Subject: Re: C460: Inverse Toeplitz Eigenproblem on Personal Computer Networks Resent-Date: Tue, 27 Jun 2000 06:49:21 -0400 Resent-From: Geoffrey Fox Resent-To: Geoffrey Fox Date: Tue, 27 Jun 2000 12:34:23 +0200 From: "Jose M. Badia" Organization: uji To: fox@csit.fsu.edu "Geoffrey Fox(Concurrency)" wrote: > > Dear Dr. Badia > I write in regard to your paper. > > C460: Inverse Toeplitz Eigenproblem on Personal Computer Networks > > My referees were satisfied with the paper and we would be happy to publish it. > Before submitting it to Wiley, I wondered if you wanted to make any changes to the > manuscript. If so, Please send the revised paper to me and I will transmit it > without delay. > In any case, I need an electronic version for submittal. Please send me a > postscript or acrobat(PDF) version even if no changes. > ... Dear Sir, Enclosed you can find a PostScript version of the papers. I hope you don't have problems with the format. Best Regards Jose M. 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w(47,57)g(2,10E-02)p 1505 3069 V 149 w(184,84)g(5,41E-03)p 2349 3069 V 150 w(165,54)g(6,04E-03)p 3193 3069 V 382 3072 2813 4 v 380 3185 4 113 v 432 3151 a(1000)p 662 3185 V 194 w(47,34)g(2,11E-02)p 1505 3185 V 149 w(185,53)g(5,39E-03)p 2349 3185 V 150 w(169,94)g(5,88E-03)p 3193 3185 V 382 3188 2813 4 v 860 3343 a(T)-8 b(able)30 b(1:)41 b(Arithmetic)29 b(p)s(erformance)h(of)g(the)h(pro)s(cessors.) 609 3618 y Fr(algorithm)37 b(dep)r(ends)h(on)f(the)h(t)n(yp)r(e)g(of)g (op)r(eration,)h(on)f(the)g(memory)f(access)609 3718 y(outline,)28 b(and)f(on)h(the)g(exploitation)f(of)g(all)g(the)h (features)f(of)h(the)g(pro)r(cessor.)734 3817 y(T)-7 b(o)32 b(analyze)g(the)h(arithmetic)f(p)r(erformance)g(of)h(the)g(pro)r (cessor)d(three)j(wi-)609 3917 y(despread)j(standard)f(algorithms)g(ha) n(v)n(e)g(b)r(een)i(used:)54 b(the)37 b(\014rst)f(one,)i Fh(DGEMV)p Fr(,)609 4016 y(p)r(erforms)28 b(the)h(matrix-v)n(ector)d (pro)r(duct)i(and)g(is)h(in)n(tegrated)e(in)i(the)g(lev)n(el)f(2)g(of) 609 4116 y(the)c(computational)e(k)n(ernel)g(BLAS.)h(The)h(second)e (one,)i Fh(DGEMM)p Fr(,)d(p)r(erforms)h(ma-)609 4216 y(trix)j(pro)r(ducts)g(and)f(b)r(elongs)h(to)g(BLAS)g(lev)n(el)f(3,)h (and)g(the)h(third)f(one,)g Fh(DGETRF)p Fr(,)609 4315 y(p)r(erforms)33 b(the)i(LU)f(decomp)r(osition)f(of)h(a)g(matrix)f(and) h(is)g(in)n(tegrated)f(in)h(the)609 4415 y(LAP)-7 b(A)n(CK)24 b(library)g([1].)36 b(In)25 b(the)g(case)g(of)g(the)g(routines)f (included)i(in)f(the)g(BLAS)609 4515 y(k)n(ernel,)32 b(w)n(e)e(ha)n(v)n(e)g(used)i(a)f(v)n(ersion)e(sp)r(ecially)i(devised)g (to)g(tak)n(e)g(adv)-5 b(an)n(tage)30 b(of)609 4614 y(P)n(en)n(tium)h (pro)r(cessors,)e(whic)n(h)i(is)g(incorp)r(orated)e(in)i(the)g(ASCI)h (Red)f(P)n(en)n(tium)609 4714 y(Pro)26 b(BLAS)i(1.1.N)f([12)o(],)h([13) o(].)734 4813 y(In)40 b(table)f(1)h(w)n(e)f(presen)n(t)g(the)h(results) f(obtained)h(b)n(y)f(the)h(previous)f(rou-)609 4913 y(tines)28 b(in)g(a)f(P)n(en)n(tium)g(I)r(I-300)f(pro)r(cessor,)f(included)j(in)g 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b(for)g(the)g(a)n(v)n (erage)e(time)j(of)f(execution)g(of)g(a)g(\015oating)609 1046 y(p)r(oin)n(t)g(op)r(eration.)65 b(While)37 b(in)h(the)g(case)e (of)h(the)h(t)n(w)n(o)f(last)g(subroutines)f(this)609 1146 y(v)-5 b(alue)34 b(is)g(appro)n(ximately)f(0,006)f Fn(\026)p Fr(s,)k(in)e(the)h(\014rst)f(one)g(the)g(v)-5 b(alue)34 b(is)h(around)609 1245 y(0,02)h Fn(\026)p Fr(s.)64 b(Th)n(us,)38 b(the)g(v)-5 b(alue)36 b(of)h(constan)n(t)f Fn(t)2062 1257 y Fl(f)2105 1245 y Fr(,)j(whic)n(h)e(is)f(more)g (appropriate)609 1345 y(to)g(mo)r(del)g(our)f(system,)j(strongly)d(dep) r(ends)h(on)g(the)g(c)n(haracteristics)e(of)i(the)609 1445 y(arithmetic)31 b(op)r(erations)e(to)h(b)r(e)h(p)r(erformed)f(and) g(on)g(the)h(exploitation)f(of)g(the)609 1544 y(di\013eren)n(t)e(lev)n (els)f(of)g(BLAS.)609 1819 y Fq(3)135 b(Description)61 b(of)f(the)h(problem)f(and)g(its)h(se-)609 1968 y(quen)l(tial)46 b(solution)609 2150 y Fr(In)19 b(this)h(section)f(w)n(e)f(brie\015y)h 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Fl(n)2037 2885 y(i;j)s Fg(=1)2213 2852 y Fn(:)734 3046 y Fr(This)33 b(kind)g(of)g(matrix)g (app)r(ears)f(in)i(the)f(solution)g(of)g(man)n(y)g(problems)f(in)609 3145 y(Ph)n(ysics)26 b(or)h(Engineering.)734 3245 y(Giv)n(en)g Fn(n)h Fr(real)e(v)-5 b(alues)28 b(suc)n(h)f(as)1518 3444 y Fn(\025)1566 3456 y Fg(1)1627 3444 y Fd(\024)22 b Fn(\025)1762 3456 y Fg(2)1823 3444 y Fd(\024)h Fn(:)14 b(:)g(:)22 b Fd(\024)h Fn(\025)2166 3456 y Fl(n)2212 3444 y Fn(;)802 b Fr(\(2\))609 3593 y(the)27 b(in)n(v)n(erse)e(eigen)n (v)-5 b(alue)26 b(problem)g(consists)g(in)g(computing)h(a)f(generator)e Fn(t)p Fr(,)j(so)609 3693 y(that)h(the)g(sp)r(ectrum)g(of)f(the)h(TRS)g (matrix)f(asso)r(ciated)g(coincides)g(with)h(\(2\).)734 3793 y(The)h(TRS)h(matrices)f(v)n(erify)g(some)g(prop)r(erties)g([5],)h ([15)o(])g(that)g(allo)n(w)f(their)609 3892 y(sp)r(ectrum)j(to)f(b)r(e) i(divided)e(in)h(t)n(w)n(o)f(parts)g(with)h(the)g(same)f(n)n(um)n(b)r (er)h(of)f(eigen-)609 3992 y(v)-5 b(alues,)25 b(kno)n(wn)g(as)f(ev)n (en)g(and)h(o)r(dd)g(eigen)n(v)-5 b(alues,)24 b(and)h(asso)r(ciating)e (them)j(with)609 4092 y(the)j(symmetric)e(and)h(sk)n(ew-symmetric)e (eigen)n(v)n(ectors,)g(resp)r(ectiv)n(ely)-7 b(.)37 b(On)28 b(the)609 4191 y(other)g(side,)h(it)g(is)f(p)r(ossible)g(to)g(compute)h (b)r(oth)g(sp)r(ectra)f(separately)-7 b(,)27 b(substan-)609 4291 y(tially)f(reducing)f(the)h(cost)g(of)f(calculating)g(the)i(eigen) n(v)-5 b(alues)24 b(and)i(eigen)n(v)n(ectors)609 4390 y(of)i(the)g(matrix.)734 4490 y(In)21 b([16)o(])f(a)h(metho)r(d)g(for)f (solving)f(the)i(in)n(v)n(erse)e(eigenproblem)h(with)h(TRS)g(ma-)609 4590 y(trices)29 b(is)h(prop)r(osed.)42 b(This)29 b(metho)r(d)i(is)e (equiv)-5 b(alen)n(t)30 b(to)f(the)h(Newton)g(metho)r(d.)609 4689 y(This)d(algorithm)f(is)g(impro)n(v)n(ed)g(in)h([20)o(])g(b)n(y)f (means)h(of)f(the)h(adequate)f(exploita-)609 4789 y(tion)i(of)f(the)h (previous)f(sp)r(ectral)g(prop)r(erties.)609 5021 y Fm(3.1)112 b(Sequen)m(tial)37 b(Algorithm)609 5175 y Fr(In)29 b(this)g(section,)g (w)n(e)g(brie\015y)f(describ)r(e)h(the)g(sequen)n(tial)f(metho)r(d)h (prop)r(osed)f(in)609 5274 y([20)o(].)57 b(W)-7 b(e)35 b(will)g(call)e Fn(p)1332 5286 y Fg(1)1369 5274 y Fr(\()p Fn(t)p Fr(\))p Fn(;)14 b(:)g(:)g(:)h(;)f(p)1691 5286 y Fl(r)1727 5274 y Fr(\()p Fn(t)p Fr(\))35 b(the)g(symmetric)f(eigen)n (v)n(ectors)e(of)i Fn(T)12 b Fr(\()p Fn(t)p Fr(\))1854 5652 y Ft(6)p eop %%Page: 7 7 7 6 bop 609 548 a Fr(and)21 b Fn(q)801 560 y Fg(1)838 548 y Fr(\()p Fn(t)p Fr(\))p Fn(;)14 b(:)g(:)g(:)g(;)g(q)1154 560 y Fl(s)1190 548 y Fr(\()p Fn(t)p Fr(\))21 b(its)g(sk)n (ew-symmetric)e(eigen)n(v)n(ectors.)32 b(On)20 b(the)h(other)g(side,) 609 648 y(w)n(e)27 b(will)h(denote)g(the)g(target)e(sp)r(ectrum)i(as) 1282 847 y(\003)23 b(=)f([)p Fn(\026)1523 859 