i)]TJ/F15 10.9091 Tf 8.613 -1.3766 Td[(to)]TJ/F27 10.9091 Tf 13.7223 0 Td[(<)]TJ/F25 7.9701 Tf 7.8788 3.9588 Td[(n)]TJ/F15 10.9091 Tf 5.6363 -3.9588 Td[(.)-551(The)]TJ -384.6175 -14.9745 Td[(matrix)]TJ/F24 10.9091 Tf 35.1334 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 3.9588 Td[(\016)]TJ 0 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 7.5744 3.0555 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(\016)]TJ -0.0843 -7.0143 Td[(i)]TJ/F24 10.9091 Tf 4.6284 3.0555 Td[(A)]TJ/F15 10.9091 Tf 8.1818 0 Td[(\050)]TJ/F24 10.9091 Tf 4.2425 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(\016)]TJ -0.0843 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 4.6284 3.0555 Td[(\051)]TJ/F25 7.9701 Tf 4.2424 3.9588 Td[(T)]TJ/F27 10.9091 Tf 9.6348 -3.9588 Td[(2)-277(<)]TJ/F25 7.9701 Tf 18.1818 3.9588 Td[(n)]TJ/F26 5.9776 Tf 5.1382 2.8128 Td[(\016)]TJ 0 -5.395 Td[(i)]TJ/F28 7.9701 Tf 4.0895 2.5822 Td[(\002)]TJ/F25 7.9701 Tf 6.5865 0 Td[(n)]TJ/F26 5.9776 Tf 5.1382 2.8128 Td[(\016)]TJ 0 -5.395 Td[(i)]TJ/F15 10.9091 Tf 7.8725 -1.3766 Td[(can)-301(b)-27(e)-302(considered)-301(as)-302(a)-301(r)1(e)-1(striction)-300(of)]TJ/F24 10.9091 Tf 172.9636 0 Td[(A)]TJ/F15 10.9091 Tf 11.4666 0 Td[(corres)-1(p)-27(ond)1(in)1(g)]TJ -345.9661 -13.5492 Td[(to)-333(the)-333(s)-1(et)]TJ/F24 10.9091 Tf 49.1515 0 Td[(W)]TJ/F25 7.9701 Tf 11.8182 3.9588 Td[(\016)]TJ -1.5152 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 6.0593 3.0555 Td[(.)]TJ -48.5774 -13.6429 Td[(The)]TJ/F20 10.9091 Tf 22.4243 0 Td[(class)1(ic)51(al)-357(o)-1(ne-l)1(evel)-357(A)-1(S)]TJ/F15 10.9091 Tf 106.7234 0 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)-333(i)1(s)-334(de\014n)1(e)-1(d)-333(b)29(y)]TJ/F24 10.9091 Tf -4.7921 -32.5163 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5879 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.022 Td[(AS)]TJ/F15 10.9091 Tf 15.5385 3.4341 Td[(=)]TJ/F25 7.9701 Tf 15.6487 13.6364 Td[(m)]TJ/F30 10.9091 Tf -4.1336 -3.2727 Td[(X)]TJ/F25 7.9701 Tf 1.0269 -23.4509 Td[(i)]TJ/F22 7.9701 Tf 2.8831 0 Td[(=1)]TJ/F15 10.9091 Tf 11.8476 13.0872 Td[(\050)]TJ/F24 10.9091 Tf 4.2424 0 Td[(R)]TJ/F25 7.9701 Tf 8.3675 4.5043 Td[(\016)]TJ -0.0843 -7.2012 Td[(i)]TJ/F15 10.9091 Tf 4.6284 2.6969 Td[(\051)]TJ/F25 7.9701 Tf 4.2424 4.5043 Td[(T)]TJ/F15 10.9091 Tf 6.6045 -4.5043 Td[(\050)]TJ/F24 10.9091 Tf 4.2425 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 4.5043 Td[(\016)]TJ 0 -7.2012 Td[(i)]TJ/F15 10.9091 Tf 4.5441 2.6969 Td[(\051)]TJ/F28 7.9701 Tf 4.2424 4.5043 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F24 10.9091 Tf 4.7323 -4.5043 Td[(R)]TJ/F25 7.9701 Tf 8.3675 4.5043 Td[(\016)]TJ -0.0843 -7.2012 Td[(i)]TJ/F24 10.9091 Tf 4.6284 2.6969 Td[(;)]TJ/F15 10.9091 Tf -268.1287 -35.9079 Td[(where)]TJ/F24 10.9091 Tf 32.3362 0 Td[(A)]TJ/F25 7.9701 Tf 8.1819 3.9588 Td[(\016)]TJ 0 -7.0143 Td[(i)]TJ/F15 10.9091 Tf 8.9712 3.0555 Td[(is)-406(ass)-1(u)1(m)-1(ed)-406(to)-405(b)-28(e)-406(non)1(s)-1(i)1(ngu)1(lar.)-662(Its)-406(ap)1(pli)1(c)-1(ation)-405(to)-406(a)-405(v)28(e)-1(ctor)]TJ/F24 10.9091 Tf 283.1864 0 Td[(v)]TJ/F27 10.9091 Tf 10.0275 0 Td[(2)-398(<)]TJ/F25 7.9701 Tf 19.4997 3.9588 Td[(n)]TJ/F15 10.9091 Tf 10.0635 -3.9588 Td[(with)1(in)-406(a)]TJ -372.2664 -13.5492 Td[(Kr)1(ylo)28(v)-333(s)-1(olv)28(er)-333(requ)1(ires)-334(the)-333(foll)1(o)27(win)1(g)-334(t)1(hree)-334(ste)-1(p)1(s)-1(:)]TJ
-ET
-1 0 0 1 86.3998 203.7409 cm
+ [(\051,)-308(where)]TJ/F22 10.9091 Tf 167.085 0 Td [(A)]TJ/F15 10.9091 Tf 11.212 0 Td [(=)-278(\050)]TJ/F22 10.9091 Tf 15.758 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 -1.636 Td [(ij)]TJ/F15 10.9091 Tf 7.265 1.636 Td [(\051)]TJ/F25 10.9091 Tf 7.273 0 Td [(2)-278(<)]TJ/F23 7.9701 Tf 18.182 3.959 Td [(n)]TJ/F26 7.9701 Tf 5.138 0 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F15 10.9091 Tf 8.934 -3.959 Td [(is)-302(a)-303(nonsingular)-302(sparse)-302(matrix)]TJ -270.136 -13.549 Td [(with)-321(a)-321(symmetric)-321(nonzero)-321(pattern,)-324(let)]TJ/F22 10.9091 Tf 184.526 0 Td [(G)]TJ/F15 10.9091 Tf 11.608 0 Td [(=)-278(\050)]TJ/F22 10.9091 Tf 15.757 0 Td [(W)28(;)-167(E)]TJ/F15 10.9091 Tf 23.531 0 Td [(\051)-321(b)-28(e)-321(the)-321(adjacency)-321(graph)-321(of)]TJ/F22 10.9091 Tf 134.378 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(,)-324(wh)1(e)-1(re)]TJ/F22 10.9091 Tf -377.982 -13.549 Td [(W)]TJ/F15 10.9091 Tf 15.419 0 Td [(=)]TJ/F25 10.9091 Tf 12.086 0 Td [(f)]TJ/F15 10.9091 Tf 5.455 0 Td [(1)]TJ/F22 10.9091 Tf 5.454 0 Td [(;)]TJ/F15 10.9091 Tf 4.849 0 Td [(2)]TJ/F22 10.9091 Tf 5.454 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(n)]TJ/F25 10.9091 Tf 30.79 0 Td [(g)]TJ/F15 10.9091 Tf 9.434 0 Td [(and)]TJ/F22 10.9091 Tf 21.554 0 Td [(E)]TJ/F15 10.9091 Tf 12.283 0 Td [(=)]TJ/F25 10.9091 Tf 12.086 0 Td [(f)]TJ/F15 10.9091 Tf 5.455 0 Td [(\050)]TJ/F22 10.9091 Tf 4.242 0 Td [(i;)-167(j)]TJ/F15 10.9091 Tf 13.724 0 Td [(\051)-330(:)]TJ/F22 10.9091 Tf 14.475 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 -1.637 Td [(ij)]TJ/F25 10.9091 Tf 10.867 1.637 Td [(6)]TJ/F15 10.9091 Tf 0 0 Td [(=)-330(0)]TJ/F25 10.9091 Tf 17.54 0 Td [(g)]TJ/F15 10.9091 Tf 9.434 0 Td [(are)-365(the)-364(v)27(ertex)-364(set)-365(and)-365(the)-365(edge)-364(set)-365(of)]TJ/F22 10.9091 Tf 184.476 0 Td [(G)]TJ/F15 10.9091 Tf 8.578 0 Td [(,)]TJ -409.421 -13.549 Td [(resp)-28(ectiv)28(ely)83(.)-466(Tw)28(o)-341(v)28(ertices)-341(are)-341(called)-340(adjacen)27(t)-340(if)-341(there)-340(is)-341(an)-341(edge)-341(conn)1(e)-1(ctin)1(g)-341(them.)-467(F)84(or)]TJ 0 -13.55 Td [(an)28(y)-238(in)28(te)-1(ger)]TJ/F22 10.9091 Tf 54.621 0 Td [(\016)-316(>)]TJ/F15 10.9091 Tf 19.806 0 Td [(0,)-257(a)]TJ/F22 10.9091 Tf 19.344 0 Td [(\016)]TJ/F15 10.9091 Tf 5.262 0 Td [(-o)28(v)28(erlap)-239(p)1(artition)-239(of)]TJ/F22 10.9091 Tf 96.34 0 Td [(W)]TJ/F15 10.9091 Tf 14.417 0 Td [(can)-238(b)-28(e)-238(de\014ned)-238(recursiv)28(ely)-239(as)-238(follo)28(ws.)-413(Giv)28(en)]TJ -209.79 -13.549 Td [(a)-344(0-o)28(v)28(erlap)-344(\050or)-343(non-o)28(v)27(erlapp)1(ing\051)-344(partition)-343(of)]TJ/F22 10.9091 Tf 216.249 0 Td [(W)]TJ/F15 10.9091 Tf 11.818 0 Td [(,)-346(i.e.)-344(a)-343(se)-1(t)-343(of)]TJ/F22 10.9091 Tf 63.378 0 Td [(m)]TJ/F15 10.9091 Tf 13.327 0 Td [(disjoin)28(t)-344(nonempt)28(y)-343(se)-1(ts)]TJ/F22 10.9091 Tf -304.772 -13.549 Td [(W)]TJ/F20 7.9701 Tf 11.818 3.959 Td [(0)]TJ/F23 7.9701 Tf -1.515 -7.015 Td [(i)]TJ/F25 10.9091 Tf 9.278 3.056 Td [(\032)]TJ/F22 10.9091 Tf 11.515 0 Td [(W)]TJ/F15 10.9091 Tf 14.857 0 Td [(suc)28(h)-279(that)]TJ/F25 10.9091 Tf 47.047 0 Td [([)]TJ/F23 7.9701 Tf 7.272 3.959 Td [(m)]TJ 0 -7.015 Td [(i)]TJ/F20 7.9701 Tf 2.884 0 Td [(=1)]TJ/F22 10.9091 Tf 11.318 3.056 Td [(W)]TJ/F20 7.9701 Tf 11.819 3.959 Td [(0)]TJ/F23 7.9701 Tf -1.516 -7.015 Td [(i)]TJ/F15 10.9091 Tf 9.278 3.056 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(W)]TJ/F15 10.9091 Tf 11.818 0 Td [(,)-290(a)]TJ/F22 10.9091 Tf 14.682 0 Td [(\016)]TJ/F15 10.9091 Tf 5.262 0 Td [(-o)28(v)28(erlap)-279(partition)-278(of)]TJ/F22 10.9091 Tf 97.661 0 Td [(W)]TJ/F15 10.9091 Tf 14.857 0 Td [(is)-279(obtai)1(ned)-279(b)28(y)-279(considering)]TJ -289.85 -13.549 Td [(the)-307(sets)]TJ/F22 10.9091 Tf 39.538 0 Td [(W)]TJ/F23 7.9701 Tf 11.818 3.959 Td [(\016)]TJ -1.515 -7.015 Td [(i)]TJ/F25 10.9091 Tf 9.089 3.056 Td [(\033)]TJ/F22 10.9091 Tf 11.515 0 Td [(W)]TJ/F23 7.9701 Tf 11.819 4.588 Td [(\016)]TJ/F26 7.9701 Tf 4.046 0 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F23 7.9701 Tf -12.147 -7.845 Td [(i)]TJ/F15 10.9091 Tf 20.224 3.257 Td [(obtained)-307(b)28(y)-306(including)-307(the)-306(v)28(e)-1(r)1(tice)-1(s)-306(that)-307(are)-306(adjacen)28(t)-307(to)-307(an)28(y)-306(v)27(ertex)]TJ -100.973 -13.549 Td [(in)]TJ/F22 10.9091 Tf 12.727 0 Td [(W)]TJ/F23 7.9701 Tf 11.818 4.587 Td [(\016)]TJ/F26 7.9701 Tf 4.046 0 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F23 7.9701 Tf -12.148 -7.844 Td [(i)]TJ/F15 10.9091 Tf 16.88 3.257 Td [(.)]TJ -22.974 -15.27 Td [(Let)]TJ/F22 10.9091 Tf 18.79 0 Td [(n)]TJ/F23 7.9701 Tf 6.548 3.959 Td [(\016)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 7.425 3.055 Td [(b)-28(e)-264(the)-264(size)-264(of)]TJ/F22 10.9091 Tf 63.706 0 Td [(W)]TJ/F23 7.9701 Tf 11.818 3.959 Td [(\016)]TJ -1.515 -7.014 Td [(i)]TJ/F15 10.9091 Tf 8.94 3.055 Td [(and)]TJ/F22 10.9091 Tf 20.456 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.959 Td [(\016)]TJ -0.084 -7.014 Td [(i)]TJ/F25 10.9091 Tf 7.658 3.055 Td [(2)-278(<)]TJ/F23 7.9701 Tf 18.182 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 2.813 Td [(\016)]TJ 0 -5.395 Td [(i)]TJ/F26 7.9701 Tf 4.09 2.582 Td [(\002)]TJ/F23 7.9701 Tf 6.586 0 Td [(n)]TJ/F15 10.9091 Tf 8.517 -3.959 Td [(the)-264(restriction)-264(op)-28(erator)-264(that)-264(maps)-264(a)-264(v)28(ector)]TJ/F22 10.9091 Tf -211.559 -14.974 Td [(v)]TJ/F25 10.9091 Tf 8.993 0 Td [(2)-304(<)]TJ/F23 7.9701 Tf 18.465 3.959 Td [(n)]TJ/F15 10.9091 Tf 9.443 -3.959 Td [(on)28(to)-349(the)-349(v)28(ector)]TJ/F22 10.9091 Tf 76.601 0 Td [(v)]TJ/F23 7.9701 Tf 5.679 3.959 Td [(\016)]TJ -0.392 -7.015 Td [(i)]TJ/F25 10.9091 Tf 8.25 3.056 Td [(2)-304(<)]TJ/F23 7.9701 Tf 18.465 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 2.812 Td [(\016)]TJ 0 -5.395 Td [(i)]TJ/F15 10.9091 Tf 8.394 -1.376 Td [(con)28(taining)-349(the)-349(comp)-28(onen)28(ts)-349(of)]TJ/F22 10.9091 Tf 144.983 0 Td [(v)]TJ/F15 10.9091 Tf 9.486 0 Td 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-1 0 0 1 -470.9919 -462.2978 cm
-BT
-/F15 10.9091 Tf 470.9919 462.2978 Td[(20)]TJ
-ET
-1 0 0 1 481.901 462.2978 cm
+ [-359(20)]TJ
0 g 0 G
-1 0 0 1 -481.901 -462.2978 cm
-BT
-/F15 10.9091 Tf 481.901 462.2978 Td[(].)-519(T)83(o)]TJ -388.3008 -13.5492 Td[(redu)1(c)-1(e)-308(th)1(e)-309(d)1(e)-1(p)-27(e)-1(n)1(dency)-308(of)-307(the)-308(n)28(u)1(m)27(b)-27(e)-1(r)-307(of)-308(iterati)1(ons)-308(on)-308(th)1(e)-308(degre)-1(e)-308(of)-307(paral)1(le)-1(l)1(is)-1(m)-308(w)28(e)-309(ma)28(y)]TJ 0 -13.5492 Td[(in)28(t)1(ro)-28(d)1(uce)-276(a)-276(glob)1(al)-275(c)-1(ou)1(plin)1(g)-276(amon)1(g)-276(th)1(e)-276(o)27(v)28(erl)1(appi)1(ng)-275(part)1(ition)1(s)-276(b)28(y)-275(de\014ni)1(ng)-275(a)-276(coarse)-1(-s)-1(p)1(ac)-1(e)]TJ 0 -13.5492 Td[(app)1(ro)28(ximation)]TJ/F24 10.9091 Tf 72.1198 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.0953 1.6887 Td[(of)-274(the)-275(matrix)]TJ/F24 10.9091 Tf 64.7833 0 Td[(A)]TJ/F15 10.9091 Tf 8.1819 0 Td[(.)-425(In)-274(a)-275(pu)1(re)-275(algebrai)1(c)-276(se)-1(tt)1(ing,)]TJ/F24 10.9091 Tf 137.1498 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.0953 1.6887 Td[(is)-275(usual)1(ly)-275(b)1(uilt)-274(with)]TJ -318.789 -13.5492 Td[(a)-375(Gal)1(e)-1(r)1(kin)-374(appr)1(oac)27(h)1(.)-569(Giv)28(en)-374(a)-375(s)-1(et)]TJ/F24 10.9091 Tf 166.3185 0 Td[(W)]TJ/F25 7.9701 Tf 10.3031 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 11.1858 1.6887 Td[(of)]TJ/F20 10.9091 Tf 12.8769 0 Td[(c)51(o)51(arse)-395(vertic)51(es)]TJ/F15 10.9091 Tf 68.9977 0 Td[(,)-385(with)-374(s)-1(i)1(z)-1(e)]TJ/F24 10.9091 Tf 53.653 0 Td[(n)]TJ/F25 7.9701 Tf 6.5481 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 7.0968 1.6887 Td[(,)-385(and)-374(a)-375(suit)1(able)]TJ -336.9799 -13.5492 Td[(res)-1(t)1(riction)-333(op)-27(e)-1(r)1(ator)]TJ/F24 10.9091 Tf 96.2426 0 Td[(R)]TJ/F25 7.9701 Tf 8.2831 -1.6887 Td[(C)]TJ/F27 10.9091 Tf 10.1271 1.6887 Td[(2)-277(<)]TJ/F25 7.9701 Tf 18.1818 3.9588 Td[(n)]TJ/F26 5.9776 Tf 5.1382 -1.3394 Td[(C)]TJ/F28 7.9701 Tf 6.1942 1.3394 Td[(\002)]TJ/F25 7.9701 Tf 6.5865 0 Td[(n)]TJ/F15 10.9091 Tf 5.6363 -3.9588 Td[(,)]TJ/F24 10.9091 Tf 6.6667 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.7332 1.6887 Td[(is)-334(d)1(e)-1(\014)1(ned)-333(as)]TJ/F24 10.9091 Tf -10.1288 -20.9733 Td[(A)]TJ/F25 7.9701 Tf 8.1818 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.1271 1.6887 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(R)]TJ/F25 7.9701 Tf 8.2831 -1.6887 Td[(C)]TJ/F24 10.9091 Tf 7.0968 1.6887 Td[(AR)]TJ/F25 7.9701 Tf 16.5493 4.5043 Td[(T)]TJ -0.0843 -7.2012 Td[(C)]TJ/F15 10.9091 Tf -233.5116 -18.2764 Td[(and)-495(the)-497(coarse)-1(-lev)28(e)-1(l)-496(corr)1(e)-1(ction)-496(matri)1(x)-496(to)-497(b)-27(e)-497(com)27(b)1(ined)-496(with)-496(a)-496(generic)-496(one-)-1(lev)28(el)-496(AS)]TJ 0 -13.5492 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)]TJ/F24 10.9091 Tf 72.1819 0 Td[(M)]TJ/F22 7.9701 Tf 10.5833 -1.6887 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F15 10.9091 Tf 9.8935 1.6887 Td[(is)-334(ob)1(tain)1(e)-1(d)-333(as)]TJ/F24 10.9091 Tf 63.8687 -20.9733 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5866 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F15 10.9091 Tf 15.5384 3.4341 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(R)]TJ/F25 7.9701 Tf 8.3675 4.5043 Td[(T)]TJ -0.0843 -7.2012 Td[(C)]TJ/F24 10.9091 Tf 7.0968 2.6969 Td[(A)]TJ/F28 7.9701 Tf 8.1818 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -6.5865 -8.0219 Td[(C)]TJ/F24 10.9091 Tf 11.3188 3.4341 Td[(R)]TJ/F25 7.9701 Tf 8.2832 -1.6887 Td[(C)]TJ/F24 10.9091 Tf 7.0968 1.6887 Td[(;)]TJ/F15 10.9091 Tf -248.6591 -20.9733 Td[(where)]TJ/F24 10.9091 Tf 31.9095 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 11.0973 1.6887 Td[(is)-367(ass)-1(u)1(m)-1(ed)-366(to)-367(b)-27(e)-367(nonsin)1(gular)1(.)-545(The)-367(ap)1(pli)1(c)-1(ati)1(on)-367(of)]TJ/F24 10.9091 Tf 240.8833 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5866 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F15 10.9091 Tf 16.5086 3.4341 Td[(to)-367(a)-366(v)28(e)-1(ctor)]TJ/F24 10.9091 Tf 56.2741 0 Td[(v)]TJ/F15 10.9091 Tf 9.6797 0 Td[(corre-)]TJ -385.1177 -13.5492 Td[(sp)-28(on)1(ds)-325(to)-326(a)-325(r)1(e)-1(striction)1(,)-327(a)-325(soluti)1(on)-325(and)-324(a)-326(p)1(rolon)1(gation)-324(s)-1(tep;)-327(the)-326(solu)1(tion)-325(step,)-327(i)1(n)28(v)28(olvi)1(ng)]TJ 0 -13.5492 Td[(th)1(e)-334(m)-1(at)1(rix)]TJ/F24 10.9091 Tf 54.2728 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 7.0968 1.6887 Td[(,)-333(ma)27(y)-333(b)-27(e)-334(carri)1(e)-1(d)-333(ou)1(t)-334(al)1(s)-1(o)-333(app)1(ro)28(ximately)83(.)]TJ -52.6149 -13.5492 Td[(The)-309(c)-1(om)28(bin)1(ation)-309(of)]TJ/F24 10.9091 Tf 96.1762 0 Td[(M)]TJ/F25 7.9701 Tf 10.5833 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.4687 1.6887 Td[(and)]TJ/F24 10.9091 Tf 20.9476 0 Td[(M)]TJ/F22 7.9701 Tf 10.5833 -1.6887 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F15 10.9091 Tf 9.629 1.6887 Td[(ma)27(y)-308(b)-28(e)-309(p)-28(erf)1(orme)-1(d)-308(in)-309(eith)1(e)-1(r)-309(an)-308(add)1(itiv)28(e)-309(or)-309(a)-309(m)27(u)1(l-)]TJ -179.5588 -13.5492 Td[(tip)1(licativ)28(e)-275(framew)27(ork)1(.)-425(In)-275(th)1(e)-276(f)1(orme)-1(r)-274(c)-1(ase)-1(,)-286(the)]TJ/F20 10.9091 Tf 217.1915 0 Td[(tw)-1(o-l)1(evel)-304(additive)]TJ/F15 10.9091 Tf 85.1983 0 Td[(Sc)28(h)28(w)28(arz)-276(p)1(rec)-1(on)1(diti)1(oner)]TJ -302.3899 -13.5492 Td[(is)-334(ob)1(tain)1(e)-1(d)1(:)]TJ/F24 10.9091 Tf 153.9707 -13.5492 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5866 0 Td[(1)]TJ -7.7759 -8.0219 Td[(2)]TJ/F25 7.9701 Tf 4.2342 0 Td[(LA)]TJ/F15 10.9091 Tf 15.6304 3.4341 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F15 10.9091 Tf 14.9324 3.4341 Td[(+)]TJ/F24 10.9091 Tf 10.9091 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7759 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F24 10.9091 Tf 8.274 3.4341 Td[(:)]TJ/F15 10.9091 Tf -255.4501 -18.2575 Td[(App)1(lyi)1(ng)]TJ/F24 10.9091 Tf 48.2012 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5866 0 Td[(1)]TJ -7.7759 -8.0219 Td[(2)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F28 7.9701 Tf 5.759 0 Td[(\000)]TJ/F25 7.9701 Tf 6.5865 0 Td[(A)]TJ/F15 10.9091 Tf 11.4059 3.4341 Td[(to)-418(a)-419(v)28(ec)-1(tor)]TJ/F24 10.9091 Tf 57.9672 0 Td[(v)]TJ/F15 10.9091 Tf 10.2441 0 Td[(with)1(in)-418(a)-419(K)1(rylo)28(v)-418(s)-1(ol)1(v)28(e)-1(r)-418(corres)-1(p)-27(on)1(ds)-419(to)-418(app)1(lyin)1(g)]TJ/F24 10.9091 Tf 234.3781 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F15 10.9091 Tf -399.9429 -11.5506 Td[(and)]TJ/F24 10.9091 Tf 21.2121 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5866 0 Td[(1)]TJ -7.7759 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2342 0 Td[(L)]TJ/F15 10.9091 Tf 11.9103 3.4341 Td[(to)]TJ/F24 10.9091 Tf 13.3334 0 Td[(v)]TJ/F15 10.9091 Tf 9.3156 0 Td[(in)1(dep)-28(end)1(e)-1(n)29(tly)-333(and)-333(th)1(e)-1(n)-332(s)-1(u)1(m)-1(min)1(g)-334(u)1(p)-333(the)-333(re)-1(sul)1(ts)-1(.)]TJ -53.6525 -13.5492 Td[(In)-355(t)1(he)-355(m)27(u)1(ltip)1(licativ)28(e)-355(c)-1(ase)-1(,)-360(th)1(e)-356(com)27(b)1(in)1(ation)-355(can)-354(b)-28(e)-355(p)-27(e)-1(r)1(formed)-355(b)28(y)-354(\014rst)-355(ap)1(ply)1(ing)-355(th)1(e)]TJ -16.9365 -13.5492 Td[(sm)-1(o)-28(ot)1(her)]TJ/F24 10.9091 Tf 47.6667 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7758 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2341 0 Td[(L)]TJ/F15 10.9091 Tf 11.9104 3.4341 Td[(and)-332(then)-333(the)-333(c)-1(oarse-)-1(lev)28(el)-334(corr)1(e)-1(ction)-333(op)-27(e)-1(r)1(ator)]TJ/F24 10.9091 Tf 217.4851 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7759 -8.0219 Td[(C)]TJ/F15 10.9091 Tf 12.5082 3.4341 Td[(:)]TJ/F24 10.9091 Tf -164.4656 -21.5786 Td[(w)]TJ/F15 10.9091 Tf 11.1338 0 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ -7.7758 -8.0219 Td[(1)]TJ/F25 7.9701 Tf 4.2341 0 Td[(L)]TJ/F24 10.9091 Tf 8.274 3.4341 Td[(v)-36(;)]TJ -45.7404 -13.7889 Td[(z)]TJ/F15 10.9091 Tf 8.5833 0 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(w)]TJ/F15 10.9091 Tf 10.5278 0 Td[(+)]TJ/F24 10.9091 Tf 10.909 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 4.5878 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F25 7.9701 Tf -7.7758 -8.0219 Td[(C)]TJ/F15 10.9091 Tf 12.5081 3.4341 Td[(\050)]TJ/F24 10.9091 Tf 4.2425 0 Td[(v)]TJ/F27 10.9091 Tf 8.1035 0 Td[(\000)]TJ/F24 10.9091 Tf 10.9091 0 Td[(Aw)]TJ/F15 10.9091 Tf 16.2854 0 Td[(\051;)]TJ