y Fg(1)1561 847 y Fn(;)14 b(\026)1648 859 y Fg(2)1685 847 y Fn(;)g(:)g(:)g(:)f(;)h (\026)1919 859 y Fl(r)1956 847 y Fn(;)g(\027)2034 859 y Fg(1)2071 847 y Fn(;)g(\027)2149 859 y Fg(2)2186 847 y Fn(;)g(:)g(:)g(:)g(;)g(\027)2412 859 y Fl(s)2447 847 y Fr(])567 b(\(3\))609 985 y(where)22 b(the)h(ev)n(en)g(and)f(o)r(dd)h (sp)r(ectra)f(ha)n(v)n(e)g(b)r(een)h(separated,)f(and)h(eac)n(h)f(one)g (has)609 1084 y(b)r(een)28 b(written)g(in)g(increasing)e(order.)734 1184 y(Let)31 b Fn(t)916 1196 y Fg(0)984 1184 y Fr(b)r(e)g(a)g Fn(n)p Fr(-v)n(ector)e(and)i(let)g(\003)f(b)r(e)i(the)f(target)f(sp)r (ectrum,)i(as)e(de\014ned)609 1283 y(in)37 b(\(3\).)63 b(By)36 b(using)g Fn(t)1302 1295 y Fg(0)1375 1283 y Fr(as)g(an)g (initial)g(generator,)g(the)h(metho)r(d)g(computes)f(a)609 1383 y(sequence)27 b Fn(t)983 1353 y Fl(m)1046 1383 y Fr(,)h Fn(m)23 b Fr(=)g(1)p Fn(;)14 b Fr(2)p Fn(;)g(:)g(:)g(:)26 b Fr(as)h(the)h(solution)f(of)h(the)g(equations:)1068 1561 y Fn(p)1110 1573 y Fl(i)1137 1561 y Fr(\()p Fn(t)1199 1531 y Fl(m)p Ff(\000)p Fg(1)1348 1561 y Fr(\))1380 1531 y Fl(T)1432 1561 y Fn(T)12 b Fr(\()p Fn(t)1555 1531 y Fl(m)1618 1561 y Fr(\))p Fn(p)1692 1573 y Fl(i)1719 1561 y Fr(\()p Fn(t)1781 1531 y Fl(m)p Ff(\000)p Fg(1)1930 1561 y Fr(\))23 b(=)g Fn(\026)2123 1573 y Fl(i)2150 1561 y Fn(;)180 b Fr(1)23 b Fd(\024)g Fn(i)f Fd(\024)h Fn(r)1061 1674 y(q)1098 1686 y Fl(j)1134 1674 y Fr(\()p Fn(t)1196 1650 y Fl(m)p Ff(\000)p Fg(1)1344 1674 y Fr(\))1376 1650 y Fl(T)1428 1674 y Fn(T)12 b Fr(\()p Fn(t)1551 1650 y Fl(m)1614 1674 y Fr(\))p Fn(q)1683 1686 y Fl(j)1718 1674 y Fr(\()p Fn(t)1780 1650 y Fl(m)p Ff(\000)p Fg(1)1928 1674 y Fr(\))24 b(=)e Fn(\027)2112 1686 y Fl(j)2147 1674 y Fn(;)180 b Fr(1)23 b Fd(\024)g Fn(j)28 b Fd(\024)22 b Fn(s)3037 1617 y Fr(\(4\))609 1806 y(where)d Fn(r)26 b Fr(=)c Fd(d)p Fn(n=)p Fr(2)p Fd(e)c Fr(and)h Fn(s)k Fr(=)g Fd(b)p Fn(n=)p Fr(2)p Fd(c)p Fr(.)33 b(The)19 b(previous)f(equations)h(can)g(b)r(e)g(written)609 1906 y(as)27 b(a)g(linear)f(system)h(of)g(dimension)h Fn(n)23 b Fr(=)f Fn(r)f Fr(+)c Fn(s)p Fr(;)28 b(and)f(w)n(e)g(can)f(obtain)h Fn(t)2884 1876 y Fl(m)2975 1906 y Fr(from)609 2006 y Fn(t)639 1975 y Fl(m)p Ff(\000)p Fg(1)787 2006 y Fr(,)h(th)n(us)g(pro)r (ducing)f(an)g(iterativ)n(e)g(metho)r(d.)734 2105 y(In)39 b(eac)n(h)f(iteration)g(of)h(the)g(algorithm,)i(w)n(e)d(start)g(b)n(y)h (constructing)f(the)609 2205 y(matrix)c(asso)r(ciated)e(with)j(the)f (linear)g(system)f(in)i(order)d(to)i(solv)n(e)f(\(4\).)57 b(This)609 2304 y(is)28 b(p)r(erformed)g(from)g(the)h(eigen)n(v)n (ectors)d(of)i(the)g(TRS)h(matrix)f(of)g(the)h(previous)609 2404 y(iteration,)i(whic)n(h)f(ha)n(v)n(e)g(b)r(een)h(computed)f(with)h (a)g(small)f(cost)g(b)n(y)g(separating)609 2504 y(the)40 b(o)r(dd)f(and)g(ev)n(en)g(sp)r(ectra.)70 b(Then,)43 b(the)c(linear)g(system)f(is)h(solv)n(ed,)i(th)n(us)609 2603 y(obtaining)24 b(a)g(new)h(generator)e(for)h(a)g(TRS)h(matrix,)g (whose)f(sp)r(ectrum)h(is)f(calcu-)609 2703 y(lated.)36 b(The)23 b(con)n(v)n(ergence)e(of)i(the)g(algorithm)f(is)i(reac)n(hed)e (when)h(the)h(di\013erence)609 2803 y(b)r(et)n(w)n(een)36 b(the)f(computed)h(sp)r(ectrum)g(and)f(the)h(target)e(sp)r(ectrum)i(is) f(smaller)609 2902 y(than)28 b(a)f(giv)n(en)g(error)e Fn(\017)1326 2914 y Fg(0)1363 2902 y Fr(.)734 3002 y(Broadly)32 b(sp)r(eaking,)i(the)g(sequen)n(tial)f(algorithm)f(w)n(e)h(ha)n(v)n(e)g (implemen)n(ted)609 3102 y(is)28 b(the)g(follo)n(wing:)609 3264 y Fc(REPEA)-7 b(T)762 3364 y Fr(\003\()p Fn(T)12 b Fr(\()p Fn(t)975 3334 y Fl(m)p Ff(\000)p Fg(1)1123 3364 y Fr(\)\))23 b Fd( )28 b Fc(Compute)f(the)g(sp)r(ectrum)h (\(eigenvalues)f(and)g(eigenvecto)n(rs\))1375 3463 y(of)g Fn(T)12 b Fr(\()p Fn(t)1592 3433 y Fl(m)p Ff(\000)p Fg(1)1740 3463 y Fr(\))762 3563 y Fn(C)30 b Fd( )d Fc(Build)h(the)f(linea)n(r)g (system)h(from)g(\(4\))f(and)h(the)f(computed)g(eigenvecto)n(rs.)762 3663 y Fn(t)792 3633 y Fl(m)878 3663 y Fd( )h Fc(Solve)f(the)h(linea)n (r)f(system)g Fn(C)6 b(t)1907 3633 y Fl(m)1994 3663 y Fr(=)22 b(\003)609 3762 y Fc(UNTIL)27 b Fd(j)p Fr(\003\()p Fn(T)12 b Fr(\()p Fn(t)1113 3732 y Fl(m)1176 3762 y Fr(\)\))19 b Fd(\000)f Fr(\003)p Fd(j)k Fn(<)h(\017)1567 3774 y Fg(0)734 4025 y Fr(The)38 b(metho)r(d)g(describ)r(ed)g(is)g(equiv)-5 b(alen)n(t)38 b(to)g(Newton's)g(metho)r(d.)68 b(Since)609 4124 y(the)39 b(Newton)g(metho)r(d)g(is)f(not)h(globally)e(con)n(v)n (ergen)n(t,)j(this)e(algorithm)g(do)r(es)609 4224 y(not)32 b(necessarily)d(con)n(v)n(erge)h(to)h(a)g(solution)g(of)g(the)h (problem.)48 b(In)32 b([20)o(])f(sev)n(eral)609 4323 y(impro)n(v)n(emen)n(ts)26 b(are)g(prop)r(osed)g(in)h(order)f(to)h(ac)n (hiev)n(e)f(the)h(con)n(v)n(ergence)e(of)i(the)609 4423 y(metho)r(d.)53 b(Sp)r(eci\014cally)-7 b(,)35 b(if)e(the)g(con)n(v)n (ergence)e(fails,)j(w)n(e)e(try)h(to)f(linearly)g(con-)609 4523 y(v)n(erge)24 b(to)i(a)f(mo)r(di\014ed)h(sp)r(ectrum)g(with)g(a)f (less)g(restrictiv)n(e)f(stopping)h(criterion.)609 4622 y(When)f(this)g(new)f(con)n(v)n(ergence)e(is)i(reac)n(hed,)g(the)g (computed)h(sp)r(ectrum)f(is)h(used)609 4722 y(as)j(a)g(starting)g(p)r (oin)n(t)h(to)f(quadratically)f(con)n(v)n(erge)f(on)j(the)g(target)e (sp)r(ectrum.)609 4951 y Fm(3.2)112 b(Analysis)37 b(of)h(the)f (theoretical)f(cost)609 5104 y Fr(Eac)n(h)26 b(iteration)g(of)h(the)g (previous)f(sequen)n(tial)g(metho)r(d)h(p)r(erforms)f(three)h(basic)609 5204 y(tasks:)55 b(the)38 b(computation)f(of)g(the)g(sp)r(ectrum)h(of)f (a)f(TRS)i(matrix,)h(the)e(con-)609 5303 y(struction)32 b(of)g(the)g(co)r(e\016cien)n(t)g(matrix)f(of)h(a)g(linear)f(system)h (and,)h(\014nally)-7 b(,)33 b(the)609 5403 y(solution)27 b(of)h(the)g(linear)f(system.)1854 5652 y Ft(7)p eop %%Page: 8 8 8 7 bop 734 548 a Fr(By)24 b(exploiting)h(the)g(sp)r(ectral)g(prop)r (erties)f(of)g(the)i(TRS)f(matrices,)g(the)g(cost)609 648 y(of)j(computing)f(their)h(sp)r(ectrum)g(is)1388 835 y Fn(t)1418 847 y Fg(1)1478 835 y Fr(=)23 b(11)p Fn(n)1700 801 y Fg(3)1736 835 y Fn(=)p Fr(6)18 b(+)g Fn(n)1971 801 y Fg(2)2008 835 y Fn(=)p Fr(4)82 b Fb(\015ops)p Fn(:)734 1006 y Fr(The)27 b(construction)g(of)h(the)g(co)r(e\016cien)n (t)f(matrix)g(implies)h(a)f(cost)g(of)1450 1194 y Fn(t)1480 1206 y Fg(2)1541 1194 y Fr(=)22 b Fn(n)1678 1160 y Fg(3)1715 1194 y Fn(=)p Fr(2)c(+)g(3)p Fn(n)1992 1160 y Fg(2)2112 1194 y Fb(\015ops)p Fn(:)734 1365 y Fr(Finally)-7 b(,)45 b(the)c(solution)g(of)g(the)h(linear)f(system)g(b)n(y)g(means)g(of)g (gaussian)609 1465 y(elimination)28 b(pro)r(duces)f(a)g(cost)g(of:)1545 1653 y Fn(t)1575 1665 y Fg(3)1635 1653 y Fr(=)c(2)p Fn(n)1815 1618 y Fg(3)1851 1653 y Fn(=)p Fr(3)82 b Fb(\015ops)q Fn(:)734 1824 y Fr(Th)n(us,)30 b(if)g(w)n(e)f(call)h Fn(it)f Fr(the)h(n)n(um)n(b)r(er)g(of)f(necessary)f(iterations)h(to)h (reac)n(h)e(the)609 1924 y(con)n(v)n(ergence,)d(the)j(total)g(cost)f (of)g(the)h(sequen)n(tial)f(algorithm)g(is)g(giv)n(en)g(b)n(y:)1018 2123 y Fn(t)1048 2135 y Fg(2)1108 