-ET
-1 0 0 1 93.6002 91.633 cm
+ [(].)-519(T)83(o)]TJ -316.09 -13.549 Td [(reduce)-308(the)-308(dep)-28(endency)-307(of)-308(the)-308(n)28(um)27(b)-27(er)-308(of)-308(iterations)-308(on)-308(the)-308(degree)-308(of)-308(parallelism)-308(w)28(e)-308(ma)28(y)]TJ 0 -13.549 Td [(in)28(tro)-28(duce)-275(a)-276(global)-275(coupling)-276(among)-275(the)-276(o)28(v)28(erlapping)-275(partitions)-276(b)28(y)-275(de\014ning)-276(a)-275(coarse-space)]TJ 0 -13.55 Td [(appro)28(ximation)]TJ/F22 10.9091 Tf 72.12 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 -1.688 Td [(C)]TJ/F15 10.9091 Tf 10.095 1.688 Td [(of)-275(the)-275(matrix)]TJ/F22 10.9091 Tf 64.783 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(.)-425(In)-275(a)-275(pure)-274(algebraic)-275(setting,)]TJ/F22 10.9091 Tf 137.15 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 -1.688 Td [(C)]TJ/F15 10.9091 Tf 10.095 1.688 Td [(is)-275(usually)-275(built)-274(with)]TJ -318.789 -13.549 Td [(a)-375(Galerkin)-375(appr)1(oac)27(h.)-569(Giv)28(en)-375(a)-374(s)-1(et)]TJ/F22 10.9091 Tf 166.319 0 Td [(W)]TJ/F23 7.9701 Tf 10.303 -1.689 Td [(C)]TJ/F15 10.9091 Tf 11.186 1.689 Td [(of)]TJ/F18 10.9091 Tf 12.876 0 Td [(c)51(o)51(arse)-396(vertic)51(es)]TJ/F15 10.9091 Tf 68.998 0 Td [(,)-385(with)-375(size)]TJ/F22 10.9091 Tf 53.653 0 Td [(n)]TJ/F23 7.9701 Tf 6.548 -1.689 Td [(C)]TJ/F15 10.9091 Tf 7.097 1.689 Td [(,)-385(and)-375(a)-375(suitable)]TJ -336.98 -13.549 Td [(restriction)-333(op)-28(erator)]TJ/F22 10.9091 Tf 96.243 0 Td [(R)]TJ/F23 7.9701 Tf 8.283 -1.689 Td [(C)]TJ/F25 10.9091 Tf 10.127 1.689 Td [(2)-278(<)]TJ/F23 7.9701 Tf 18.182 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.34 Td [(C)]TJ/F26 7.9701 Tf 6.194 1.34 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F15 10.9091 Tf 5.636 -3.959 Td [(,)]TJ/F22 10.9091 Tf 6.667 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 -1.689 Td [(C)]TJ/F15 10.9091 Tf 10.733 1.689 Td [(is)-333(de\014ned)-334(as)]TJ/F22 10.9091 Tf -10.129 -20.973 Td [(A)]TJ/F23 7.9701 Tf 8.182 -1.689 Td [(C)]TJ/F15 10.9091 Tf 10.127 1.689 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(R)]TJ/F23 7.9701 Tf 8.283 -1.689 Td [(C)]TJ/F22 10.9091 Tf 7.097 1.689 Td [(AR)]TJ/F23 7.9701 Tf 16.549 4.504 Td [(T)]TJ -0.084 -7.201 Td [(C)]TJ/F15 10.9091 Tf -233.512 -18.277 Td [(and)-496(the)-496(c)-1(oar)1(s)-1(e-lev)28(el)-496(correction)-496(matrix)-497(to)-496(b)-28(e)-496(com)28(bined)-496(with)-497(a)-496(generic)-496(one-lev)28(el)-497(AS)]TJ 0 -13.549 Td [(preconditioner)]TJ/F22 10.9091 Tf 72.182 0 Td [(M)]TJ/F20 7.9701 Tf 10.583 -1.688 Td [(1)]TJ/F23 7.9701 Tf 4.235 0 Td [(L)]TJ/F15 10.9091 Tf 9.893 1.688 Td [(is)-333(obtained)-334(as)]TJ/F22 10.9091 Tf 63.869 -20.973 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(C)]TJ/F15 10.9091 Tf 15.539 3.434 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 4.504 Td [(T)]TJ -0.084 -7.201 Td [(C)]TJ/F22 10.9091 Tf 7.097 2.697 Td [(A)]TJ/F26 7.9701 Tf 8.182 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F23 7.9701 Tf -6.586 -8.022 Td [(C)]TJ/F22 10.9091 Tf 11.318 3.434 Td [(R)]TJ/F23 7.9701 Tf 8.284 -1.689 Td [(C)]TJ/F22 10.9091 Tf 7.096 1.689 Td [(;)]TJ/F15 10.9091 Tf -248.659 -20.973 Td [(where)]TJ/F22 10.9091 Tf 31.91 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 -1.689 Td [(C)]TJ/F15 10.9091 Tf 11.097 1.689 Td [(is)-367(assumed)-367(t)1(o)-367(b)-28(e)-367(nonsingular)1(.)-545(The)-367(application)-366(of)]TJ/F22 10.9091 Tf 240.883 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F23 7.9701 Tf -7.775 -8.022 Td [(C)]TJ/F15 10.9091 Tf 16.508 3.434 Td [(to)-367(a)-366(v)27(ector)]TJ/F22 10.9091 Tf 56.274 0 Td [(v)]TJ/F15 10.9091 Tf 9.68 0 Td [(corre-)]TJ -385.118 -13.549 Td [(sp)-28(onds)-325(to)-325(a)-325(restriction,)-327(a)-325(solution)-325(and)-325(a)-325(prolongation)-325(step;)-328(the)-325(solution)-325(step,)-327(in)28(v)27(olvi)1(ng)]TJ 0 -13.55 Td [(the)-333(matrix)]TJ/F22 10.9091 Tf 54.273 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 -1.688 Td [(C)]TJ/F15 10.9091 Tf 7.097 1.688 Td [(,)-333(ma)27(y)-333(b)-28(e)-333(carried)-333(out)-334(also)-333(appro)28(ximately)83(.)]TJ -52.615 -13.549 Td [(The)-309(com)28(bination)-309(of)]TJ/F22 10.9091 Tf 96.176 0 Td [(M)]TJ/F23 7.9701 Tf 10.583 -1.689 Td [(C)]TJ/F15 10.9091 Tf 10.469 1.689 Td [(and)]TJ/F22 10.9091 Tf 20.948 0 Td [(M)]TJ/F20 7.9701 Tf 10.583 -1.689 Td [(1)]TJ/F23 7.9701 Tf 4.234 0 Td [(L)]TJ/F15 10.9091 Tf 9.629 1.689 Td [(ma)28(y)-309(b)-28(e)-309(p)-28(erformed)-309(in)-309(either)-309(an)-309(additiv)28(e)-309(or)-309(a)-309(m)27(ul-)]TJ -179.559 -13.549 Td [(tiplicativ)28(e)-275(framew)28(ork.)-425(In)-275(the)-275(former)-275(c)-1(ase,)-286(the)]TJ/F18 10.9091 Tf 217.192 0 Td [(two-level)-304(additive)]TJ/F15 10.9091 Tf 85.198 0 Td [(Sc)28(h)28(w)27(arz)-275(preconditioner)]TJ -302.39 -13.549 Td [(is)-333(obtained:)]TJ/F22 10.9091 Tf 153.971 -13.549 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.587 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ -7.776 -8.022 Td [(2)]TJ/F23 7.9701 Tf 4.234 0 Td [(LA)]TJ/F15 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-13.5492 Td[(The)-321(c)-1(or)1(res)-1(p)-27(ond)1(ing)-321(p)1(rec)-1(on)1(dit)1(ioners,)-323(c)-1(all)1(e)-1(d)]TJ/F20 10.9091 Tf 202.455 0 Td[(mu)1(lti-l)1(evel)]TJ/F15 10.9091 Tf 53.6944 0 Td[(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(s)-1(,)-323(can)-321(signi)1(\014can)28(tly)]TJ -256.1494 -13.5492 Td[(redu)1(c)-1(e)-282(the)-283(com)-1(p)1(utat)1(ional)-282(cos)-1(t)-282(of)-282(pr)1(e)-1(cond)1(ition)1(ing)-282(with)-282(r)1(e)-1(sp)-28(ec)-1(t)-282(to)-282(the)-282(t)28(w)27(o-lev)28(e)-1(l)-282(c)-1(ase)-283(\050se)-1(e)]TJ 0 -13.5492 Td[([)]TJ
-ET
-1 0 0 1 89.4301 370.2545 cm
+/F15 10.9091 Tf -203.265 -35.866 Td [(this)-278(corresp)-28(onds)-278(to)-278(the)-278(follo)28(wing)]TJ/F18 10.9091 Tf 156.322 0 Td [(two-level)-307(hybrid)-307(pr)51(e-smo)52(o)-1(the)52(d)]TJ/F15 10.9091 Tf 143.004 0 Td [(Sc)28(h)28(w)27(arz)-278(preconditioner:)]TJ/F22 10.9091 Tf -190.113 -24.572 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ -7.776 -8.022 Td [(2)]TJ/F23 7.9701 Tf 4.234 0 Td [(LH)]TJ/F26 7.9701 Tf 13.33 0 Td [(\000)]TJ/F23 7.9701 Tf 6.586 0 Td [(P)-148(R)-6(E)]TJ/F15 10.9091 Tf 23.154 3.434 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(C)]TJ/F15 10.9091 Tf 14.933 3.434 Td [(+)]TJ/F28 10.9091 Tf 10.909 8.836 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.836 Td [(I)]TJ/F25 10.9091 Tf 8.075 0 Td [(\000)]TJ/F22 10.9091 Tf 10.91 0 Td [(M)]TJ/F26 7.9701 Tf 11.772 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(C)]TJ/F22 10.9091 Tf 12.508 3.434 Td [(A)]TJ/F28 10.9091 Tf 8.182 8.836 Td [(\001)]TJ/F22 10.9091 Tf 6.818 -8.836 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ -7.776 -8.022 Td [(1)]TJ/F23 7.9701 Tf 4.235 0 Td [(L)]TJ/F22 10.9091 Tf 8.274 3.434 Td [(:)]TJ/F15 10.9091 Tf -300.208 -24.572 Td [(On)-368(the)-369(other)-368(hand,)-377(b)27(y)-368(applying)-368(the)-369(smo)-28(other)-368(after)-369(the)-368(coarse-lev)27(el)-368(correction,)-377(i.e.)-369(b)28(y)]TJ 0 -13.55 Td [(computing)]TJ/F22 10.9091 Tf 150.505 -12.905 Td [(w)]TJ/F15 10.9091 Tf 11.134 0 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.587 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.021 Td [(C)]TJ/F22 10.9091 Tf 12.508 3.434 Td [(v)-36(;)]TJ -45.741 -13.789 Td [(z)]TJ/F15 10.9091 Tf 8.584 0 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(w)]TJ/F15 10.9091 Tf 10.528 0 Td [(+)]TJ/F22 10.9091 Tf 10.909 0 Td [(M)]TJ/F26 7.9701 Tf 11.772 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ -7.776 -8.022 Td [(1)]TJ/F23 7.9701 Tf 4.234 0 Td [(L)]TJ/F15 10.9091 Tf 8.274 3.434 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(v)]TJ/F25 10.9091 Tf 8.103 0 Td [(\000)]TJ/F22 10.9091 Tf 10.909 0 Td [(Aw)]TJ/F15 10.9091 Tf 16.286 0 Td [(\051)]TJ/F22 10.9091 Tf 4.242 0 Td [(;)]TJ/F15 10.9091 Tf -258.915 -19.188 Td [(the)]TJ/F18 10.9091 Tf 18.788 0 Td [(two-level)-358(hybrid)-357(p)51(ost-smo)51(othe)51(d)]TJ/F15 10.9091 Tf 148.756 0 Td [(Sc)28(h)28(w)27(arz)-333(preconditioner)-333(is)-334(ob)1(tained:)]TJ/F22 10.9091 Tf -60.972 -24.572 Td [(M)]TJ/F26 7.9701 Tf 11.772 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ -7.776 -8.022 Td [(2)]TJ/F23 7.9701 Tf 4.234 0 Td [(LH)]TJ/F26 7.9701 Tf 13.329 0 Td [(\000)]TJ/F23 7.9701 Tf 6.587 0 Td [(P)-148(O)-29(S)-56(T)]TJ/F15 10.9091 Tf 28.436 3.434 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ -7.775 -8.022 Td [(1)]TJ/F23 7.9701 Tf 4.234 0 Td [(L)]TJ/F15 10.9091 Tf 10.698 3.434 Td [(+)]TJ/F28 10.9091 Tf 10.909 8.836 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.836 Td [(I)]TJ/F25 10.9091 Tf 8.076 0 Td [(\000)]TJ/F22 10.9091 Tf 10.909 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ -7.776 -8.022 Td [(1)]TJ/F23 7.9701 Tf 4.234 0 Td [(L)]TJ/F22 10.9091 Tf 8.274 3.434 Td [(A)]TJ/F28 10.9091 Tf 8.182 8.836 Td [(\001)]TJ/F22 10.9091 Tf 6.818 -8.836 Td [(M)]TJ/F26 7.9701 Tf 11.773 4.588 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F23 7.9701 Tf -7.776 -8.022 Td [(C)]TJ/F22 10.9091 Tf 12.508 3.434 Td [(:)]TJ/F15 10.9091 Tf -302.849 -24.572 Td [(One)-222(more)-222(v)55(arian)28(t)-222(of)-222(t)27(w)28(o-lev)28(el)-222(h)27(yb)1(rid)-223(precondi)1(tioner)-223(is)-222(obtained)-222(b)28(y)-222(applying)-222(the)-223(smo)-27(other)]TJ 0 -13.55 Td [(b)-28(efore)-239(and)-239(after)-239(the)-240(coarse-lev)28(el)-239(correction.)-413(In)-239(this)-240(case,)-258(the)-239(preconditioner)-239(is)-239(symme)-1(tr)1(ic)]TJ 0 -13.549 Td [(if)]TJ/F22 10.9091 Tf 10 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(,)]TJ/F22 10.9091 Tf 6.666 0 Td [(M)]TJ/F20 7.9701 Tf 10.584 -1.688 Td [(1)]TJ/F23 7.9701 Tf 4.234 0 Td [(L)]TJ/F15 10.9091 Tf 9.893 1.688 Td [(and)]TJ/F22 10.9091 Tf 21.213 0 Td [(M)]TJ/F23 7.9701 Tf 10.583 -1.688 Td [(C)]TJ/F15 10.9091 Tf 10.733 1.688 Td [(are)-333(symmetric.)]TJ -75.152 -13.57 Td [(As)-533(previou)1(s)-1(l)1(y)-533(noted,)-582(on)-533(parallel)-532(computers)-533(the)-532(n)28(um)27(b)-27(er)-533(of)-532(submatrices)-533(usually)]TJ -16.936 -13.55 Td [(matc)28(hes)-307(the)-307(n)28(um)28(b)-28(er)-307(of)-307(a)28(v)56(ailable)-307(pro)-28(cessors.)-436(When)-307(t)1(he)-307(size)-307(of)-307(the)-307(system)-307(to)-307(b)-28(e)-306(precon-)]TJ 0 -13.549 Td [(ditioned)-300(is)-300(v)28(ery)-300(large,)-307(the)-300(use)-301(of)-300(man)28(y)-300(pro)-28(cessors,)-307(i.e.)-300(of)-300(man)28(y)-300(small)-300(submatrices,)-307(often)]TJ 0 -13.549 Td [(leads)-314(to)-314(a)-314(large)-314(coarse-lev)28(el)-314(syste)-1(m,)-317(whose)-314(s)-1(olu)1(tion)-314(ma)27(y)-314(b)-27(e)-314(com)-1(p)1(utationally)-314(exp)-28(ensiv)28(e.)]TJ 0 -13.549 Td [(On)-385(the)-384(other)-385(hand,)-397(the)-385(use)-385(of)-384(few)-385(pro)-28(cessors)-385(often)-384(leads)-385(to)-385(lo)-27(cal)-385(sumatrices)-385(that)-384(are)]TJ 0 -13.549 Td [(to)-28(o)-424(exp)-28(ensiv)28(e)-424(to)-424(b)-28(e)-424(pro)-27(cesse)-1(d)-423(on)-424(single)-424(pro)-28(cessors,)-447(b)-28(ecause)-424(of)-424(memory)-424(and/or)-424(com-)]TJ 0 -13.55 Td [(puting)-324(requiremen)28(ts.)-441(Therefore,)-326(it)-324(seems)-324(natural)-324(to)-324(use)-324(a)-323(rec)-1(u)1(rs)-1(i)1(v)27(e)-324(approac)28(h,)-326(i)1(n)-324(whic)27(h)]TJ 0 -13.549 Td [(the)-394(coarse-)-1(l)1(e)-1(v)28(el)-394(correction)-395(is)-394(re-applied)-394(starting)-395(from)-394(the)-394(curren)27(t)-394(coarse-lev)28(el)-395(system.)]TJ 0 -13.549 Td [(The)-321(corresp)-28(onding)-320(preconditioners,)-324(called)]TJ/F18 10.9091 Tf 202.455 0 Td [(multi-level)]TJ/F15 10.9091 Tf 53.694 0 Td [(preconditioners,)-323(can)-321(signi\014can)28(tly)]TJ -256.149 -13.549 Td [(reduce)-282(the)-283(computational)-282(cost)-282(of)-283(preconditioning)-282(with)-282(resp)-28(ect)-283(to)-282(the)-282(t)28(w)27(o-lev)28(el)-282(case)-283(\050see)]TJ 0 -13.549 Td [([)]TJ
1 0 0 rg 1 0 0 RG
-1 0 0 1 -89.4301 -370.2545 cm
-BT
-/F15 10.9091 Tf 89.4301 370.2545 Td[(20)]TJ
-ET
-1 0 0 1 100.3392 370.2545 cm
+ [(20)]TJ
0 g 0 G
-1 0 0 1 -100.3392 -370.2545 cm
-BT
-/F15 10.9091 Tf 100.3392 370.2545 Td[(,)-320(Chap)1(te)-1(r)-317(3])1(\051.)-439(Add)1(itiv)28(e)-317(and)-316(h)28(ybr)1(id)-317(m)28(ult)1(ilev)28(e)-1(l)-317(p)1(re)-1(cond)1(ition)1(e)-1(r)1(s)-318(ar)1(e)-318(obt)1(ained)-317(as)-317(dir)1(e)-1(ct)]TJ -13.9394 -13.5492 Td[(exte)-1(n)1(s)-1(i)1(ons)-359(of)-359(th)1(e)-360(t)28(w)28(o-)-1(lev)28(el)-359(c)-1(ou)1(n)28(terpar)1(ts)-1(.)-521(F)84(or)-359(a)-359(detailed)-359(d)1(e)-1(sc)-1(r)1(ition)-358(of)-359(th)1(e)-1(m,)-365(the)-359(reader)]TJ 0 -13.5492 Td[(is)-321(referred)-321(to)-321([)]TJ
-ET
-1 0 0 1 153.7247 343.1561 cm
+ [(,)-320(Chapter)-317(3]\051.)-439(Additiv)28(e)-318(and)-317(h)28(ybrid)-317(m)28(ultilev)28(el)-317(preconditioners)-317(are)-317(obtained)-317(as)-318(direct)]TJ 0 -13.549 Td [(extensions)-359(of)-359(the)-359(t)28(w)27(o-lev)28(el)-359(coun)28(terparts.)-522(F)84(or)-359(a)-359(detailed)-359(descrition)-359(of)-359(them,)-366(the)-359(reader)]TJ 0 -13.55 Td [(is)-321(referred)-321(to)-321([)]TJ
1 0 0 rg 1 0 0 RG
-1 0 0 1 -153.7247 -343.1561 cm
-BT
-/F15 10.9091 Tf 153.7247 343.1561 Td[(20)]TJ
-ET
-1 0 0 1 164.6338 343.1561 cm
+ [(20)]TJ
0 g 0 G
-1 0 0 1 -164.6338 -343.1561 cm
-BT
-/F15 10.9091 Tf 164.6338 343.1561 Td[(,)-323(C)-1(h)1(apter)-321(3])1(.)-441(Th)1(e)-322(algor)1(ith)1(m)-322(f)1(or)-321(the)-321(app)1(lication)-320(of)-321(a)-321(m)27(u)1(lti-lev)28(e)-1(l)-320(h)28(ybr)1(id)]TJ -78.234 -13.5492 Td[(p)-27(os)-1(t-sm)-1(o)-28(ot)1(hed)-335(pr)1(e)-1(cond)1(ition)1(e)-1(r)]TJ/F24 10.9091 Tf 145.6758 0 Td[(M)]TJ/F15 10.9091 Tf 15.4287 0 Td[(to)-335(a)-335(v)28(e)-1(ctor)]TJ/F24 10.9091 Tf 55.2408 0 Td[(v)]TJ/F15 10.9091 Tf 5.6793 0 Td[(,)-335(i.e.)-335(for)-335(th)1(e)-336(comput)1(ation)-335(of)]TJ/F24 10.9091 Tf 135.9109 0 Td[(w)]TJ/F15 10.9091 Tf 11.1665 0 Td[(=)]TJ/F24 10.9091 Tf 11.5479 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F24 10.9091 Tf 4.7323 -3.9588 Td[(v)]TJ/F15 10.9091 Tf 5.6793 0 Td[(,)]TJ -409.4208 -13.5492 Td[(is)-311(rep)-27(orted,)-315(f)1(or)-311(example,)-315(in)-310(Figu)1(re)]TJ
-ET
-1 0 0 1 252.6225 316.0577 cm
+ [(,)-324(Ch)1(apter)-321(3].)-441(The)-321(algorithm)-321(for)-321(the)-321(application)-321(of)-321(a)-321(m)28(ulti-lev)28(el)-321(h)28(ybrid)]TJ 0 -13.549 Td [(p)-28(ost-smo)-28(othed)-335(preconditioner)]TJ/F22 10.9091 Tf 145.676 0 Td [(M)]TJ/F15 10.9091 Tf 15.428 0 Td [(to)-335(a)-335(v)28(e)-1(ctor)]TJ/F22 10.9091 Tf 55.241 0 Td [(v)]TJ/F15 10.9091 Tf 5.68 0 Td [(,)-336(i.)1(e)-1(.)-335(for)-335(the)-335(computation)-335(of)]TJ/F22 10.9091 Tf 135.91 0 Td [(w)]TJ/F15 10.9091 Tf 11.167 0 Td [(=)]TJ/F22 10.9091 Tf 11.548 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 3.959 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.732 -3.959 Td [(v)]TJ/F15 10.9091 Tf 5.68 0 Td [(,)]TJ -409.421 -13.549 Td [(is)-311(rep)-27(orted,)-316(for)-310(example,)-315(in)-311(Figure)]TJ
0 0 1 rg 0 0 1 RG
-1 0 0 1 -252.6225 -316.0577 cm
-BT
-/F15 10.9091 Tf 252.6225 316.0577 Td[(1)]TJ
-ET
-1 0 0 1 258.077 316.0577 cm
+ [-311(1)]TJ
0 g 0 G
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+/F22 9.9626 Tf -322.451 -45.783 Td [(v)]TJ/F7 6.9738 Tf 4.829 -1.495 Td [(1)]TJ/F15 9.9626 Tf 7.237 1.495 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(v)]TJ/F15 9.9626 Tf 5.187 0 Td [(;)]TJ/F43 9.9626 Tf -27.769 -17.625 Td [(for)]TJ/F22 9.9626 Tf 17.767 0 Td [(l)]TJ/F15 9.9626 Tf 5.936 0 Td [(=)-278(2)]TJ/F22 9.9626 Tf 15.497 0 Td [(;)-167(nl)-19(ev)]TJ/F43 9.9626 Tf 27.221 0 Td [(do)]TJ/F15 9.9626 Tf -56.458 -14.789 Td [(!)-444(transfer)]TJ/F22 9.9626 Tf 44.112 0 Td [(v)]TJ/F10 6.9738 Tf 4.829 -1.495 Td [(l)]TJ/F13 6.9738 Tf 2.56 0 Td [(\000)]TJ/F7 6.9738 Tf 6.227 0 Td [(1)]TJ/F15 9.9626 Tf 7.79 1.495 Td [(to)-333(the)-334(next)-333(coarser)-333(lev)27(el)]TJ/F22 9.9626 Tf -65.518 -11.956 Td [(v)]TJ/F10 6.9738 Tf 4.829 -1.494 Td [(l)]TJ/F15 9.9626 Tf 5.825 1.494 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(R)]TJ/F10 6.9738 Tf 7.565 -1.494 Td [(l)]TJ/F22 9.9626 Tf 3.058 1.494 Td [(v)]TJ/F10 6.9738 Tf 4.829 -1.494 Td [(l)]TJ/F13 6.9738 Tf 2.56 0 Td [(\000)]TJ/F7 6.9738 Tf 6.227 0 Td [(1)]TJ/F43 9.9626 Tf -55.372 -13.295 Td [(endfor)]TJ/F15 9.9626 Tf 0 -17.625 Td [(!)-444(apply)-334(the)-333(coarsest-lev)28(el)-334(correction)]TJ/F22 9.9626 Tf 0 -15.177 Td [(y)]TJ/F10 6.9738 Tf 4.885 -1.494 Td [(nl)-6(ev)]TJ/F15 9.9626 Tf 18.765 1.494 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(A)]TJ/F13 6.9738 Tf 7.472 4.262 Td [(\000)]TJ/F7 6.9738 Tf 6.227 0 Td [(1)]TJ/F10 6.9738 Tf -6.227 -7.267 Td [(nl)-6(ev)]TJ/F22 9.9626 Tf 15.998 3.005 Td [(v)]TJ/F10 6.9738 Tf 4.829 -1.494 Td [(nl)-6(ev)]TJ/F43 9.9626 Tf -62.465 -16.131 Td [(for)]TJ/F22 9.9626 Tf 17.767 0 Td [(l)]TJ/F15 9.9626 Tf 5.936 0 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(nl)-20(ev)]TJ/F25 9.9626 Tf 21.188 0 Td [(\000)]TJ/F15 9.9626 Tf 9.962 0 Td [(1)]TJ/F22 9.9626 Tf 4.982 0 Td [(;)]TJ/F15 9.9626 Tf 4.428 0 Td [(1)]TJ/F22 9.9626 Tf 4.981 0 Td [(;)]TJ/F25 9.9626 Tf 4.428 0 Td [(\000)]TJ/F15 9.9626 Tf 7.749 0 Td [(1)]TJ/F43 9.9626 Tf 8.8 0 Td [(do)]TJ/F15 9.9626 Tf -90.774 -14.789 Td [(!)-444(transfer)]TJ/F22 9.9626 Tf 44.112 0 Td [(y)]TJ/F10 6.9738 Tf 4.885 -1.495 Td [(l)]TJ/F7 6.9738 Tf 2.56 0 Td [(+1)]TJ/F15 9.9626 Tf 13.906 1.495 Td [(to)-333(the)-334(next)-333(\014ner)-333(lev)28(e)-1(l)]TJ/F22 9.9626 Tf -65.463 -11.968 Td [(y)]TJ/F10 6.9738 Tf 4.884 -1.494 Td [(l)]TJ/F15 9.9626 Tf 5.826 1.494 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(R)]TJ/F10 6.9738 Tf 7.641 3.616 Td [(T)]TJ -0.077 -6.437 Td [(l)]TJ/F7 6.9738 Tf 2.56 0 Td [(+1)]TJ/F22 9.9626 Tf 10.586 2.821 Td [(y)]TJ/F10 6.9738 Tf 4.884 -1.494 Td [(l)]TJ/F7 6.9738 Tf 2.56 0 Td [(+1)]TJ/F15 9.9626 Tf 10.585 1.494 Td [(;)]TJ -59.965 -14.855 Td [(!)