2123 y Fr(=)c(\()p Fn(t)1258 2135 y Fg(1)1314 2123 y Fr(+)18 b Fn(t)1427 2135 y Fg(2)1483 2123 y Fr(+)g Fn(t)1596 2135 y Fg(3)1633 2123 y Fr(\))23 b(=)g(\(3)p Fn(n)1900 2089 y Fg(3)1955 2123 y Fr(+)18 b(13)p Fn(n)2172 2089 y Fg(2)2208 2123 y Fn(=)p Fr(4\))g Fd(\003)g Fn(it)83 b Fb(\015ops)p Fn(:)303 b Fr(\(5\))609 2395 y Fq(4)135 b(Outline)46 b(of)f(the)g(parallel)i (algorithm)609 2577 y Fr(The)25 b(parallelization)f(of)h(the)g(sequen)n (tial)g(metho)r(d)g(is)g(based)g(on)g(the)g(use)g(of)g(the)609 2677 y(ScaLAP)-7 b(A)n(CK)35 b(parallel)g(linear)g(algebra)f(library)h ([3)o(].)62 b(In)36 b(this)g(en)n(vironmen)n(t)609 2776 y(the)24 b(algorithms)f(use)h(a)f(SPMD)h(mo)r(del)g(and)g(a)f(blo)r(c)n (k)g(cyclic)h(data)f(distribution)609 2876 y(among)k(the)h(pro)r (cessors)d(of)i(a)h(logical)e(bidimensional)h(mesh.)734 2976 y(The)35 b(algorithm)f(parallelizes)f(the)i(three)g(main)g(steps)g (of)g(the)g(sequen)n(tial)609 3075 y(v)n(ersion)d(and)i(p)r(erforms)e (the)i(comm)n(unications)f(needed)g(to)h(redistribute)f(the)609 3175 y(data)20 b(appropriately)e(in)i(order)e(to)i(start)f(eac)n(h)g (step.)35 b(A)20 b(v)n(ery)f(simpli\014ed)h(v)n(ersion)609 3275 y(of)28 b(the)g(parallel)e(algorithm)h(is)g(summarized)g(in)h(the) g(follo)n(wing)e(pseudo)r(co)r(de:)609 3442 y Fc(WHILE)i(not)f (converged)762 3542 y(Compute)g(the)h(o)r(dd)f(and)g(even)h(sp)r(ectra) g(of)f Fn(T)12 b Fr(\()p Fn(t)2228 3512 y Fl(m)p Ff(\000)p Fg(1)2376 3542 y Fr(\))28 b Fc(in)f(pa)n(rallel.)762 3641 y(Gather)h(the)g(eigenvecto)n(rs)e(in)h(the)h(\014rst)f(ro)n(w)g (of)h(p)n(ro)r(cesso)n(rs.)762 3741 y(IF)g(the)g(p)n(ro)r(cesso)n(r)e (is)i(in)f(the)g(\014rst)h(ro)n(w)f(of)g(the)h(mesh)f(THEN)915 3841 y(Compute)g(the)h(co)n(rresp)r(onding)e(ro)n(ws)h(of)g(matrix)h Fn(C)34 b Fc(in)27 b(the)h(linea)n(r)f(system.)762 3940 y(Redistribute)g(matrix)h Fn(C)34 b Fc(among)26 b(all)i(the)f(p)n(ro)r (cesso)n(rs)f(in)i(the)f(mesh.)762 4040 y(Solve)h(the)f(linea)n(r)g (system)h(in)f(pa)n(rallel.)734 4307 y Fr(In)37 b(order)e(to)i (accomplish)g(the)g(parallel)f(solution)g(of)h(the)h(linear)e(system) 609 4407 y(and)22 b(the)h(computation)f(of)g(the)g(sp)r(ectra,)h(w)n(e) f(ha)n(v)n(e)f(used)h(sev)n(eral)f(ScaLAP)-7 b(A)n(CK)609 4506 y(routines,)29 b(namely)-7 b(,)30 b Fh(PDGETRF)p Fr(,)d Fh(PDGETRS)g Fr(and)i Fh(PDSYEV)p Fr(.)e(T)-7 b(o)29 b(compute)h(the)f(co-)609 4606 y(e\016cien)n(t)k(matrix)e(for)h (the)h(linear)e(system)h(w)n(e)g(ha)n(v)n(e)f(exploited)h(the)h(fact)f (that)609 4706 y(eac)n(h)h(ro)n(w)f(dep)r(ends)i(on)f(one)g(eigen)n(v)n (ector,)g(and)h(therefore,)g(all)f(matrix)g(ro)n(ws)609 4805 y(can)27 b(b)r(e)h(computed)g(in)g(parallel.)734 4905 y(Due)f(to)f(the)h(mesh)f(top)r(ology)f(of)h(the)h(en)n(vironmen)n (t)e(and)i(the)f(data)g(dep)r(en-)609 5005 y(dencies)e(of)g(the)h (problem,)f(w)n(e)g(ha)n(v)n(e)f(to)h(p)r(erform)f(some)h (redistributions)f(of)h(the)609 5104 y(data)g(in)h(eac)n(h)e(iteration) h(of)g(the)h(algorithm.)34 b(These)24 b(comm)n(unications)g(greatly)609 5204 y(increase)j(the)g(cost)h(of)f(the)h(parallel)f(algorithm.)734 5303 y(First,)43 b(in)e(eac)n(h)f(iteration,)j(w)n(e)d(m)n(ust)g (gather)g(the)g(eigen)n(v)n(ectors)e(in)j(the)609 5403 y(\014rst)23 b(ro)n(w)g(of)g(pro)r(cessors)e(in)j(order)e(to)h(compute) h(the)g(co)r(e\016cien)n(t)f(matrix)g(of)g(the)1854 5652 y Ft(8)p eop 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(sho)n(wn)f(in)i(\014gure)f(2.)931 2306 y @beginspecial 1 @llx 3 @lly 413 @urx 185 @ury 2267 @rwi 1133 @rhi @setspecial %%BeginDocument: fig2bis.eps %!PS-Adobe-2.0 EPSF-1.2 %%Title: fig2bis.eps %%Creator: Canvas 3.5 %%For: badia %%CreationDate: vier., 3 sept 1999 12:58 %%BoundingBox: 1 3 413 185 %%DocumentProcSets: CanvasDict %%DocumentSuppliedProcSets: CanvasDict %%Copyright ©1988-91 Deneba Systems, Inc. - All Rights Reserved Worldwide %%DocumentFonts: Times-Roman %%DocumentNeededFonts: Times-Roman %%EndComments %%BeginProcSet:CanvasDict /CanvasDict where not{/CanvasDict 250 dict def}{pop}ifelse CanvasDict begin systemdict/setpacking known{/origpack currentpacking def true setpacking}if /bdf{bind def}bind def /_cv 20 dict begin /isDeviceColor {/currentcolorspace where{pop currentcolorspace 0 get dup /DeviceGray eq exch dup /DeviceRGB eq exch /DeviceCMYK eq or or}{T}ifelse}bdf /stg{isDeviceColor{setgray}{pop}ifelse}bdf /strgb{isDeviceColor{setrgbcolor}{pop pop pop}ifelse}bdf 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@b -.004 -.004 1920.004 762.004 .004 .004 2616 536 @b -.004 -.004 528.004 1375.004 .004 .004 1224 1084 @b -.004 -.004 1224.004 1084.004 .004 .004 1920 836 @b -.004 -.004 1920.004 836.004 .004 .004 2616 649 @b -.004 -.004 528.004 1393.004 .004 .004 1224 1137 @b -.004 -.004 1224.004 1137.004 .004 .004 1920 936 @b -.004 -.004 1920.004 936.004 .004 .004 2616 797 @b -.004 -.004 528.004 1421.004 .004 .004 1224 1218 @b -.004 -.004 1224.004 1218.004 .004 .004 1920 1077 @b -.004 -.004 1920.004 1077.004 .004 .004 2616 991 @b -.004 -.004 528.004 1477.004 .004 .004 1224 1351 @b -.004 -.004 1224.004 1351.004 .004 .004 1920 1281 @b -.004 -.004 1920.004 1281.004 .004 .004 2616 1248 @b -.004 -.004 528.004 1603.004 .004 .004 1224 1574 @b -.004 -.004 1224.004 1574.004 .004 .004 1920 1565 @b 696.004 1920 1565.002 @c 516 1352 24 24 rF 1 G 515.5 1351.5 25 25 rS 0 G 1212 1035 24 24 rF 1 G 1211.5 1034.5 25 25 rS 0 G 1908 750 24 24 rF 1 G 1907.5 749.5 25 25 rS 0 G 2604 524 24 24 rF 1 G 2603.5 523.5 25 25 rS 516 1363 24 24 rF 0 G 515.5 1362.5 25 25 rS 1 G 1212 1072 24 24 rF 0 G 1211.5 1071.5 25 25 rS 1 G 1908 824 24 24 rF 0 G 1907.5 823.5 25 25 rS 1 G 2604 637 24 24 rF 0 G 2603.5 636.5 25 25 rS np 528 1406 :M 541 1393 :L 528 1380 :L 515 1393 :L 528 1406 :L eofill 1 G -.004 -.004 528.004 1406.004 .004 .004 541 1393 @b 528 1380.004 -.004 .004 541.004 1393 .004 528 1380 @a -.004 -.004 515.004 1393.004 .004 .004 528 1380 @b 515 1393.004 -.004 .004 528.004 1406 .004 515 1393 @a -.004 -.004 528.004 1406.004 .004 .004 541 1393 @b 528 1380.004 -.004 .004 541.004 1393 .004 528 1380 @a -.004 -.004 515.004 1393.004 .004 .004 528 1380 @b 515 1393.004 -.004 .004 528.004 1406 .004 515 1393 @a 0 G np 1224 1150 :M 1237 1137 :L 1224 1124 :L 1211 1137 :L 1224 1150 :L eofill 1 G -.004 -.004 1224.004 1150.004 .004 .004 1237 1137 @b 1224 1124.004 -.004 .004 1237.004 1137 .004 1224 1124 @a -.004 -.004 1211.004 1137.004 .004 .004 1224 1124 @b 1211 1137.004 -.004 .004 1224.004 1150 .004 1211 1137 @a -.004 -.004 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-.004 2616.004 810.004 .004 .004 2629 797 @b 2616 784.004 -.004 .004 2629.004 797 .004 2616 784 @a -.004 -.004 2603.004 797.004 .004 .004 2616 784 @b 2603 797.004 -.004 .004 2616.004 810 .004 2603 797 @a np 528 1434 :M 541 1421 :L 528 1408 :L 515 1421 :L 528 1434 :L eofill 0 G -.004 -.004 528.004 1434.004 .004 .004 541 1421 @b 528 1408.004 -.004 .004 541.004 1421 .004 528 1408 @a -.004 -.004 515.004 1421.004 .004 .004 528 1408 @b 515 1421.004 -.004 .004 528.004 1434 .004 515 1421 @a -.004 -.004 528.004 1434.004 .004 .004 541 1421 @b 528 1408.004 -.004 .004 541.004 1421 .004 528 1408 @a -.004 -.004 515.004 1421.004 .004 .004 528 1408 @b 515 1421.004 -.004 .004 528.004 1434 .004 515 1421 @a 1 G np 1224 1231 :M 1237 1218 :L 1224 1205 :L 1211 1218 :L 1224 1231 :L eofill 0 G -.004 -.004 1224.004 1231.004 .004 .004 1237 1218 @b 1224 1205.004 -.004 .004 1237.004 1218 .004 1224 1205 @a -.004 -.004 1211.004 1218.004 .004 .004 1224 1205 @b 1211 1218.004 -.004 .004 1224.004 1231 .004 1211 1218 @a -.004 -.004 1224.004 1231.004 .004 .004 1237 1218 @b 1224 1205.004 -.004 .004 1237.004 1218 .004 1224 1205 @a -.004 -.004 1211.004 1218.004 .004 .004 1224 1205 @b 1211 1218.004 -.004 .004 1224.004 1231 .004 1211 1218 @a 1 G np 1920 1090 :M 1933 1077 :L 1920 1064 :L 1907 1077 :L 1920 1090 :L eofill 0 G -.004 -.004 1920.004 1090.004 .004 .004 1933 1077 @b 1920 1064.004 -.004 .004 1933.004 1077 .004 1920 1064 @a -.004 -.004 1907.004 1077.004 .004 .004 1920 1064 @b 1907 1077.004 -.004 .004 1920.004 1090 .004 1907 1077 @a -.004 -.004 1920.004 1090.004 .004 .004 1933 1077 @b 1920 1064.004 -.004 .004 1933.004 1077 .004 1920 1064 @a -.004 -.004 1907.004 1077.004 .004 .004 1920 1064 @b 1907 1077.004 -.004 .004 1920.004 1090 .004 1907 1077 @a 1 G np 2616 1004 :M 2629 991 :L 2616 978 :L 2603 991 :L 2616 1004 :L eofill 0 G -.004 -.004 2616.004 1004.004 .004 .004 2629 991 @b 2616 978.004 -.004 .004 2629.004 991 .004 2616 978 @a -.004 -.004 2603.004 991.004 .004 .004 2616 978 @b 2603 991.004 -.004 .004 2616.004 1004 .004 2603 991 @a -.004 -.004 2616.004 1004.004 .004 .004 2629 991 @b 2616 978.004 -.004 .004 2629.004 991 .004 2616 978 @a -.004 -.004 2603.004 991.004 .004 .004 2616 978 @b 2603 991.004 -.004 .004 2616.004 1004 .004 2603 991 @a np 528 1464 :M 541 1490 :L 515 1490 :L 528 1464 :L eofill 1 G 528 1464.004 -.004 .004 541.004 1490 .004 528 1464 @a 26.004 515 1490.002 @c -.004 -.004 515.004 1490.004 .004 .004 528 1464 @b 528 1464.004 -.004 .004 541.004 1490 .004 528 1464 @a 26.004 515 1490.002 @c -.004 -.004 515.004 1490.004 .004 .004 528 1464 @b 0 G np 1224 1338 :M 1237 1364 :L 1211 1364 :L 1224 1338 :L eofill 1 G 1224 1338.004 -.004 .004 1237.004 1364 .004 1224 1338 @a 26.004 1211 1364.002 @c -.004 -.004 1211.004 1364.004 .004 .004 1224 1338 @b 1224 1338.004 -.004 .004 1237.004 1364 .004 1224 1338 @a 26.004 1211 1364.002 @c -.004 -.004 1211.004 1364.004 .004 .004 1224 1338 @b 0 G np 1920 1268 :M 1933 1294 :L 1907 1294 :L 1920 1268 :L eofill 1 G 1920 1268.004 -.004 .004 1933.004 1294 .004 1920 1268 @a 26.004 1907 1294.002 @c -.004 -.004 1907.004 1294.004 .004 .004 1920 1268 @b 1920 1268.004 -.004 .004 1933.004 1294 .004 1920 1268 @a 26.004 1907 1294.002 @c -.004 -.004 1907.004 1294.004 .004 .004 1920 1268 @b 0 G np 2616 1235 :M 2629 1261 :L 2603 1261 :L 2616 1235 :L eofill 1 G 2616 1235.004 -.004 .004 2629.004 1261 .004 2616 1235 @a 26.004 2603 1261.002 @c -.004 -.004 2603.004 1261.004 .004 .004 2616 1235 @b 2616 1235.004 -.004 .004 2629.004 1261 .004 2616 1235 @a 26.004 2603 1261.002 @c -.004 -.004 2603.004 1261.004 .004 .004 2616 1235 @b np 528 1590 :M 541 1616 :L 515 1616 :L 528 1590 :L eofill 0 G 528 1590.004 -.004 .004 541.004 1616 .004 528 1590 @a 26.004 515 1616.002 @c -.004 -.004 515.004 1616.004 .004 .004 528 1590 @b 528 1590.004 -.004 .004 541.004 1616 .004 528 1590 @a 26.004 515 1616.002 @c -.004 -.004 515.004 1616.004 .004 .004 528 1590 @b 1 G np 1224 1561 :M 1237 1587 :L 1211 1587 :L 1224 1561 :L eofill 0 G 1224 1561.004 -.004 .004 1237.004 1587 .004 1224 1561 @a 26.004 1211 1587.002 @c -.004 -.004 1211.004 1587.004 .004 .004 1224 1561 @b 1224 1561.004 -.004 .004 1237.004 1587 .004 1224 1561 @a 26.004 1211 1587.002 @c -.004 -.004 1211.004 1587.004 .004 .004 1224 1561 @b 1 G np 1920 1552 :M 1933 1578 :L 1907 1578 :L 1920 1552 :L eofill 0 G 1920 1552.004 -.004 .004 1933.004 1578 .004 1920 1552 @a 26.004 1907 1578.002 @c -.004 -.004 1907.004 1578.004 .004 .004 1920 1552 @b 1920 1552.004 -.004 .004 1933.004 1578 .004 1920 1552 @a 26.004 1907 1578.002 @c -.004 -.004 1907.004 1578.004 .004 .004 1920 1552 @b 1 G np 2616 1552 :M 2629 1578 :L 2603 1578 :L 2616 1552 :L eofill 0 G 2616 1552.004 -.004 .004 2629.004 1578 .004 2616 1552 @a 26.004 2603 1578.002 @c -.004 -.004 2603.004 1578.004 .004 .004 2616 1552 @b 2616 1552.004 -.004 .004 2629.004 1578 .004 2616 1552 @a 26.004 2603 1578.002 @c -.004 -.004 2603.004 1578.004 .004 .004 2616 1552 @b gR gS 1426 1926 293 126 rC 1435 2009 :M f147 sf (Speedup)S gR .004 lw gS 0 0 3407 2368 rC gS 360 1269 :T 270 rotate -360 -1269 :T 360 1269 :M f147 sf (Processors)S gR 185 240 3036 1889 rC 443 1805 :M f175 sf (0)S 443 1668 :M (1)S 443 1530 :M (2)S 443 1392 :M (3)S 443 1254 :M (4)S 443 1117 :M (5)S 443 979 :M (6)S 443 841 :M (7)S 443 703 :M (8)S 443 566 :M (9)S 414 428 :M (10)S 514 1900 :M (4)S 1210 1900 :M (9)S 1891 1900 :M (16)S 2588 1900 :M 1 G 0 G (25)S 1 G 2714 741 414 716 rF 0 G 2740 767 361 90 rC 206.004 2748 805.002 @c 2839 793 24 24 rF 1 G 2838.5 792.5 25 25 rS 2978 830 :M 0 G (1200)S gR gS 2740 882 361 89 rC 206.004 2748 920.002 @c 1 G 2839 908 24 24 rF 0 G 2838.5 907.5 25 25 rS 2978 945 :M f175 sf (1000)S gR gS 2740 996 333 90 rC 206.004 2748 1034.002 @c np 2851 1047 :M 2864 1034 :L 2851 1021 :L 2838 1034 :L 2851 1047 :L eofill 1 G -.004 -.004 2851.004 1047.004 .004 .004 2864 1034 @b 2851 1021.004 -.004 .004 2864.004 1034 .004 2851 1021 @a -.004 -.004 2838.004 1034.004 .004 .004 2851 1021 @b 2838 1034.004 -.004 .004 2851.004 1047 .004 2838 1034 @a -.004 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1276 .004 2851 1250 @a 26.004 2838 1276.002 @c -.004 -.004 2838.004 1276.004 .004 .004 2851 1250 @b 2851 1250.004 -.004 .004 2864.004 1276 .004 2851 1250 @a 26.004 2838 1276.002 @c -.004 -.004 2838.004 1276.004 .004 .004 2851 1250 @b 2978 1288 :M 0 G f175 sf (400)S gR gS 2740 1340 333 90 rC 206.004 2748 1378.002 @c 1 G np 2851 1365 :M 2864 1391 :L 2838 1391 :L 2851 1365 :L eofill 0 G 2851 1365.004 -.004 .004 2864.004 1391 .004 2851 1365 @a 26.004 2838 1391.002 @c -.004 -.004 2838.004 1391.004 .004 .004 2851 1365 @b 2851 1365.004 -.004 .004 2864.004 1391 .004 2851 1365 @a 26.004 2838 1391.002 @c -.004 -.004 2838.004 1391.004 .004 .004 2851 1365 @b 2978 1403 :M f175 sf (200)S gR gS 2710 603 391 126 rC 2719 686 :M f147 sf (Matrix Size)S gR endp showpage %%Trailer end %%EOF %%EndDocument @endspecial 859 5019 a Ft(Figure)j(3:)41 b(P)m(erformance)31 b(using)e(the)i(theoretical)f(mo)s(del.)734 5303 y Fr(Figure)e(4)h(sho) n(ws)e(the)j(in\015uence)f(of)g(the)g(three)g(factors)f(in)h(\(6\))g (in)g(the)g(the-)609 5403 y(oretical)g(p)r(erformance)g(of)i(the)f (algorithms.)44 b(Sp)r(eci\014cally)-7 b(,)31 