-444(com)-1(p)1(ute)-334(the)-333(residual)-333(at)-334(the)-333(curren)28(t)-334(lev)28(el)]TJ/F22 9.9626 Tf 0 -12.342 Td [(r)]TJ/F10 6.9738 Tf 4.495 -1.495 Td [(l)]TJ/F15 9.9626 Tf 5.825 1.495 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(v)]TJ/F10 6.9738 Tf 4.829 -1.495 Td [(l)]TJ/F25 9.9626 Tf 5.272 1.495 Td [(\000)]TJ/F22 9.9626 Tf 9.963 0 Td [(A)]TJ/F13 6.9738 Tf 7.472 4.262 Td [(\000)]TJ/F7 6.9738 Tf 6.226 0 Td [(1)]TJ/F10 6.9738 Tf -6.226 -7.268 Td [(l)]TJ/F22 9.9626 Tf 10.696 3.006 Td [(y)]TJ/F10 6.9738 Tf 4.884 -1.495 Td [(l)]TJ/F15 9.9626 Tf 3.058 1.495 Td [(;)]TJ -67.01 -14.79 Td [(!)-444(apply)-334(the)-333(basic)-333(Sc)27(h)28(w)28(arz)-333(preconditioner)-334(to)-333(the)-333(residual)]TJ/F22 9.9626 Tf 0 -12.343 Td [(r)]TJ/F10 6.9738 Tf 4.495 -1.494 Td [(l)]TJ/F15 9.9626 Tf 5.825 1.494 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(M)]TJ/F13 6.9738 Tf 10.751 4.262 Td [(\000)]TJ/F7 6.9738 Tf 6.227 0 Td [(1)]TJ/F10 6.9738 Tf -7.313 -7.267 Td [(l)]TJ/F22 9.9626 Tf 11.782 3.005 Td [(r)]TJ/F10 6.9738 Tf 4.495 -1.494 Td [(l)]TJ/F15 9.9626 Tf -46.778 -13.295 Td [(!)-444(up)-28(date)]TJ/F22 9.9626 Tf 40.681 0 Td [(y)]TJ/F10 6.9738 Tf 4.884 -1.495 Td [(l)]TJ/F22 9.9626 Tf -45.565 -10.461 Td [(y)]TJ/F10 6.9738 Tf 4.884 -1.494 Td [(l)]TJ/F15 9.9626 Tf 5.826 1.494 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(y)]TJ/F10 6.9738 Tf 4.884 -1.494 Td [(l)]TJ/F15 9.9626 Tf 5.272 1.494 Td [(+)]TJ/F22 9.9626 Tf 9.963 0 Td [(r)]TJ/F10 6.9738 Tf 4.495 -1.494 Td [(l)]TJ/F43 9.9626 Tf -55.803 -10.461 Td [(endfor)]TJ/F22 9.9626 Tf 0 -14.79 Td [(w)]TJ/F15 9.9626 Tf 10.168 0 Td [(=)]TJ/F22 9.9626 Tf 10.516 0 Td [(y)]TJ/F7 6.9738 Tf 4.885 -1.494 Td [(1)]TJ/F15 9.9626 Tf 4.469 1.494 Td [(;)]TJ
+ET
q
-[]0 d
-0 J
-0.3985 w
-0 0.1992 m
-268.3164 0.1992 l
-S
+1 0 0 1 165.668 714.896 cm
+[]0 d 0 J 0.398 w 0 0 m 268.316 0 l S
Q
-1 0 0 1 0 -271.2158 cm
q
-[]0 d
-0 J
-0.3985 w
-0.1992 0 m
-0.1992 271.2158 l
-S
+1 0 0 1 165.867 443.282 cm
+[]0 d 0 J 0.398 w 0 0 m 0 271.614 l S
Q
-1 0 0 1 1.5691 259.8584 cm
-0 g 0 G
-0 g 0 G
-1 0 0 1 -167.2367 -703.3393 cm
-BT
-/F24 9.9626 Tf 172.6912 694.2188 Td[(v)]TJ/F7 6.9738 Tf 4.8291 -1.4944 Td[(1)]TJ/F15 9.9626 Tf 7.2367 1.4944 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(v)]TJ/F15 9.9626 Tf 5.1866 0 Td[(;)]TJ/F45 9.9626 Tf -27.7685 -17.6244 Td[(for)]TJ/F24 9.9626 Tf 17.7666 0 Td[(l)]TJ/F15 9.9626 Tf 5.936 0 Td[(=)-278(2)]TJ/F24 9.9626 Tf 15.4974 0 Td[(;)-166(n)-1(l)-20(e)1(v)]TJ/F45 9.9626 Tf 27.2209 0 Td[(do)]TJ/F15 9.9626 Tf -56.4583 -14.7898 Td[(!)-444(tran)1(s)-1(f)1(e)-1(r)]TJ/F24 9.9626 Tf 44.1125 0 Td[(v)]TJ/F10 6.9738 Tf 4.8291 -1.4944 Td[(l)]TJ/F13 6.9738 Tf 2.5599 0 Td[(\000)]TJ/F7 6.9738 Tf 6.2266 0 Td[(1)]TJ/F15 9.9626 Tf 7.7903 1.4944 Td[(to)-333(the)-333(next)-333(c)-1(oarser)-333(le)-1(v)28(el)]TJ/F24 9.9626 Tf -65.5183 -11.9552 Td[(v)]TJ/F10 6.9738 Tf 4.8291 -1.4944 Td[(l)]TJ/F15 9.9626 Tf 5.8253 1.4944 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(R)]TJ/F10 6.9738 Tf 7.5645 -1.4944 Td[(l)]TJ/F24 9.9626 Tf 3.058 1.4944 Td[(v)]TJ/F10 6.9738 Tf 4.8291 -1.4944 Td[(l)]TJ/F13 6.9738 Tf 2.5599 0 Td[(\000)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1)]TJ/F45 9.9626 Tf -55.3714 -13.2954 Td[(endfor)]TJ/F15 9.9626 Tf 0 -17.6244 Td[(!)-444(app)1(ly)-333(the)-333(c)-1(oarses)-1(t-lev)28(e)-1(l)-333(correc)-1(ti)1(on)]TJ/F24 9.9626 Tf 0 -15.1773 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(nl)-6(ev)]TJ/F15 9.9626 Tf 18.7654 1.4944 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(A)]TJ/F13 6.9738 Tf 7.472 4.2618 Td[(\000)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1)]TJ/F10 6.9738 Tf -6.2267 -7.2671 Td[(nl)-6(ev)]TJ/F24 9.9626 Tf 15.9981 3.0053 Td[(v)]TJ/F10 6.9738 Tf 4.8291 -1.4944 Td[(nl)-6(ev)]TJ/F45 9.9626 Tf -62.4652 -16.13 Td[(for)]TJ/F24 9.9626 Tf 17.7666 0 Td[(l)]TJ/F15 9.9626 Tf 5.936 0 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(nl)-20(ev)]TJ/F27 9.9626 Tf 21.1879 0 Td[(\000)]TJ/F15 9.9626 Tf 9.9626 0 Td[(1)]TJ/F24 9.9626 Tf 4.9813 0 Td[(;)]TJ/F15 9.9626 Tf 4.4278 0 Td[(1)]TJ/F24 9.9626 Tf 4.9814 0 Td[(;)]TJ/F27 9.9626 Tf 4.4278 0 Td[(\000)]TJ/F15 9.9626 Tf 7.7487 0 Td[(1)]TJ/F45 9.9626 Tf 8.8003 0 Td[(do)]TJ/F15 9.9626 Tf -90.7739 -14.7898 Td[(!)-444(tran)1(s)-1(f)1(e)-1(r)]TJ/F24 9.9626 Tf 44.1124 0 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(l)]TJ/F7 6.9738 Tf 2.5599 0 Td[(+1)]TJ/F15 9.9626 Tf 13.9062 1.4944 Td[(to)-333(the)-333(next)-333(\014ner)-333(lev)28(e)-1(l)]TJ/F24 9.9626 Tf -65.463 -11.9674 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(l)]TJ/F15 9.9626 Tf 5.8253 1.4944 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(R)]TJ/F10 6.9738 Tf 7.6415 3.6153 Td[(T)]TJ -0.077 -6.4368 Td[(l)]TJ/F7 6.9738 Tf 2.5599 0 Td[(+1)]TJ/F24 9.9626 Tf 10.5854 2.8215 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(l)]TJ/F7 6.9738 Tf 2.5598 0 Td[(+1)]TJ/F15 9.9626 Tf 10.5854 1.4944 Td[(;)]TJ -59.9654 -14.855 Td[(!)-444(c)-1(ompu)1(te)-334(th)1(e)-334(res)-1(i)1(du)1(al)-334(at)-333(th)1(e)-334(c)-1(u)1(rr)1(e)-1(n)28(t)-333(lev)28(e)-1(l)]TJ/F24 9.9626 Tf 0 -12.3426 Td[(r)]TJ/F10 6.9738 Tf 4.4947 -1.4944 Td[(l)]TJ/F15 9.9626 Tf 5.8253 1.4944 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(v)]TJ/F10 6.9738 Tf 4.8291 -1.4944 Td[(l)]TJ/F27 9.9626 Tf 5.2719 1.4944 Td[(\000)]TJ/F24 9.9626 Tf 9.9626 0 Td[(A)]TJ/F13 6.9738 Tf 7.472 4.2618 Td[(\000)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1)]TJ/F10 6.9738 Tf -6.2267 -7.2671 Td[(l)]TJ/F24 9.9626 Tf 10.696 3.0053 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(l)]TJ/F15 9.9626 Tf 3.058 1.4944 Td[(;)]TJ -67.0102 -14.7898 Td[(!)-444(app)1(ly)-333(the)-333(basic)-334(Sc)28(h)28(w)28(arz)-334(p)1(re)-1(cond)1(ition)1(e)-1(r)-333(to)-333(th)1(e)-334(res)-1(i)1(dual)]TJ/F24 9.9626 Tf 0 -12.3426 Td[(r)]TJ/F10 6.9738 Tf 4.4947 -1.4944 Td[(l)]TJ/F15 9.9626 Tf 5.8253 1.4944 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(M)]TJ/F13 6.9738 Tf 10.7513 4.2618 Td[(\000)]TJ/F7 6.9738 Tf 6.2267 0 Td[(1)]TJ/F10 6.9738 Tf -7.3128 -7.2671 Td[(l)]TJ/F24 9.9626 Tf 11.7822 3.0053 Td[(r)]TJ/F10 6.9738 Tf 4.4947 -1.4944 Td[(l)]TJ/F15 9.9626 Tf -46.7782 -13.2954 Td[(!)-444(up)-27(date)]TJ/F24 9.9626 Tf 40.6809 0 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(l)]TJ/F24 9.9626 Tf -45.5654 -10.4608 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(l)]TJ/F15 9.9626 Tf 5.8253 1.4944 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(y)]TJ/F10 6.9738 Tf 4.8845 -1.4944 Td[(l)]TJ/F15 9.9626 Tf 5.2719 1.4944 Td[(+)]TJ/F24 9.9626 Tf 9.9626 0 Td[(r)]TJ/F10 6.9738 Tf 4.4947 -1.4944 Td[(l)]TJ/F45 9.9626 Tf -55.8022 -10.4608 Td[(endfor)]TJ/F24 9.9626 Tf 0 -14.7898 Td[(w)]TJ/F15 9.9626 Tf 10.1678 0 Td[(=)]TJ/F24 9.9626 Tf 10.5161 0 Td[(y)]TJ/F7 6.9738 Tf 4.8845 -1.4944 Td[(1)]TJ/F15 9.9626 Tf 4.4694 1.4944 Td[(;)]TJ
-ET
-1 0 0 1 433.5855 443.4809 cm
q
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-0 J
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-0.1992 0 m
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+1 0 0 1 433.785 443.282 cm
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Q
-1 0 0 1 -267.9179 -0.3985 cm
q
-[]0 d
-0 J
-0.3985 w
-0 0.1992 m
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-S
+1 0 0 1 165.668 443.282 cm
+[]0 d 0 J 0.398 w 0 0 m 268.316 0 l S
Q
-1 0 0 1 -51.4935 -18.5347 cm
0 g 0 G
-1 0 0 1 -114.1741 -424.5477 cm
BT
-/F15 10.9091 Tf 114.1741 424.5477 Td[(Figu)1(re)-334(1:)-444(Ap)1(pli)1(c)-1(ation)-332(of)-333(the)-334(m)28(ul)1(ti-le)-1(v)28(el)-333(h)28(yb)1(rid)-333(p)-27(ost-)-1(smo)-28(othed)-333(pr)1(e)-1(cond)1(ition)1(e)-1(r)1(.)]TJ
-ET
-1 0 0 1 485.4775 424.5477 cm
+/F15 10.9091 Tf 114.174 424.548 Td [(Figure)-333(1:)-445(Application)-333(of)-333(the)-333(m)27(ulti-lev)28(el)-333(h)28(ybrid)-334(p)-27(ost-smo)-28(othed)-334(p)1(re)-1(cond)1(itioner.)]TJ
0 g 0 G
-1 0 0 1 -391.8773 -14.0763 cm
0 g 0 G
-1 0 0 1 0 -32.1522 cm
0 g 0 G
-1 0 0 1 -93.6002 -378.3192 cm
-BT
-/F15 10.9091 Tf 106.9336 378.3192 Td[(2.)]TJ
-ET
-1 0 0 1 115.4185 378.3192 cm
+ -7.24 -46.229 Td [(2.)]TJ
0 g 0 G
-1 0 0 1 -115.4185 -378.3192 cm
-BT
-/F15 10.9091 Tf 120.873 378.3192 Td[(construction)-333(of)-332(the)-334(p)1(rolon)1(gator)]TJ/F24 10.9091 Tf 152.5759 0 Td[(R)]TJ/F25 7.9701 Tf 8.3675 3.9588 Td[(T)]TJ -0.0843 -7.1916 Td[(C)]TJ/F15 10.9091 Tf 7.0968 3.2328 Td[(;)]TJ
-ET
-1 0 0 1 93.6002 353.2676 cm
+ [-500(construction)-333(of)-333(the)-334(prolongator)]TJ/F22 10.9091 Tf 166.515 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.959 Td [(T)]TJ -0.084 -7.192 Td [(C)]TJ/F15 10.9091 Tf 7.097 3.233 Td [(;)]TJ
0 g 0 G
-1 0 0 1 -93.6002 -353.2676 cm
-BT
-/F15 10.9091 Tf 106.9336 353.2676 Td[(3.)]TJ
-ET
-1 0 0 1 115.4185 353.2676 cm
+ -181.895 -25.051 Td [(3.)]TJ
0 g 0 G
-1 0 0 1 -115.4185 -353.2676 cm
-BT
-/F15 10.9091 Tf 120.873 353.2676 Td[(app)1(lication)-333(of)]TJ/F24 10.9091 Tf 68.788 0 Td[(R)]TJ/F25 7.9701 Tf 8.2831 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 10.7332 1.6887 Td[(and)]TJ/F24 10.9091 Tf 21.2121 0 Td[(R)]TJ/F25 7.9701 Tf 8.3674 3.9588 Td[(T)]TJ -0.0842 -7.1916 Td[(C)]TJ/F15 10.9091 Tf 10.7331 3.2328 Td[(to)-333(bu)1(ild)]TJ/F24 10.9091 Tf 41.2122 0 Td[(A)]TJ/F25 7.9701 Tf 8.1818 -1.6887 Td[(C)]TJ/F15 10.9091 Tf 7.0968 1.6887 Td[(.)]TJ -194.8598 -25.0516 Td[(T)83(o)-426(p)-28(erf)1(orm)-426(the)-427(coarse)-1(n)1(ing)-426(ste)-1(p)1(,)-449(w)27(e)-427(h)1(a)28(v)28(e)-427(impleme)-1(n)28(t)1(e)-1(d)-426(th)1(e)-427(aggregation)-426(al)1(gorith)1(m)]TJ -16.9365 -13.5492 Td[(sk)28(e)-1(tc)28(hed)-389(i)1(n)-388([)]TJ
-ET
-1 0 0 1 154.5615 314.6668 cm
+ [-500(application)-333(of)]TJ/F22 10.9091 Tf 82.727 0 Td [(R)]TJ/F23 7.9701 Tf 8.283 -1.689 Td [(C)]TJ/F15 10.9091 Tf 10.733 1.689 Td [(and)]TJ/F22 10.9091 Tf 21.212 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.958 Td [(T)]TJ -0.084 -7.191 Td [(C)]TJ/F15 10.9091 Tf 10.733 3.233 Td [(to)-333(build)]TJ/F22 10.9091 Tf 41.212 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 -1.689 Td [(C)]TJ/F15 10.9091 Tf 7.097 1.689 Td [(.)]TJ -194.86 -25.052 Td [(T)83(o)-426(p)-28(erform)-426(the)-426(coarsening)-426(s)-1(tep,)-449(w)28(e)-427(ha)28(v)28(e)-426(implemen)27(ted)-426(the)-426(aggregation)-426(algorithm)]TJ -16.937 -13.549 Td [(sk)28(etc)27(hed)-388(in)-388([)]TJ
1 0 0 rg 1 0 0 RG
-1 0 0 1 -154.5615 -314.6668 cm
-BT
-/F15 10.9091 Tf 154.5615 314.6668 Td[(4)]TJ
-ET
-1 0 0 1 160.016 314.6668 cm
+ [(4)]TJ
0 g 0 G
-1 0 0 1 -160.016 -314.6668 cm
-BT
-/F15 10.9091 Tf 160.016 314.6668 Td[(].)-610(Accordi)1(ng)-388(to)-389([)]TJ
-ET
-1 0 0 1 242.1465 314.6668 cm
+ [(].)-610(According)-389(to)-388([)]TJ
1 0 0 rg 1 0 0 RG
-1 0 0 1 -242.1465 -314.6668 cm
-BT
-/F15 10.9091 Tf 242.1465 314.6668 Td[(24)]TJ
-ET
-1 0 0 1 253.0556 314.6668 cm
+ [(24)]TJ
0 g 0 G
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0 g 0 G
-1 0 0 1 -93.6002 -555.4954 cm
-BT
-/F15 10.9091 Tf 106.9336 555.4954 Td[(2.)]TJ
-ET
-1 0 0 1 115.4185 555.4954 cm
+ -13.939 -21.934 Td [(2.)]TJ
0 g 0 G
-1 0 0 1 -115.4185 -555.4954 cm
-BT
-/F20 10.9091 Tf 120.873 555.4954 Td[(A)25(l)-50(lo)51(c)51(ate)-364(an)-1(d)-364(ini)1(t)-1(i)1(aliz)1(e)-365(the)-364(pr)51(e)51(c)52(o)-1(nditi)1(o)-1(ner)-364(data)-364(structur)52(e,)-365(a)-1(c)52(c)51(or)51(ding)-364(to)-364(a)-365(pr)52(e)51(c)51(ond)-1(i)1(-)]TJ 0 -13.5492 Td[(tioner)-376(typ)51(e)-375(cho)-1(s)1(en)-377(by)-375(the)-376(user)]TJ/F15 10.9091 Tf 144.0045 0 Td[(.)-503(This)-353(is)-354(p)-27(erfor)1(m)-1(ed)-353(b)29(y)-353(the)-353(rou)1(tin)1(e)]TJ/F46 10.9091 Tf 169.417 0 Td[(mld_precinit)]TJ/F15 10.9091 Tf 68.7265 0 Td[(,)]TJ -382.148 -13.5492 Td[(whic)28(h)-406(al)1(s)-1(o)-405(s)-1(ets)-406(defaul)1(ts)-406(for)-405(e)-1(ac)28(h)-406(p)1(rec)-1(on)1(diti)1(oner)-406(t)28(yp)-27(e)-406(s)-1(elec)-1(ted)-405(b)28(y)-406(th)1(e)-406(use)-1(r)1(.)-662(The)]TJ 0 -13.5492 Td[(defau)1(lts)-404(ass)-1(o)-28(ciated)-404(to)-403(e)-1(ac)28(h)-404(p)1(re)-1(cond)1(ition)1(e)-1(r)-403(t)28(yp)-27(e)-405(ar)1(e)-404(giv)28(e)-1(n)-403(in)-403(T)83(abl)1(e)]TJ
-ET
-1 0 0 1 445.5028 514.8478 cm
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0 0 1 rg 0 0 1 RG
-1 0 0 1 -445.5028 -514.8478 cm
-BT
-/F15 10.9091 Tf 445.5028 514.8478 Td[(1)]TJ
-ET
-1 0 0 1 450.9574 514.8478 cm
+ [-404(1)]TJ
0 g 0 G
-1 0 0 1 -450.9574 -514.8478 cm
-BT
-/F15 10.9091 Tf 450.9574 514.8478 Td[(,)-421(where)-404(the)]TJ -330.0844 -13.5492 Td[(strin)1(gs)-306(use)-1(d)-304(b)28(y)]TJ/F46 10.9091 Tf 74.4455 0 Td[(mld_precinit)]TJ/F15 10.9091 Tf 72.0568 0 Td[(to)-305(iden)28(ti)1(fy)-305(th)1(e)-306(pr)1(e)-1(cond)1(ition)1(e)-1(r)-305(t)28(yp)-27(es)-306(are)-306(al)1(s)-1(o)-305(giv)28(en.)]TJ -146.5023 -13.5492 Td[(Note)-353(th)1(at)-352(thes)-1(e)-352(s)-1(tr)1(ings)-353(ar)1(e)-353(v)56(alid)-352(al)1(s)-1(o)-352(if)-352(u)1(pp)-27(e)-1(r)1(c)-1(ase)-353(letters)-353(are)-352(s)-1(u)1(bstitu)1(te)-1(d)-352(b)29(y)-353(cor-)]TJ 0 -13.5492 Td[(res)-1(p)-27(on)1(din)1(g)-334(l)1(o)27(w)28(e)-1(r)1(c)-1(ase)-334(on)1(e)-1(s.)]TJ
-ET
-1 0 0 1 93.6002 452.2668 cm
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0 g 0 G
-1 0 0 1 -93.6002 -452.2668 cm
-BT
-/F15 10.9091 Tf 106.9336 452.2668 Td[(3.)]TJ
-ET
-1 0 0 1 115.4185 452.2668 cm
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0 g 0 G
-1 0 0 1 -115.4185 -452.2668 cm
-BT
-/F20 10.9091 Tf 120.873 452.2668 Td[(Mo)51(di)1(fy)-340(the)-341(sel)1(e)51(cte)51(d)-341(pr)51(e)52(c)51(on)-1(di)1(t)-1(i)1(on)-1(er)-340(typ)51(e,)-343(by)-340(pr)51(op)51(erl)1(y)-340(setting)-341(pr)51(e)52(c)51(on)-1(di)1(t)-1(i)1(on)-1(er)-340(p)51(ar)52(am-)]TJ 0 -13.5492 Td[(eters.)]TJ/F15 10.9091 Tf 30.6685 0 Td[(This)-266(i)1(s)-266(p)-28(erf)1(orme)-1(d)-265(b)28(y)-265(th)1(e)-266(routi)1(ne)]TJ/F46 10.9091 Tf 155.1693 0 Td[(mld_precset)]TJ/F15 10.9091 Tf 62.9993 0 Td[(.)-422(Th)1(is)-266(rou)1(tine)-266(m)28(ust)-266(b)-27(e)-266(called)]TJ -248.8371 -13.5492 Td[(onl)1(y)-365(i)1(f)-364(the)-365(u)1(s)-1(er)-364(w)27(an)28(ts)-365(t)1(o)-365(mo)-28(di)1(fy)-364(the)-365(d)1(e)-1(f)1(aul)1(t)-365(v)56(alu)1(e)-1(s)-365(of)-364(th)1(e)-365(paramete)-1(r)1(s)-365(as)-1(so)-28(ciated)]TJ 0 -13.5492 Td[(to)-468(t)1(he)-468(se)-1(lec)-1(t)1(e)-1(d)-467(pr)1(e)-1(cond)1(ition)1(e)-1(r)-467(t)28(yp)-27(e)-1(,)-501(to)-467(obtai)1(n)-468(a)-467(v)56(arian)28(t)-467(of)-467(the)-468(p)1(re)-1(cond)1(ition)1(e)-1(r)1(.)]TJ 0 -13.5492 Td[(Ex)1(am)-1(p)1(les)-396(of)-394(u)1(s)-1(e)-395(of)]TJ/F46 10.9091 Tf 95.9773 0 Td[(mld_precset)]TJ/F15 10.9091 Tf 67.3042 0 Td[(are)-395(giv)28(en)-394(in)-394(Sec)-1(t)1(ion)]TJ
-ET
-1 0 0 1 384.4348 398.07 cm
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0 0 1 rg 0 0 1 RG
-1 0 0 1 -384.4348 -398.07 cm
-BT
-/F15 10.9091 Tf 384.4348 398.07 Td[(5.1)]TJ
-ET
-1 0 0 1 398.3742 398.07 cm
+ [-395(5.1)]TJ
0 g 0 G
-1 0 0 1 -398.3742 -398.07 cm
-BT
-/F15 10.9091 Tf 398.3742 398.07 Td[(;)-425(a)-395(complete)-395(list)-395(of)-394(all)]TJ -277.5012 -13.5492 Td[(th)1(e)-304(p)1(rec)-1(on)1(diti)1(oner)-303(p)1(arame)-1(ters)-303(an)1(d)-303(th)1(e)-1(i)1(r)-303(all)1(o)27(w)28(e)-1(d)-302(an)1(d)-303(d)1(e)-1(f)1(ault)-302(v)56(alues)-303(is)-303(pro)28(vi)1(ded)-303(i)1(n)]TJ 0 -13.5492 Td[(Section)]TJ
-ET
-1 0 0 1 159.0549 370.9716 cm
+ [(;)-425(a)-394(com)-1(p)1(le)-1(t)1(e)-395(list)-395(of)-394(all)]TJ -163.281 -13.549 Td [(the)-303(preconditioner)-302(parameters)-303(and)-303(their)-302(allo)27(w)28(ed)-303(and)-302(default)-303(v)56(alues)-303(is)-303(pro)28(vided)-303(in)]TJ 0 -13.549 Td [(Section)]TJ
0 0 1 rg 0 0 1 RG
-1 0 0 1 -159.0549 -370.9716 cm
-BT
-/F15 10.9091 Tf 159.0549 370.9716 Td[(6)]TJ
-ET
-1 0 0 1 164.5094 370.9716 cm
+ [-333(6)]TJ
0 g 0 G
-1 0 0 1 -164.5094 -370.9716 cm
-BT
-/F15 10.9091 Tf 164.5094 370.9716 Td[(,)-333(T)83(abl)1(e)-1(s)]TJ
-ET
-1 0 0 1 205.4791 370.9716 cm
+ [(,)-334(T)84(ables)]TJ
0 0 1 rg 0 0 1 RG
-1 0 0 1 -205.4791 -370.9716 cm
-BT
-/F15 10.9091 Tf 205.4791 370.9716 Td[(2)]TJ
-ET
-1 0 0 1 210.9337 370.9716 cm
+ [-334(2)]TJ
0 g 0 G
-1 0 0 1 -210.9337 -370.9716 cm
-BT
-/F15 10.9091 Tf 210.9337 370.9716 Td[(-)]TJ
-ET
-1 0 0 1 214.57 370.9716 cm
+ [(-)]TJ
0 0 1 rg 0 0 1 RG
-1 0 0 1 -214.57 -370.9716 cm
-BT
-/F15 10.9091 Tf 214.57 370.9716 Td[(5)]TJ
-ET
-1 0 0 1 220.0246 370.9716 cm
+ [(5)]TJ
0 g 0 G
-1 0 0 1 -220.0246 -370.9716 cm
-BT
-/F15 10.9091 Tf 220.0246 370.9716 Td[(.)]TJ
-ET
-1 0 0 1 93.6002 349.0382 cm
+ [(.)]TJ
0 g 0 G
-1 0 0 1 -93.6002 -349.0382 cm
-BT
-/F15 10.9091 Tf 106.9336 349.0382 Td[(4.)]TJ
-ET
-1 0 0 1 115.4185 349.0382 cm
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0 g 0 G
-1 0 0 1 -115.4185 -349.0382 cm
-BT
-/F20 10.9091 Tf 120.873 349.0382 Td[(Bui)1(ld)-457(the)-457(pr)51(e)51(c)51(onditioner)-457(f)1(or)-457(a)-457(given)-457(matrix.)]TJ/F15 10.9091 Tf 223.3234 0 Td[(This)-441(is)-442(p)-27(erfor)1(m)-1(ed)-441(b)28(y)-441(th)1(e)-442(r)1(outin)1(e)]TJ/F46 10.9091 Tf -223.3234 -13.5492 Td[(mld_precbld)]TJ/F15 10.9091 Tf 62.9993 0 Td[(.)]TJ
-ET
-1 0 0 1 93.6002 313.5556 cm
+/F18 10.9091 Tf 13.939 0 Td [(Build)-457(the)-457(pr)51(e)51(c)51(onditioner)-457(for)-457(a)-457(given)-457(matrix.)]TJ/F15 10.9091 Tf 223.323 0 Td [(This)-441(is)-442(p)-27(erformed)-441(b)27(y)-441(the)-441(routine)]TJ/F44 10.9091 Tf -223.323 -13.549 Td [(mld_precbld)]TJ/F15 10.9091 Tf 62.999 0 Td [(.)]TJ
0 g 0 G
-1 0 0 1 -93.6002 -313.5556 cm
-BT
-/F15 10.9091 Tf 106.9336 313.5556 Td[(5.)]TJ
-ET
-1 0 0 1 115.4185 313.5556 cm
+ -76.938 -21.933 Td [(5.)]TJ
0 g 0 G
-1 0 0 1 -115.4185 -313.5556 cm
-BT