b(this)f(\014gure)g (repre-)1831 5652 y Ft(10)p eop %%Page: 11 11 11 10 bop 609 548 a Fr(sen)n(ts)29 b(their)h(v)-5 b(alue)29 b(using)h(25)e(pro)r(cessors)g(for)h(matrices)f(of)i(sev)n(eral)e (sizes.)42 b(W)-7 b(e)609 648 y(can)30 b(see)g(that)h(the)f (computation)g(cost)g(gro)n(ws)f(more)g(quic)n(kly)h(than)g(the)h(com-) 609 747 y(m)n(unications)20 b(cost,)i(but)f(it)f(is)h(larger)d(only)i (with)h(matrices)f(of)g(size)h(600)e(or)g(more.)609 847 y(Therefore,)25 b(the)h(o)n(v)n(erload)d(due)i(to)h(the)g(comm)n (unications)e(limits)i(the)g(sp)r(eedup)609 946 y(that)34 b(w)n(e)g(can)g(obtain)f(with)i(the)f(parallel)f(algorithm)g(except)h (for)f(v)n(ery)g(large)609 1046 y(matrices.)459 2562 y @beginspecial 81 @llx 94 @lly 763 @urx 492 @ury 3401 @rwi 1700 @rhi @setspecial %%BeginDocument: fig4.eps %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 81 94 763 492 %%Title: 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%%BeginFile: lw8_feature-1.01 %%Copyright: Copyright 1990-1998 Adobe Systems Incorporated and Apple Computer Incorporated. All Rights Reserved. /bd{bind def}bind def /ld{load def}bd /xs{exch store}bd /Z{0 def}bd /T true def /F false def /level2 /languagelevel where { pop languagelevel 2 ge }{ F }ifelse def /odictstk Z /oopstk Z /fcl { count oopstk sub dup 0 gt { {pop}repeat }{ pop }ifelse countdictstack odictstk sub dup 0 gt { {end}repeat }{ pop }ifelse }bd /sfcl2 { /odictstk countdictstack store count/oopstk xs }bd /efcl2 { stopped{$error/newerror F put}if fcl }bd /noload Z /startnoload { {/noload save store}if }bd /endnoload { {noload restore}if }bd /setcopies{ level2 { 1 dict begin/NumCopies exch def currentdict end setpagedevice }{ userdict/#copies 3 -1 roll put }ifelse }def level2 startnoload /ststpgdev{}def /dopgdev{}def /stpgdev{}def /buf Z /didstop T def /sfcl { /didstop T store /odictstk countdictstack store count/oopstk xs currentfile cvx stopped { $error/newerror F put didstop { save/didstop xs /buf vmstatus exch sub exch pop dup 0 lt{pop 0}if dup 64000 gt{pop 64000}if string store { currentfile buf readline { (}efcl)eq{exit}if }{ /UnexpectedEOF errordict/rangecheck get exec }ifelse }loop didstop restore }if }if fcl }bd /efcl { /didstop F store exec stop }bd level2 endnoload level2 not startnoload /setpagedevice where{pop/realstpgdev/setpagedevice ld}if /SC_topddict Z /SC_spdict Z /$spusrdict F def /dopgdev { userdict/setpagedevice undef $spusrdict { userdict/setpagedevice/realstpgdev load put /$spusrdict F store }if SC_topddict realstpgdev }bd /stpgdev { SC_topddict dup 3 -1 roll { SC_spdict 2 index known { SC_spdict 2 index get dup 3 -1 roll { put dup }forall pop put dup }{ put dup }ifelse }forall pop pop }bd /ststpgdev { /setpagedevice where { userdict eq { /$spusrdict T store }if }if userdict/setpagedevice/stpgdev load put /SC_topddict 0 dict store /SC_spdict 3 dict begin /InputAttributes 0 dict def /Policies 0 dict def /OutputAttributes 0 dict def currentdict end store }def /sfcl/sfcl2 ld /efcl/efcl2 ld level2 not endnoload %%EndFile %%BeginFile: lw8_basic-3.0 /xdf{exch def}bd /:L/lineto /lw/setlinewidth /:M/moveto /rl/rlineto /rm/rmoveto /:C/curveto /:T/translate /:K/closepath /:mf/makefont /gS/gsave /gR/grestore /np/newpath 12{ld}repeat /framewidth -1 def /QDframwid -1 def /numframes Z /mTS matrix def /$m matrix def /av 86 def /por T def /normland F def /psb-nosave{}def /pse-nosave{}def /us Z /psb{/us save store}bd /pse{us restore}bd /level3 /languagelevel where { pop languagelevel 3 ge }{ F }ifelse def level2 startnoload /setjob { statusdict/jobname 3 -1 roll put }bd /devg/DeviceGray def /devr/DeviceRGB def /devc/DeviceCMYK def level2 endnoload level2 not startnoload /setjob { 1 dict begin/JobName xdf currentdict end setuserparams }bd /devg[/DeviceGray]def /devr[/DeviceRGB]def /devc[/DeviceCMYK]def level2 not endnoload /pm Z /mT Z /sD Z /initializepage { mT $m currentmatrix mTS concatmatrix pop /pm save store mT concat }bd /endp { pm restore }bd /adjRect { dup 2 mul 6 2 roll 4 index sub exch 5 -1 roll sub exch 4 2 roll 4 index add exch 5 -1 roll add exch 4 2 roll }bd /frame1up { gS mTS setmatrix QDframwid lw /setstrokeadjust where{pop T setstrokeadjust}if clippath pathbbox 2 index sub exch 3 index sub exch currentlinewidth framewidth mul adjRect numframes dup 0 lt{pop 0}if { 4 copy rS currentlinewidth framewidth mul 4 mul adjRect }repeat pop pop pop pop gR }bd /$c devr def /rectclip where { pop/rC/rectclip ld }{ /rC { np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K clip np }bd }ifelse /rectfill where { pop/rF/rectfill ld }{ /rF { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl fill gR }bd }ifelse /rectstroke where { pop/rS/rectstroke ld }{ /rS { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K stroke gR }bd }ifelse %%EndFile level3 startnoload %%BeginFile: lw8_safeclipL12-1.0 /rectclip where { pop/rCa/rectclip ld }{ /rCa { np 0 1 index length 4 idiv { 2 copy 4 getinterval aload pop 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K 4 add }repeat clip np pop pop }bd }ifelse /savedstack Z /subsavedstack Z /execstring Z /saferCa { /execstring xs /odictstk countdictstack store /oopstk 0 store count 0 ne { savedstack 0 eq{ count 100 gt{count}{100}ifelse array/savedstack xs }{ count savedstack length gt { count array/savedstack xs }if }ifelse count savedstack 0 3 -1 roll getinterval astore/subsavedstack xs }{ /subsavedstack 0 store }ifelse execstring cvx stopped { $error/newerror F put newpath }if fcl subsavedstack 0 ne{ subsavedstack aload pop }if }bd %%EndFile level3 endnoload level3 not startnoload %%BeginFile: lw8_safeclipL3-1.0 /saferCa/pop ld %%EndFile level3 not endnoload %%BeginFile: lw8_level1_colorspace-2.0 /G/setgray ld /:F1/setgray ld /:F/setrgbcolor ld /:F4/setcmykcolor where { pop /setcmykcolor ld }{ { 3 { dup 3 -1 roll add dup 1 gt{pop 1}if 1 exch sub 4 1 roll }repeat pop setrgbcolor }bd }ifelse /:Fx { counttomark {0{G}0{:F}{:F4}} exch get exec pop }bd /$cs Z /:rg{devr :ss}bd /:sc{$cs :ss}bd /:dc { dup type/arraytype eq{0 get}if dup/DeviceCMYK eq { pop devc }{ /DeviceGray eq { devg }{ devr }ifelse }ifelse /$cs xdf }bd /:sgl{}def /:dr{}bd /:fCRD{pop}bd /:ckcs{}bd /:ss{/$c xdf}bd %%EndFile %%BeginFile: lw8_uniform_graphics-2.0 /@a { np :M 0 rl :L 0 exch rl 0 rl :L fill }bd /@b { np :M 0 rl 0 exch rl :L 0 rl 0 exch rl fill }bd /@c { moveto 0 rlineto stroke }bd /@w { moveto 0 exch rlineto stroke }bd /arct where { pop }{ /arct { arcto pop pop pop pop }bd }ifelse /x1 Z /x2 Z /y1 Z /y2 Z /rad Z /@q { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct fill }bd /@s { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct :K stroke }bd /@i { np 0 360 arc fill }bd /@j { gS np :T scale 0 0 .5 0 360 arc fill gR }bd /@e { np 0 360 arc :K stroke }bd /@f { np $m currentmatrix pop :T scale 0 0 .5 0 360 arc :K $m setmatrix stroke }bd /@k { gS np :T 0 0 :M 0 0 5 2 roll arc fill gR }bd /@l { gS np :T 0 0 :M scale 0 0 .5 5 -2 roll arc fill gR }bd /@m { np arc stroke }bd /@n { np $m currentmatrix pop :T scale 0 0 .5 5 -2 roll arc $m setmatrix stroke }bd %%EndFile %%BeginFile: lw8_basic_text-2.1 /S/show ld /A{ 0.0 exch ashow }bd /R{ 0.0 exch 32 exch widthshow }bd /W{ 0.0 3 1 roll widthshow }bd /J{ 0.0 32 4 2 roll 0.0 exch awidthshow }bd /V{ 0.0 4 1 roll 0.0 exch awidthshow }bd /fcflg T def /fc{ fcflg{ vmstatus exch sub 50000 lt{ (%%[ Warning: Running out of memory ]%%\r)print flush/fcflg F store }if pop }if }bd /$f[1 0 0 -1 0 0]def /:ff{$f :mf}bd /MacEncoding StandardEncoding 256 array copy def MacEncoding dup 9/space put dup 39/quotesingle put 96/grave put /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis 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Z /sf1pass { lU setfont }bd /scf1pass { exch lU exch scalefont fD }bd /scf2pass { scalefont fD }bd md/fontname known not{ /fontname/customfont def }if /Encoding Z /:mre { copyfontdict /Encoding MacEncoding def fontname currentdict end definefont :ff fD }bd /:bsr { copyfontdict /Encoding Encoding 256 array copy def Encoding dup }bd /pd{put dup}bd /:esr { pop pop fontname currentdict end definefont :ff fD }bd /ps Z /fz{/ps xs}bd /cF/currentfont ld /mbf { /makeblendedfont where { pop makeblendedfont /ABlend exch definefont }{ pop }ifelse fD }def %%EndFile /currentpacking where {pop sc_oldpacking setpacking}if end %%EndProlog %%BeginSetup md begin /fD/def ld/sf/setfont ld /scf/scf2pass ld /mT[.24 0 0 -.24 12.24 581.773]def /sD 16 dict def %%IncludeFont: Times-Bold /f133/Times-Bold :mre /f147 f133 75 scf %%IncludeFont: Times-Roman /f160/Times-Roman :mre /f175 f160 58 scf /Courier findfont[10 0 0 -10 0 0]:mf setfont %%EndSetup %%Page: 1 1 %%BeginPageSetup initializepage %%EndPageSetup gS 0 0 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gR gS 1317 1926 390 126 rC 1326 2009 :M f147 sf (Matrix Size)S gR 1 G .004 lw gS 0 0 3407 2368 rC gS 360 1273 :T 270 rotate -360 -1273 :T 360 1273 :M 0 G f147 sf (Time \(sec.\))S gR 185 240 3036 1889 rC 436 1805 :M 0 G f175 sf (0)S 436 1576 :M (1)S 436 1346 :M (2)S 436 1117 :M (3)S 436 887 :M (4)S 436 657 :M (5)S 436 428 :M (6)S 478 1900 :M (200)S 874 1900 :M (400)S 1270 1900 :M (600)S 1666 1900 :M (800)S 2048 1900 :M (1000)S 2445 1900 :M 1 G 0 G (1200)S 1 G 2629 913 499 372 rF 0 G 2655 939 446 90 rC 206.004 2663 977.002 @c 2754 965 24 24 rF 1 G 2753.5 964.5 25 25 rS 2893 1002 :M 0 G (Comput.)S gR 0 G gS 2655 1054 423 89 rC 206.004 2663 1092.002 @c 1 G 2754 1080 24 24 rF 0 G 2753.5 1079.5 25 25 rS 2893 1117 :M f175 sf (Bandw.)S gR gS 2655 1168 431 90 rC 206.004 2663 1206.002 @c np 2766 1219 :M 2779 1206 :L 2766 1193 :L 2753 1206 :L 2766 1219 :L eofill 1 G -.004 -.004 2766.004 1219.004 .004 .004 2779 1206 @b 2766 1193.004 -.004 .004 2779.004 1206 .004 2766 1193 @a -.004 -.004 2753.004 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(the)g(parallel)e(algorithm.)459 4637 y @beginspecial 81 @llx 92 @lly 763 @urx 493 @ury 3401 @rwi 1700 @rhi @setspecial %%BeginDocument: fig6.eps %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 81 92 763 493 %%Title: ([invtoep.wbk]Escalado isotemporal MPI GM Gr\207fico 2) %%Creator: (Microsoft Excel: LaserWriter 8 E1-8.6) %%CreationDate: (17:08 lunes, 20 septiembre 1999) %%For: (badia) %%Routing: (mailto:\000email@domain.com) %%Pages: 1 %%DocumentFonts: Times-Bold Times-Roman %%DocumentNeededFonts: Times-Bold Times-Roman %%DocumentSuppliedFonts: %%DocumentData: Clean7Bit %%PageOrder: Ascend %%Orientation: Landscape %%DocumentMedia: Default 841.92 594.96 0 () () %RBINumCopies: 1 %RBINupNess: 1 1 %ADO_ImageableArea: 12.24 13.44 829.92 581.76 %%EndComments %%BeginDefaults %%ViewingOrientation: 0 -1 1 0 %%EndDefaults userdict/dscInfo 5 dict dup begin /Title([invtoep.wbk]Escalado isotemporal MPI GM Gr\207fico 2)def /Creator(Microsoft Excel: LaserWriter 8 E1-8.6)def /CreationDate(17:08 lunes, 20 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8.004 2331 404.002 @c 8.004 2355 404.002 @c 8.004 2379 404.002 @c 8.004 2403 404.002 @c 8.004 2427 404.002 @c 8.004 2451 404.002 @c 8.004 2475 404.002 @c 8.004 2499 404.002 @c 8.004 2523 404.002 @c 8.004 2547 404.002 @c 8.004 2571 404.002 @c 8.004 2595 404.002 @c 8.004 2619 404.002 @c 8.004 2643 404.002 @c gR gS 280 315 2846 1738 rC .004 lw 1431.004 483.002 404 @w 41.004 462 1835.002 @c 41.004 462 1677.002 @c 41.004 462 1517.002 @c 41.004 462 1358.002 @c 41.004 462 1199.002 @c 41.004 462 1040.002 @c 41.004 462 881.002 @c 41.004 462 722.002 @c 41.004 462 563.002 @c 41.004 462 404.002 @c 2175.004 483 1835.002 @c 41.004 483.002 1815 @w 41.004 1208.002 1815 @w 41.004 1933.002 1815 @w 41.004 2658.002 1815 @w 468 389 2205 1461 rC -.004 -.004 483.004 1564.004 .004 .004 1208 1294 @b -.004 -.004 1208.004 1294.004 .004 .004 1933 956 @b -.004 -.004 1933.004 956.004 .004 .004 2658 489 @b -.004 -.004 483.004 1574.004 .004 .004 1208 1342 @b -.004 -.004 1208.004 1342.004 .004 .004 1933 963 @b -.004 -.004 1933.004 963.004 .004 .004 2658 838 @b -.004 -.004 483.004 1682.004 .004 .004 1208 1592 @b -.004 -.004 1208.004 1592.004 .004 .004 1933 1414 @b -.004 -.004 1933.004 1414.004 .004 .004 2658 1390 @b -.004 -.004 483.004 1753.004 .004 .004 1208 1737 @b -.004 -.004 1208.004 1737.004 .004 .004 1933 1685 @b -.004 -.004 1933.004 1685.004 .004 .004 2658 1648 @b 471 1552 24 24 rF 1 G 470.5 1551.5 25 25 rS 0 G 1196 1282 24 24 rF 1 G 1195.5 1281.5 25 25 rS 0 G 1921 944 24 24 rF 1 G 1920.5 943.5 25 25 rS 0 G 2646 477 24 24 rF 1 G 2645.5 476.5 25 25 rS 471 1562 24 24 rF 0 G 470.5 1561.5 25 25 rS 1 G 1196 1330 24 24 rF 0 G 1195.5 1329.5 25 25 rS 1 G 1921 951 24 24 rF 0 G 1920.5 950.5 25 25 rS 1 G 2646 826 24 24 rF 0 G 2645.5 825.5 25 25 rS np 483 1695 :M 496 1682 :L 483 1669 :L 470 1682 :L 483 1695 :L eofill 1 G -.004 -.004 483.004 1695.004 .004 .004 496 1682 @b 483 1669.004 -.004 .004 496.004 1682 .004 483 1669 @a -.004 -.004 470.004 1682.004 .004 .004 483 1669 @b 470 1682.004 -.004 .004 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1401 @b 1920 1414.004 -.004 .004 1933.004 1427 .004 1920 1414 @a -.004 -.004 1933.004 1427.004 .004 .004 1946 1414 @b 1933 1401.004 -.004 .004 1946.004 1414 .004 1933 1401 @a -.004 -.004 1920.004 1414.004 .004 .004 1933 1401 @b 1920 1414.004 -.004 .004 1933.004 1427 .004 1920 1414 @a 0 G np 2658 1403 :M 2671 1390 :L 2658 1377 :L 2645 1390 :L 2658 1403 :L eofill 1 G -.004 -.004 2658.004 1403.004 .004 .004 2671 1390 @b 2658 1377.004 -.004 .004 2671.004 1390 .004 2658 1377 @a -.004 -.004 2645.004 1390.004 .004 .004 2658 1377 @b 2645 1390.004 -.004 .004 2658.004 1403 .004 2645 1390 @a -.004 -.004 2658.004 1403.004 .004 .004 2671 1390 @b 2658 1377.004 -.004 .004 2671.004 1390 .004 2658 1377 @a -.004 -.004 2645.004 1390.004 .004 .004 2658 