-/F20 10.9091 Tf 120.873 313.5556 Td[(App)-1(l)1(y)-281(th)-1(e)-281(pr)51(e)51(c)51(onditioner)-281(at)-282(e)51(ach)-282(iter)52(at)-1(i)1(on)-282(of)-282(a)-281(Kryl)1(ov)-282(sol)1(ver.)]TJ/F15 10.9091 Tf 288.3113 0 Td[(This)-251(i)1(s)-251(p)-28(erf)1(orme)-1(d)-250(b)28(y)]TJ -288.3113 -13.5492 Td[(th)1(e)-341(rou)1(tine)]TJ/F46 10.9091 Tf 56.5445 0 Td[(mld_precaply)]TJ/F15 10.9091 Tf 68.7265 0 Td[(.)-465(Wh)1(e)-1(n)-339(using)-340(th)1(e)-341(PS)1(B)-1(LAS)-339(Kryl)1(o)27(v)-340(solv)28(ers,)-342(thi)1(s)-341(s)-1(t)1(e)-1(p)-340(i)1(s)]TJ -125.271 -13.5492 Td[(com)-1(p)1(le)-1(t)1(e)-1(ly)-301(tran)1(s)-1(p)1(aren)28(t)-302(to)-301(the)-302(u)1(s)-1(er,)-308(sin)1(c)-1(e)]TJ/F46 10.9091 Tf 196.4514 0 Td[(mld_precaply)]TJ/F15 10.9091 Tf 72.0175 0 Td[(is)-302(called)-302(b)29(y)-302(th)1(e)-302(PS)1(B)-1(LAS)]TJ -268.4689 -13.5492 Td[(rou)1(tin)1(e)-334(impleme)-1(n)28(ti)1(ng)-333(the)-333(Kryl)1(o)27(v)-333(solv)28(er)-333(\050)]TJ/F46 10.9091 Tf 195.9093 0 Td[(psb_krylov)]TJ/F15 10.9091 Tf 57.2721 0 Td[(\051.)]TJ
-ET
-1 0 0 1 93.6002 250.9746 cm
+/F18 10.9091 Tf 13.939 0 Td [(Apply)-282(the)-281(pr)51(e)51(c)51(onditioner)-281(at)-282(e)51(ach)-282(iter)52(ation)-282(of)-282(a)-281(Krylov)-282(solver.)]TJ/F15 10.9091 Tf 288.311 0 Td [(This)-251(is)-250(p)-28(erformed)-250(b)27(y)]TJ -288.311 -13.55 Td [(the)-340(routine)]TJ/F44 10.9091 Tf 56.544 0 Td [(mld_precaply)]TJ/F15 10.9091 Tf 68.727 0 Td [(.)-465(When)-340(using)-340(the)-341(PSBLAS)-340(Krylo)28(v)-340(solv)28(ers,)-342(this)-341(step)-340(is)]TJ -125.271 -13.549 Td [(completely)-302(transparen)28(t)-301(to)-302(the)-302(user,)-308(since)]TJ/F44 10.9091 Tf 196.451 0 Td [(mld_precaply)]TJ/F15 10.9091 Tf 72.018 0 Td [(is)-302(called)-301(b)27(y)-301(the)-302(PSBLAS)]TJ -268.469 -13.549 Td [(routine)-333(implemen)28(ting)-334(the)-333(Krylo)28(v)-334(solv)28(er)-333(\050)]TJ/F44 10.9091 Tf 195.909 0 Td [(psb_krylov)]TJ/F15 10.9091 Tf 57.272 0 Td [(\051.)]TJ
0 g 0 G
-1 0 0 1 -93.6002 -250.9746 cm
-BT
-/F15 10.9091 Tf 106.9336 250.9746 Td[(6.)]TJ
-ET
-1 0 0 1 115.4185 250.9746 cm
+ -267.12 -21.933 Td [(6.)]TJ
0 g 0 G
-1 0 0 1 -115.4185 -250.9746 cm
-BT
-/F20 10.9091 Tf 120.873 250.9746 Td[(F)77(r)51(e)51(e)-459(the)-459(pr)51(e)51(c)52(o)-1(nditi)1(o)-1(ner)-459(data)-459(structur)52(e)]TJ/F15 10.9091 Tf 183.9714 0 Td[(.)-775(Th)1(is)-444(is)-444(p)-27(e)-1(r)1(for)1(m)-1(ed)-443(b)28(y)-443(the)-444(r)1(outin)1(e)]TJ/F46 10.9091 Tf 178.2981 0 Td[(mld_)]TJ -362.2695 -13.5492 Td[(precfree)]TJ/F15 10.9091 Tf 45.8177 0 Td[(.)-501(T)-1(h)1(is)-353(step)-352(is)-353(com)-1(p)1(lem)-1(en)28(tary)-352(to)-352(ste)-1(p)-352(1)-352(and)-352(shou)1(ld)-352(b)-27(e)-353(p)-27(e)-1(r)1(formed)-352(w)-1(h)1(e)-1(n)]TJ -45.8177 -13.5492 Td[(th)1(e)-334(pr)1(e)-1(cond)1(ition)1(e)-1(r)-333(is)-333(no)-333(m)-1(or)1(e)-334(use)-1(d)1(.)]TJ -27.2728 -21.0602 Td[(A)-384(detail)1(e)-1(d)-384(d)1(e)-1(scrip)1(tion)-384(of)-383(the)-384(ab)-28(o)28(v)28(e)-384(rout)1(ines)-385(i)1(s)-385(giv)28(en)-384(in)-383(Sec)-1(ti)1(on)]TJ
-ET
-1 0 0 1 402.9592 202.8161 cm
+/F18 10.9091 Tf 13.939 0 Td [(F)77(r)51(e)51(e)-459(the)-459(pr)51(e)51(c)51(onditioner)-459(data)-459(structur)51(e)]TJ/F15 10.9091 Tf 183.971 0 Td [(.)-775(This)-443(is)-444(p)-28(erfor)1(m)-1(ed)-443(b)28(y)-444(the)-443(routine)]TJ/F44 10.9091 Tf 178.298 0 Td [(mld_)]TJ -362.269 -13.55 Td [(precfree)]TJ/F15 10.9091 Tf 45.818 0 Td [(.)-501(This)-353(step)-352(is)-353(complemen)28(tary)-352(to)-353(step)-352(1)-352(and)-353(should)-352(b)-28(e)-352(p)-28(erformed)-352(when)]TJ -45.818 -13.549 Td [(the)-333(preconditioner)-334(i)1(s)-334(no)-333(more)-334(u)1(s)-1(ed.)]TJ -27.273 -21.06 Td [(A)-384(detailed)-384(description)-384(of)-384(the)-384(ab)-28(o)28(v)28(e)-384(routines)-384(is)-384(giv)27(en)-384(in)-384(Section)]TJ
0 0 1 rg 0 0 1 RG
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-/F19 14.3462 Tf 86.3998 704.1363 Td[(6)-1125(User)-376(In)32(t)-1(er)-1(f)1(ace)]TJ/F15 10.9091 Tf 0 -24.3522 Td[(The)-396(basic)-397(u)1(s)-1(er)-396(in)29(te)-1(r)1(face)-397(of)-396(M)1(LD2P4)-396(consists)-397(of)-396(six)-396(r)1(outin)1(e)-1(s.)-632(T)-1(h)1(e)-397(f)1(our)-396(r)1(outin)1(e)-1(s)]TJ/F46 10.9091 Tf 389.5423 0 Td[(mld_)]TJ -389.5423 -13.5492 Td[(precinit)]TJ/F15 10.9091 Tf 45.8177 0 Td[(,)]TJ/F46 10.9091 Tf 6.0956 0 Td[(mld_precset)]TJ/F15 10.9091 Tf 62.9993 0 Td[(,)]TJ/F46 10.9091 Tf 6.0956 0 Td[(mld_precbld)]TJ/F15 10.9091 Tf 65.9218 0 Td[(and)]TJ/F46 10.9091 Tf 20.4983 0 Td[(mld_precaply)]TJ/F15 10.9091 Tf 71.649 0 Td[(encapsulate)-268(all)-267(the)-268(fu)1(nction-)]TJ -279.0773 -13.5492 Td[(alit)1(ie)-1(s)-323(f)1(or)-323(th)1(e)-323(s)-1(etup)-322(an)1(d)-323(t)1(he)-323(app)1(lication)-322(of)-322(an)28(y)-322(one-)-1(lev)28(el)-323(an)1(d)-323(m)28(ult)1(i-)-1(l)1(e)-1(v)28(el)-323(p)1(rec)-1(on)1(diti)1(oner)]TJ 0 -13.5492 Td[(impl)1(e)-1(me)-1(n)29(te)-1(d)-292(in)-292(th)1(e)-293(pac)27(k)56(age.)-431(The)-293(rou)1(tin)1(e)]TJ/F46 10.9091 Tf 198.5716 0 Td[(mld_precfree)]TJ/F15 10.9091 Tf 71.9188 0 Td[(deallo)-27(c)-1(ates)-293(the)-293(p)1(rec)-1(on)1(diti)1(oner)]TJ -270.4904 -13.5492 Td[(dat)1(a)-388(structu)1(re)-1(,)-400(w)-1(h)1(ile)]TJ/F46 10.9091 Tf 105.0723 0 Td[(mld_precdescr)]TJ/F15 10.9091 Tf 78.6811 0 Td[(pr)1(in)28(ts)-388(a)-387(des)-1(cri)1(ption)-387(of)-387(th)1(e)-388(pr)1(e)-1(cond)1(ition)1(e)-1(r)-387(s)-1(etu)1(p)]TJ -183.7534 -13.5492 Td[(b)28(y)-333(th)1(e)-334(use)-1(r)1(.)]TJ 16.9365 -13.5492 Td[(F)84(or)-222(e)-1(ac)28(h)-222(rou)1(tine,)-244(the)-222(s)-1(ame)-223(u)1(s)-1(er)-222(in)28(terf)1(ac)-1(e)-222(is)-223(o)28(v)28(e)-1(r)1(loaded)-222(with)-221(re)-1(sp)-27(e)-1(ct)-222(to)-222(the)-223(r)1(e)-1(al)1(/c)-1(omplex)]TJ -16.9365 -13.5492 Td[(cas)-1(e)-371(and)-370(th)1(e)-372(sin)1(gle)-1(/d)1(oub)1(le)-371(pr)1(e)-1(cis)-1(i)1(on;)-389(argu)1(m)-1(en)28(ts)-371(with)-370(app)1(rop)1(riate)-371(dat)1(a)-371(t)28(yp)-27(e)-1(s)-371(m)27(u)1(s)-1(t)-370(b)-28(e)]TJ 0 -13.5492 Td[(passe)-1(d)-333(to)-333(th)1(e)-334(rou)1(tine,)-333(i.e.)]TJ
-ET
-1 0 0 1 86.3998 548.875 cm
+/F17 14.3462 Tf -203.265 -35.866 Td [(6)-1125(User)-375(In)31(terface)]TJ/F15 10.9091 Tf 0 -24.352 Td [(The)-396(basic)-396(user)-396(in)28(terface)-397(of)-396(MLD2P4)-396(consists)-396(of)-396(six)-396(routines.)-633(The)-396(four)-396(routines)]TJ/F44 10.9091 Tf 389.542 0 Td [(mld_)]TJ -389.542 -13.549 Td [(precinit)]TJ/F15 10.9091 Tf 45.817 0 Td [(,)]TJ/F44 10.9091 Tf 6.096 0 Td [(mld_precset)]TJ/F15 10.9091 Tf 62.999 0 Td [(,)]TJ/F44 10.9091 Tf 6.096 0 Td [(mld_precbld)]TJ/F15 10.9091 Tf 65.922 0 Td [(and)]TJ/F44 10.9091 Tf 20.498 0 Td [(mld_precaply)]TJ/F15 10.9091 Tf 71.649 0 Td [(encapsulate)-268(all)-268(the)-268(function-)]TJ -279.077 -13.549 Td [(alities)-323(for)-322(the)-323(setup)-323(an)1(d)-323(the)-323(application)-322(of)-323(an)28(y)-323(one-lev)28(el)-323(and)-322(m)28(ulti-lev)27(el)-322(preconditioner)]TJ 0 -13.549 Td [(implemen)28(ted)-293(in)-292(the)-293(pac)28(k)55(age.)-431(The)-292(routine)]TJ/F44 10.9091 Tf 198.571 0 Td [(mld_precfree)]TJ/F15 10.9091 Tf 71.919 0 Td [(deallo)-28(cates)-293(th)1(e)-293(preconditioner)]TJ -270.49 -13.55 Td [(data)-387(structure,)-401(while)]TJ/F44 10.9091 Tf 105.072 0 Td [(mld_precdescr)]TJ/F15 10.9091 Tf 78.681 0 Td [(prin)28(ts)-388(a)-387(description)-388(of)-387(the)-388(p)1(re)-1(cond)1(itioner)-388(setup)]TJ -183.753 -13.549 Td [(b)28(y)-333(the)-334(user.)]TJ 16.936 -13.549 Td [(F)83(or)-222(eac)28(h)-222(routine,)-245(the)-222(same)-222(user)-222(in)27(terface)-222(is)-222(o)28(v)27(erloaded)-222(with)-222(resp)-28(ect)-222(to)-222(the)-222(real/complex)]TJ -16.936 -13.549 Td [(case)-371(and)-371(t)1(he)-371(single/double)-371(precision;)-389(argumen)28(ts)-371(with)-371(appropriate)-370(data)-371(t)28(yp)-28(es)-371(m)28(ust)-371(b)-28(e)]TJ 0 -13.549 Td [(passed)-333(to)-334(the)-333(routine,)-333(i.e.)]TJ
0 g 0 G
-1 0 0 1 -86.3998 -548.875 cm
-BT
-/F27 10.9091 Tf 102.7635 548.875 Td[(\017)]TJ
-ET
-1 0 0 1 108.218 548.875 cm
+/F25 10.9091 Tf 16.363 -22.516 Td [(\017)]TJ
0 g 0 G
-1 0 0 1 -108.218 -548.875 cm
-BT
-/F15 10.9091 Tf 113.6726 548.875 Td[(th)1(e)-464(sparse)-464(matri)1(x)-463(data)-463(stru)1(c)-1(tu)1(re,)-496(con)28(tain)1(in)1(g)-464(t)1(he)-464(matr)1(ix)-463(to)-463(b)-27(e)-464(pr)1(e)-1(cond)1(ition)1(e)-1(d)1(,)]TJ 0 -13.5492 Td[(m)27(u)1(s)-1(t)-423(b)-28(e)-424(of)-423(t)28(yp)-28(e)]TJ/F46 10.9091 Tf 82.7978 0 Td[(mld_)]TJ/F20 10.9091 Tf 22.9088 0 Td[(x)]TJ/F46 10.9091 Tf 6.3742 0 Td[(spmat_type)]TJ/F15 10.9091 Tf 61.8958 0 Td[(with)]TJ/F20 10.9091 Tf 25.8358 0 Td[(x)]TJ/F15 10.9091 Tf 10.9979 0 Td[(=)]TJ/F46 10.9091 Tf 13.1085 0 Td[(s)]TJ/F15 10.9091 Tf 10.3509 0 Td[(for)-423(real)-424(singl)1(e)-425(p)1(rec)-1(ision)1(,)]TJ/F20 10.9091 Tf 121.0749 0 Td[(x)]TJ/F15 10.9091 Tf 10.9979 0 Td[(=)]TJ/F46 10.9091 Tf 13.1085 0 Td[(d)]TJ/F15 10.9091 Tf -379.4511 -13.5492 Td[(for)-370(real)-371(dou)1(ble)-371(pr)1(e)-1(cision,)]TJ/F20 10.9091 Tf 123.4131 0 Td[(x)]TJ/F15 10.9091 Tf 10.4215 0 Td[(=)]TJ/F46 10.9091 Tf 12.5322 0 Td[(c)]TJ/F15 10.9091 Tf 9.7744 0 Td[(for)-370(c)-1(ompl)1(e)-1(x)-371(sin)1(gle)-372(p)1(rec)-1(i)1(s)-1(ion)1(,)]TJ/F20 10.9091 Tf 140.1102 0 Td[(x)]TJ/F15 10.9091 Tf 10.4214 0 Td[(=)]TJ/F46 10.9091 Tf 12.5322 0 Td[(z)]TJ/F15 10.9091 Tf 9.7745 0 Td[(for)-370(c)-1(ompl)1(e)-1(x)]TJ -328.9795 -13.5492 Td[(dou)1(bl)1(e)-334(pr)1(e)-1(cis)-1(i)1(on;)]TJ
-ET
-1 0 0 1 86.3998 485.7118 cm
+/F15 10.9091 Tf 10.91 0 Td [(the)-463(sparse)-463(matrix)-463(data)-463(structure,)-496(con)28(taining)-463(the)-463(matrix)-463(to)-463(b)-28(e)-463(preconditioned,)]TJ 0 -13.549 Td [(m)28(ust)-424(b)-28(e)-424(of)-423(t)27(yp)-27(e)]TJ/F44 10.9091 Tf 82.797 0 Td [(mld_)]TJ/F18 10.9091 Tf 22.909 0 Td [(x)]TJ/F44 10.9091 Tf 6.374 0 Td [(spmat_type)]TJ/F15 10.9091 Tf 61.896 0 Td [(with)]TJ/F18 10.9091 Tf 25.836 0 Td [(x)]TJ/F15 10.9091 Tf 10.998 0 Td [(=)]TJ/F44 10.9091 Tf 13.108 0 Td [(s)]TJ/F15 10.9091 Tf 10.351 0 Td [(for)-424(real)-424(single)-423(precision,)]TJ/F18 10.9091 Tf 121.075 0 Td [(x)]TJ/F15 10.9091 Tf 10.998 0 Td [(=)]TJ/F44 10.9091 Tf 13.109 0 Td [(d)]TJ/F15 10.9091 Tf -379.451 -13.549 Td [(for)-371(real)-371(double)-371(precision,)]TJ/F18 10.9091 Tf 123.413 0 Td [(x)]TJ/F15 10.9091 Tf 10.421 0 Td [(=)]TJ/F44 10.9091 Tf 12.532 0 Td [(c)]TJ/F15 10.9091 Tf 9.775 0 Td [(for)-371(complex)-371(single)-371(precision,)]TJ/F18 10.9091 Tf 140.11 0 Td [(x)]TJ/F15 10.9091 Tf 10.421 0 Td [(=)]TJ/F44 10.9091 Tf 12.533 0 Td [(z)]TJ/F15 10.9091 Tf 9.774 0 Td [(for)-371(complex)]TJ -328.979 -13.55 Td [(double)-333(precision;)]TJ
0 g 0 G
-1 0 0 1 -86.3998 -485.7118 cm
-BT
-/F27 10.9091 Tf 102.7635 485.7118 Td[(\017)]TJ
-ET
-1 0 0 1 108.218 485.7118 cm
+/F25 10.9091 Tf -10.91 -22.515 Td [(\017)]TJ
0 g 0 G
-1 0 0 1 -108.218 -485.7118 cm
-BT
-/F15 10.9091 Tf 113.6726 485.7118 Td[(th)1(e)-338(p)1(re)-1(cond)1(ition)1(e)-1(r)-336(data)-337(stru)1(c)-1(tu)1(re)-337(m)27(u)1(s)-1(t)-337(b)-27(e)-338(of)-336(t)28(yp)-27(e)]TJ/F46 10.9091 Tf 241.7755 0 Td[(mld_)]TJ/F20 10.9091 Tf 22.9088 0 Td[(x)]TJ/F46 10.9091 Tf 6.3742 0 Td[(prec_type)]TJ/F15 10.9091 Tf 51.5449 0 Td[(,)-338(with)]TJ/F20 10.9091 Tf 31.6054 0 Td[(x)]TJ/F15 10.9091 Tf 10.0506 0 Td[(=)]TJ/F46 10.9091 Tf 12.1613 0 Td[(s)]TJ/F15 10.9091 Tf 5.7273 0 Td[(,)]TJ/F46 10.9091 Tf -382.148 -13.5492 Td[(d)]TJ/F15 10.9091 Tf 5.7272 0 Td[(,)]TJ/F46 10.9091 Tf 6.6667 0 Td[(c)]TJ/F15 10.9091 Tf 5.7272 0 Td[(,)]TJ/F46 10.9091 Tf 6.6666 0 Td[(z)]TJ/F15 10.9091 Tf 5.7272 0 Td[(,)-333(ac)-1(cord)1(ing)-333(to)-333(the)-333(s)-1(p)1(arse)-334(m)-1(at)1(rix)-333(dat)1(a)-334(stru)1(c)-1(tu)1(re;)]TJ
-ET
-1 0 0 1 86.3998 449.647 cm
+/F15 10.9091 Tf 10.91 0 Td [(the)-337(preconditioner)-337(data)-337(structure)-337(m)28(ust)-337(b)-28(e)-337(of)-337(t)28(yp)-28(e)]TJ/F44 10.9091 Tf 241.775 0 Td [(mld_)]TJ/F18 10.9091 Tf 22.909 0 Td [(x)]TJ/F44 10.9091 Tf 6.374 0 Td [(prec_type)]TJ/F15 10.9091 Tf 51.545 0 Td [(,)-338(with)]TJ/F18 10.9091 Tf 31.605 0 Td [(x)]TJ/F15 10.9091 Tf 10.051 0 Td [(=)]TJ/F44 10.9091 Tf 12.161 0 Td [(s)]TJ/F15 10.9091 Tf 5.728 0 Td [(,)]TJ/F44 10.9091 Tf -382.148 -13.549 Td [(d)]TJ/F15 10.9091 Tf 5.727 0 Td [(,)]TJ/F44 10.9091 Tf 6.666 0 Td [(c)]TJ/F15 10.9091 Tf 5.728 0 Td [(,)]TJ/F44 10.9091 Tf 6.666 0 Td [(z)]TJ/F15 10.9091 Tf 5.728 0 Td [(,)-333(according)-334(to)-333(the)-333(sparse)-334(matrix)-333(data)-333(structure;)]TJ
0 g 0 G
-1 0 0 1 -86.3998 -449.647 cm
-BT
-/F27 10.9091 Tf 102.7635 449.647 Td[(\017)]TJ
-ET
-1 0 0 1 108.218 449.647 cm
+/F25 10.9091 Tf -41.425 -22.516 Td [(\017)]TJ
0 g 0 G
-1 0 0 1 -108.218 -449.647 cm
-BT
-/F15 10.9091 Tf 113.6726 449.647 Td[(th)1(e)-322(arr)1(a)27(y)1(s)-322(con)28(tain)1(ing)-321(th)1(e)-322(v)28(ec)-1(t)1(ors)]TJ/F24 10.9091 Tf 159.8466 0 Td[(v)]TJ/F15 10.9091 Tf 9.1819 0 Td[(and)]TJ/F24 10.9091 Tf 21.0784 0 Td[(w)]TJ/F15 10.9091 Tf 11.6062 0 Td[(in)28(v)28(ol)1(v)28(e)-1(d)-320(in)-321(th)1(e)-322(p)1(rec)-1(on)1(diti)1(oner)-321(app)1(lica-)]TJ -201.7131 -13.5492 Td[(tion)]TJ/F24 10.9091 Tf 21.3155 0 Td[(w)]TJ/F15 10.9091 Tf 11.1338 0 Td[(=)]TJ/F24 10.9091 Tf 11.5151 0 Td[(M)]TJ/F28 7.9701 Tf 11.7727 3.9588 Td[(\000)]TJ/F22 7.9701 Tf 6.5865 0 Td[(1)]TJ/F24 10.9091 Tf 4.7323 -3.9588 Td[(v)]TJ/F15 10.9091 Tf 8.2069 0 Td[(m)27(u)1(s)-1(t)-231(b)-27(e)-233(of)-231(t)28(yp)-27(e)]TJ/F20 10.9091 Tf 74.4135 0 Td[(typ)51(e)]TJ/F46 10.9091 Tf 19.7771 0 Td[(\050)]TJ/F20 10.9091 Tf 5.7272 0 Td[(kind)]TJ
+/F15 10.9091 Tf 10.91 0 Td [(the)-321(arra)28(ys)-321(con)27(t)1(aining)-321(the)-322(v)28(ectors)]TJ/F22 10.9091 Tf 159.846 0 Td [(v)]TJ/F15 10.9091 Tf 9.182 0 Td [(and)]TJ/F22 10.9091 Tf 21.078 0 Td [(w)]TJ/F15 10.9091 Tf 11.607 0 Td [(in)28(v)28(olv)27(ed)-321(in)-321(the)-321(precondition)1(e)-1(r)-321(app)1(lica-)]TJ -201.713 -13.549 Td [(tion)]TJ/F22 10.9091 Tf 21.315 0 Td [(w)]TJ/F15 10.9091 Tf 11.134 0 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(M)]TJ/F26 7.9701 Tf 11.773 3.959 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.733 -3.959 Td [(v)]TJ/F15 10.9091 Tf 8.206 0 Td [(m)28(ust)-232(b)-28(e)-231(of)-232(t)28(yp)-28(e)]TJ/F18 10.9091 Tf 74.414 0 Td [(typ)51(e)]TJ/F44 10.9091 Tf 19.777 0 Td [(\050)]TJ/F18 10.9091 Tf 5.727 0 Td [(kind)]TJ
ET
-1 0 0 1 309.5948 436.0978 cm
q
-[]0 d
-0 J
-0.3985 w
-0 0.1992 m
-3.3455 0.1992 l
-S
+1 0 0 1 309.595 436.297 cm
+[]0 d 0 J 0.398 w 0 0 m 3.345 0 l S
Q
-1 0 0 1 -309.5948 -436.0978 cm
BT
-/F20 10.9091 Tf 312.9403 436.0978 Td[(p)51(ar)51(ameter)]TJ/F46 10.9091 Tf 48.5678 0 Td[(\051)]TJ/F15 10.9091 Tf 5.7272 0 Td[(,)-252(with)]TJ/F20 10.9091 Tf 29.5194 0 Td[(typ)51(e)]TJ/F15 10.9091 Tf 22.3047 0 Td[(=)]TJ/F46 10.9091 Tf 11.0125 0 Td[(real)]TJ/F15 10.9091 Tf 22.9088 0 Td[(,)]TJ/F46 10.9091 Tf 5.7797 0 Td[(complex)]TJ/F15 10.9091 Tf -345.0879 -13.5492 Td[(and)]TJ/F20 10.9091 Tf 21.3918 0 Td[(kind)]TJ
+/F18 10.9091 Tf 312.94 436.098 Td [(p)51(ar)51(ameter)]TJ/F44 10.9091 Tf 48.568 0 Td [(\051)]TJ/F15 10.9091 Tf 5.727 0 Td [(,)-252(with)]TJ/F18 10.9091 Tf 29.52 0 Td [(typ)51(e)]TJ/F15 10.9091 Tf 22.304 0 Td [(=)]TJ/F44 10.9091 Tf 11.013 0 Td [(real)]TJ/F15 10.9091 Tf 22.909 0 Td [(,)]TJ/F44 10.9091 Tf 5.779 0 Td [(complex)]TJ/F15 10.9091 Tf -345.087 -13.549 Td [(and)]TJ/F18 10.9091 Tf 21.391 0 Td [(kind)]TJ
ET
-1 0 0 1 155.8059 422.5487 cm
q
-[]0 d
-0 J
-0.3985 w
-0 0.1992 m
-3.3455 0.1992 l
-S
+1 0 0 1 155.806 422.748 cm
+[]0 d 0 J 0.398 w 0 0 m 3.345 0 l S
Q
-1 0 0 1 -155.8059 -422.5487 cm
BT
-/F20 10.9091 Tf 159.1514 422.5487 Td[(p)51(ar)51(ameter)]TJ/F15 10.9091 Tf 52.3838 0 Td[(=)]TJ/F46 10.9091 Tf 12.3008 0 Td[(kind\0501.e0\051)]TJ/F15 10.9091 Tf 57.2721 0 Td[(,)]TJ/F46 10.9091 Tf 6.8912 0 Td[(kind\0501.d0\051)]TJ/F15 10.9091 Tf 57.2721 0 Td[(,)-354(acc)-1(or)1(din)1(g)-350(to)-350(th)1(e)-350(s)-1(p)1(arse)-351(matri)1(x)]TJ -231.5989 -13.5492 Td[(and)-304(pr)1(e)-1(cond)1(ition)1(e)-1(r)-305(d)1(ata)-305(s)-1(tr)1(uctur)1(e)-1(;)-314(note)-305(th)1(at)-305(the)-305(PSBLAS)-305(mo)-28(d)1(ule)]TJ/F46 10.9091 Tf 316.4518 0 Td[(psb_base_mod)]TJ/F15 10.9091 Tf -316.4518 -13.5492 Td[(pr)1(o)28(vides)-334(th)1(e)-334(constan)28(ts)]TJ/F46 10.9091 Tf 110.2123 0 Td[(psb_spk_)]TJ/F15 10.9091 Tf 49.454 0 Td[(=)]TJ/F46 10.9091 Tf 12.1212 0 Td[(kind\0501.e0\051)]TJ/F15 10.9091 Tf 60.9085 0 Td[(and)]TJ/F46 10.9091 Tf 21.2121 0 Td[(psb_dpk_)]TJ/F15 10.9091 Tf 49.4541 0 Td[(=)]TJ/F46 10.9091 Tf 12.1212 0 Td[(kind\0501.d0\051)]TJ/F15 10.9091 Tf 57.2721 0 Td[(;)]TJ
-ET
-1 0 0 1 86.3998 372.9347 cm
+/F18 10.9091 Tf 159.151 422.549 Td [(p)51(ar)51(ameter)]TJ/F15 10.9091 Tf 52.384 0 Td [(=)]TJ/F44 10.9091 Tf 12.301 0 Td [(kind\0501.e0\051)]TJ/F15 10.9091 Tf 57.272 0 Td [(,)]TJ/F44 10.9091 Tf 6.891 0 Td [(kind\0501.d0\051)]TJ/F15 10.9091 Tf 57.272 0 Td [(,)-354(according)-350(to)-349(the)-350(sparse)-350(matrix)]TJ -231.598 -13.55 Td [(and)-305(preconditioner)-305(data)-305(structure;)-314(note)-306(t)1(hat)-306(th)1(e)-306(PSBLAS)-305(mo)-27(dule)]TJ/F44 10.9091 Tf 316.451 0 Td [(psb_base_mod)]TJ/F15 10.9091 Tf -316.451 -13.549 Td [(pro)28(vides)-333(the)-334(constan)28(ts)]TJ/F44 10.9091 Tf 110.212 0 Td [(psb_spk_)]TJ/F15 10.9091 Tf 49.454 0 Td [(=)]TJ/F44 10.9091 Tf 12.121 0 Td [(kind\0501.e0\051)]TJ/F15 10.9091 Tf 60.908 0 Td [(and)]TJ/F44 10.9091 Tf 21.213 0 Td [(psb_dpk_)]TJ/F15 10.9091 Tf 49.454 0 Td [(=)]TJ/F44 10.9091 Tf 12.121 0 Td [(kind\0501.d0\051)]TJ/F15 10.9091 Tf 57.272 0 Td [(;)]TJ
0 g 0 G
-1 0 0 1 -86.3998 -372.9347 cm
-BT
-/F27 10.9091 Tf 102.7635 372.9347 Td[(\017)]TJ
-ET
-1 0 0 1 108.218 372.9347 cm
+/F25 10.9091 Tf -383.665 -22.515 Td [(\017)]TJ
0 g 0 G
-1 0 0 1 -108.218 -372.9347 cm
-BT
-/F15 10.9091 Tf 113.6726 372.9347 Td[(real)-450(p)1(arame)-1(ters)-450(de\014n)1(ing)-450(th)1(e)-451(p)1(rec)-1(on)1(dit)1(ioner)-450(m)28(ust)-450(b)-28(e)-450(declared)-450(acc)-1(or)1(din)1(g)-450(to)-450(the)]TJ 0 -13.5492 Td[(pr)1(e)-1(cision)-245(of)-245(the)-245(s)-1(p)1(arse)-246(m)-1(atr)1(ix)-245(and)-245(p)1(rec)-1(on)1(diti)1(oner)-246(d)1(ata)-245(s)-1(tr)1(uctures)-246(\050se)-1(e)-245(Sec)-1(tion)]TJ
-ET
-1 0 0 1 477.6387 359.3855 cm
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