1377 @b 2645 1390.004 -.004 .004 2658.004 1403 .004 2645 1390 @a np 483 1766 :M 496 1753 :L 483 1740 :L 470 1753 :L 483 1766 :L eofill 0 G -.004 -.004 483.004 1766.004 .004 .004 496 1753 @b 483 1740.004 -.004 .004 496.004 1753 .004 483 1740 @a -.004 -.004 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:T 344 1314 :M f147 sf (Scaled Speedup)S gR 185 240 3036 1889 rC 402 1848 :M f175 sf (0)S 402 1689 :M (2)S 402 1529 :M (4)S 402 1370 :M (6)S 402 1211 :M (8)S 377 1052 :M (10)S 377 893 :M (12)S 377 734 :M (14)S 377 575 :M (16)S 377 416 :M (18)S 445 1936 :M (2x2)S 1170 1936 :M (3x3)S 1895 1936 :M (4x4)S 2620 1936 :M 1 G 0 G (5x5)S 1 G 2762 901 366 438 rF 0 G 2788 927 313 78 rC 207.004 2795 959.002 @c 2887 947 24 24 rF 1 G 2886.5 946.5 25 25 rS 3020 978 :M 0 G (800)S gR gS 2788 1030 313 77 rC 207.004 2795 1062.002 @c 1 G 2887 1050 24 24 rF 0 G 2886.5 1049.5 25 25 rS 3020 1080 :M f175 sf (600)S gR gS 2788 1132 313 78 rC 207.004 2795 1164.002 @c np 2899 1177 :M 2912 1164 :L 2899 1151 :L 2886 1164 :L 2899 1177 :L eofill 1 G -.004 -.004 2899.004 1177.004 .004 .004 2912 1164 @b 2899 1151.004 -.004 .004 2912.004 1164 .004 2899 1151 @a -.004 -.004 2886.004 1164.004 .004 .004 2899 1151 @b 2886 1164.004 -.004 .004 2899.004 1177 .004 2886 1164 @a -.004 -.004 2899.004 1177.004 .004 .004 2912 1164 @b 2899 1151.004 -.004 .004 2912.004 1164 .004 2899 1151 @a -.004 -.004 2886.004 1164.004 .004 .004 2899 1151 @b 2886 1164.004 -.004 .004 2899.004 1177 .004 2886 1164 @a 3020 1183 :M 0 G f175 sf (400)S gR gS 2788 1235 313 77 rC 207.004 2795 1267.002 @c 1 G np 2899 1280 :M 2912 1267 :L 2899 1254 :L 2886 1267 :L 2899 1280 :L eofill 0 G -.004 -.004 2899.004 1280.004 .004 .004 2912 1267 @b 2899 1254.004 -.004 .004 2912.004 1267 .004 2899 1254 @a -.004 -.004 2886.004 1267.004 .004 .004 2899 1254 @b 2886 1267.004 -.004 .004 2899.004 1280 .004 2886 1267 @a -.004 -.004 2899.004 1280.004 .004 .004 2912 1267 @b 2899 1254.004 -.004 .004 2912.004 1267 .004 2899 1254 @a -.004 -.004 2886.004 1267.004 .004 .004 2899 1254 @b 2886 1267.004 -.004 .004 2899.004 1280 .004 2886 1267 @a 3020 1285 :M f175 sf (200)S gR gS 2798 797 302 89 rC 2805 859 :M f147 sf (Matrix Size)S gR endp showpage %%Trailer end %%EOF %%EndDocument @endspecial 792 4833 a Ft(Figure)k(6:)41 b(Scaled)30 b(sp)s(eedup)f(using)g(an)h(isotemp)s(oral)f(metric.)734 5104 y Fr(Figure)k(7)g(sho)n(ws)f(the)i(sp)r(eedups)g(obtained)f (without)h(scaling,)g(that)g(is,)h(if)609 5204 y(w)n(e)g(main)n(tain)g (the)h(size)f(of)g(the)h(problem)f(while)h(increasing)e(the)h(n)n(um)n (b)r(er)h(of)609 5303 y(pro)r(cessors.)64 b(If)38 b(w)n(e)f(compare)f (this)i(\014gure)e(with)i(\014gure)f(6,)i(w)n(e)e(can)g(see)g(the)609 5403 y(e\013ect)28 b(of)g(scaling)e(in)i(the)g(p)r(erformance)f(of)g (the)h(algorithm.)1831 5652 y Ft(13)p eop %%Page: 14 14 14 13 bop 459 1874 a @beginspecial 81 @llx 94 @lly 763 @urx 492 @ury 3401 @rwi 1700 @rhi @setspecial %%BeginDocument: fig7.eps %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 81 94 763 492 %%Title: ([invtoep.wbk]Nuevos resultados GM \(25 pr.\) Gr\207fico 3) %%Creator: (Microsoft Excel: LaserWriter 8 E1-8.6) %%CreationDate: (09:44 mi\216rcoles, 15 septiembre 1999) %%For: (badia) %%Routing: (mailto:\000email@domain.com) %%Pages: 1 %%DocumentFonts: Times-Bold Times-Roman 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All Rights Reserved. /bd{bind def}bind def /ld{load def}bd /xs{exch store}bd /Z{0 def}bd /T true def /F false def /level2 /languagelevel where { pop languagelevel 2 ge }{ F }ifelse def /odictstk Z /oopstk Z /fcl { count oopstk sub dup 0 gt { {pop}repeat }{ pop }ifelse countdictstack odictstk sub dup 0 gt { {end}repeat }{ pop }ifelse }bd /sfcl2 { /odictstk countdictstack store count/oopstk xs }bd /efcl2 { stopped{$error/newerror F put}if fcl }bd /noload Z /startnoload { {/noload save store}if }bd /endnoload { {noload restore}if }bd /setcopies{ level2 { 1 dict begin/NumCopies exch def currentdict end setpagedevice }{ userdict/#copies 3 -1 roll put }ifelse }def level2 startnoload /ststpgdev{}def /dopgdev{}def /stpgdev{}def /buf Z /didstop T def /sfcl { /didstop T store /odictstk countdictstack store count/oopstk xs currentfile cvx stopped { $error/newerror F put didstop { save/didstop xs /buf vmstatus exch sub exch pop dup 0 lt{pop 0}if dup 64000 gt{pop 64000}if string store { currentfile buf readline { (}efcl)eq{exit}if }{ /UnexpectedEOF errordict/rangecheck get exec }ifelse }loop didstop restore }if }if fcl }bd /efcl { /didstop F store exec stop }bd level2 endnoload level2 not startnoload /setpagedevice where{pop/realstpgdev/setpagedevice ld}if /SC_topddict Z /SC_spdict Z /$spusrdict F def /dopgdev { userdict/setpagedevice undef $spusrdict { userdict/setpagedevice/realstpgdev load put /$spusrdict F store }if SC_topddict realstpgdev }bd /stpgdev { SC_topddict dup 3 -1 roll { SC_spdict 2 index known { SC_spdict 2 index get dup 3 -1 roll { put dup }forall pop put dup }{ put dup }ifelse }forall pop pop }bd /ststpgdev { /setpagedevice where { userdict eq { /$spusrdict T store }if }if userdict/setpagedevice/stpgdev load put /SC_topddict 0 dict store /SC_spdict 3 dict begin /InputAttributes 0 dict def /Policies 0 dict def /OutputAttributes 0 dict def currentdict end store }def /sfcl/sfcl2 ld /efcl/efcl2 ld level2 not endnoload %%EndFile %%BeginFile: lw8_basic-3.0 /xdf{exch def}bd /:L/lineto /lw/setlinewidth /:M/moveto /rl/rlineto /rm/rmoveto /:C/curveto /:T/translate /:K/closepath /:mf/makefont /gS/gsave /gR/grestore /np/newpath 12{ld}repeat /framewidth -1 def /QDframwid -1 def /numframes Z /mTS matrix def /$m matrix def /av 86 def /por T def /normland F def /psb-nosave{}def /pse-nosave{}def /us Z /psb{/us save store}bd /pse{us restore}bd /level3 /languagelevel where { pop languagelevel 3 ge }{ F }ifelse def level2 startnoload /setjob { statusdict/jobname 3 -1 roll put }bd /devg/DeviceGray def /devr/DeviceRGB def /devc/DeviceCMYK def level2 endnoload level2 not startnoload /setjob { 1 dict begin/JobName xdf currentdict end setuserparams }bd /devg[/DeviceGray]def /devr[/DeviceRGB]def /devc[/DeviceCMYK]def level2 not endnoload /pm Z /mT Z /sD Z /initializepage { mT $m currentmatrix mTS concatmatrix pop /pm save store mT concat }bd /endp { pm restore }bd /adjRect { dup 2 mul 6 2 roll 4 index sub exch 5 -1 roll sub exch 4 2 roll 4 index add exch 5 -1 roll add exch 4 2 roll }bd /frame1up { gS mTS setmatrix QDframwid lw /setstrokeadjust where{pop T setstrokeadjust}if clippath pathbbox 2 index sub exch 3 index sub exch currentlinewidth framewidth mul adjRect numframes dup 0 lt{pop 0}if { 4 copy rS currentlinewidth framewidth mul 4 mul adjRect }repeat pop pop pop pop gR }bd /$c devr def /rectclip where { pop/rC/rectclip ld }{ /rC { np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K clip np }bd }ifelse /rectfill where { pop/rF/rectfill ld }{ /rF { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl fill gR }bd }ifelse /rectstroke where { pop/rS/rectstroke ld }{ /rS { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K stroke gR }bd }ifelse %%EndFile level3 startnoload %%BeginFile: lw8_safeclipL12-1.0 /rectclip where { pop/rCa/rectclip ld }{ /rCa { np 0 1 index length 4 idiv { 2 copy 4 getinterval aload pop 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K 4 add }repeat clip np pop pop }bd }ifelse /savedstack Z /subsavedstack Z /execstring Z /saferCa { /execstring xs /odictstk countdictstack store /oopstk 0 store count 0 ne { savedstack 0 eq{ count 100 gt{count}{100}ifelse array/savedstack xs }{ count savedstack length gt { count array/savedstack xs }if }ifelse count savedstack 0 3 -1 roll getinterval astore/subsavedstack xs }{ /subsavedstack 0 store }ifelse execstring cvx stopped { $error/newerror F put newpath }if fcl subsavedstack 0 ne{ subsavedstack aload pop }if }bd %%EndFile level3 endnoload level3 not startnoload %%BeginFile: lw8_safeclipL3-1.0 /saferCa/pop ld %%EndFile level3 not endnoload %%BeginFile: lw8_level1_colorspace-2.0 /G/setgray ld /:F1/setgray ld /:F/setrgbcolor ld /:F4/setcmykcolor where { pop /setcmykcolor ld }{ { 3 { dup 3 -1 roll add dup 1 gt{pop 1}if 1 exch sub 4 1 roll }repeat pop setrgbcolor }bd }ifelse /:Fx { counttomark {0{G}0{:F}{:F4}} exch get exec pop }bd /$cs Z /:rg{devr :ss}bd /:sc{$cs :ss}bd /:dc { dup type/arraytype eq{0 get}if dup/DeviceCMYK eq { pop devc }{ /DeviceGray eq { devg }{ devr }ifelse }ifelse /$cs xdf }bd /:sgl{}def /:dr{}bd /:fCRD{pop}bd /:ckcs{}bd /:ss{/$c xdf}bd %%EndFile %%BeginFile: lw8_uniform_graphics-2.0 /@a { np :M 0 rl :L 0 exch rl 0 rl :L fill }bd /@b { np :M 0 rl 0 exch rl :L 0 rl 0 exch rl fill }bd /@c { moveto 0 rlineto stroke }bd /@w { moveto 0 exch rlineto stroke }bd /arct where { pop }{ /arct { arcto pop pop pop pop }bd }ifelse /x1 Z /x2 Z /y1 Z /y2 Z /rad Z /@q { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct fill }bd /@s { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct :K stroke }bd /@i { np 0 360 arc fill }bd /@j { gS np :T scale 0 0 .5 0 360 arc fill gR }bd /@e { np 0 360 arc :K stroke }bd /@f { np $m currentmatrix pop :T scale 0 0 .5 0 360 arc :K $m setmatrix stroke }bd /@k { gS np :T 0 0 :M 0 0 5 2 roll arc fill gR }bd /@l { gS np :T 0 0 :M 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1717 :L 1220 1691 :L eofill 0 G 1220 1691.004 -.004 .004 1233.004 1717 .004 1220 1691 @a 26.004 1207 1717.002 @c -.004 -.004 1207.004 1717.004 .004 .004 1220 1691 @b 1220 1691.004 -.004 .004 1233.004 1717 .004 1220 1691 @a 26.004 1207 1717.002 @c -.004 -.004 1207.004 1717.004 .004 .004 1220 1691 @b 1 G np 1911 1697 :M 1924 1723 :L 1898 1723 :L 1911 1697 :L eofill 0 G 1911 1697.004 -.004 .004 1924.004 1723 .004 1911 1697 @a 26.004 1898 1723.002 @c -.004 -.004 1898.004 1723.004 .004 .004 1911 1697 @b 1911 1697.004 -.004 .004 1924.004 1723 .004 1911 1697 @a 26.004 1898 1723.002 @c -.004 -.004 1898.004 1723.004 .004 .004 1911 1697 @b 1 G np 2602 1709 :M 2615 1735 :L 2589 1735 :L 2602 1709 :L eofill 0 G 2602 1709.004 -.004 .004 2615.004 1735 .004 2602 1709 @a 26.004 2589 1735.002 @c -.004 -.004 2589.004 1735.004 .004 .004 2602 1709 @b 2602 1709.004 -.004 .004 2615.004 1735 .004 2602 1709 @a 26.004 2589 1735.002 @c -.004 -.004 2589.004 1735.004 .004 .004 2602 1709 @b gR gS 1385 1926 360 126 rC 1395 2009 :M f147 sf (Processors)S gR .004 lw gS 0 0 3407 2368 rC gS 360 1236 :T 270 rotate -360 -1236 :T 360 1236 :M f147 sf (Speedup)S gR 185 240 3036 1889 rC 443 1805 :M f175 sf (0)S 443 1668 :M (1)S 443 1530 :M (2)S 443 1392 :M (3)S 443 1254 :M (4)S 443 1117 :M (5)S 443 979 :M (6)S 443 841 :M (7)S 443 703 :M (8)S 443 566 :M (9)S 414 428 :M (10)S 485 1900 :M (2x2)S 1176 1900 :M (3x3)S 1867 1900 :M (4x4)S 2558 1900 :M 1 G 0 G (5x5)S 1 G 2714 741 414 716 rF 0 G 2740 767 361 90 rC 206.004 2748 805.002 @c 2839 793 24 24 rF 1 G 2838.5 792.5 25 25 rS 2978 830 :M 0 G (1200)S gR gS 2740 882 361 89 rC 206.004 2748 920.002 @c 1 G 2839 908 24 24 rF 0 G 2838.5 907.5 25 25 rS 2978 945 :M f175 sf (1000)S gR gS 2740 996 333 90 rC 206.004 2748 1034.002 @c np 2851 1047 :M 2864 1034 :L 2851 1021 :L 2838 1034 :L 2851 1047 :L eofill 1 G -.004 -.004 2851.004 1047.004 .004 .004 2864 1034 @b 2851 1021.004 -.004 .004 2864.004 1034 .004 2851 1021 @a -.004 -.004 2838.004 1034.004 .004 .004 2851 1021 @b 2838 1034.004 -.004 .004 2851.004 1047 .004 2838 1034 @a -.004 -.004 2851.004 1047.004 .004 .004 2864 1034 @b 2851 1021.004 -.004 .004 2864.004 1034 .004 2851 1021 @a -.004 -.004 2838.004 1034.004 .004 .004 2851 1021 @b 2838 1034.004 -.004 .004 2851.004 1047 .004 2838 1034 @a 2978 1059 :M 0 G f175 sf (800)S gR gS 2740 1111 333 89 rC 206.004 2748 1149.002 @c 1 G np 2851 1162 :M 2864 1149 :L 2851 1136 :L 2838 1149 :L 2851 1162 :L eofill 0 G -.004 -.004 2851.004 1162.004 .004 .004 2864 1149 @b 2851 1136.004 -.004 .004 2864.004 1149 .004 2851 1136 @a -.004 -.004 2838.004 1149.004 .004 .004 2851 1136 @b 2838 1149.004 -.004 .004 2851.004 1162 .004 2838 1149 @a -.004 -.004 2851.004 1162.004 .004 .004 2864 1149 @b 2851 1136.004 -.004 .004 2864.004 1149 .004 2851 1136 @a -.004 -.004 2838.004 1149.004 .004 .004 2851 1136 @b 2838 1149.004 -.004 .004 2851.004 1162 .004 2838 1149 @a 2978 1174 :M f175 sf (600)S gR gS 2740 1225 333 90 rC 206.004 2748 1263.002 @c np 2851 1250 :M 2864 1276 :L 2838 1276 :L 2851 1250 :L eofill 1 G 2851 1250.004 -.004 .004 2864.004 1276 .004 2851 1250 @a 26.004 2838 1276.002 @c -.004 -.004 2838.004 1276.004 .004 .004 2851 1250 @b 2851 1250.004 -.004 .004 2864.004 1276 .004 2851 1250 @a 26.004 2838 1276.002 @c -.004 -.004 2838.004 1276.004 .004 .004 2851 1250 @b 2978 1288 :M 0 G f175 sf (400)S gR gS 2740 1340 333 90 rC 206.004 2748 1378.002 @c 1 G np 2851 1365 :M 2864 1391 :L 2838 1391 :L 2851 1365 :L eofill 0 G 2851 1365.004 -.004 .004 2864.004 1391 .004 2851 1365 @a 26.004 2838 1391.002 @c -.004 -.004 2838.004 1391.004 .004 .004 2851 1365 @b 2851 1365.004 -.004 .004 2864.004 1391 .004 2851 1365 @a 26.004 2838 1391.002 @c -.004 -.004 2838.004 1391.004 .004 .004 2851 1365 @b 2978 1403 :M f175 sf (200)S gR gS 2707 635 391 126 rC 2716 718 :M f147 sf (Matrix Size)S gR endp showpage %%Trailer end %%EOF %%EndDocument @endspecial 1178 2070 a Ft(Figure)30 b(7:)41 b(Sp)s(eedup)28 b(without)i(scaling.)734 2420 y Fr(Figure)h(7)h(also)e(allo)n(ws)h(us)h 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(\014gure)f(1,)g(on)g(the)h(size)g(of)f(the)h(message.)38 b(All)29 b(these)609 4113 y(asp)r(ects)h(of)g(the)h(implemen)n(tation)f (de\014ne)g(the)h(real)e(cost)h(of)g(the)h(comm)n(unica-)609 4213 y(tions)j(and)f(can)g(justify)i(the)f(p)r(ossible)f(di\013erences) h(b)r(et)n(w)n(een)f(the)h(theoretical)609 4313 y(and)28 b(exp)r(erimen)n(tal)f(results.)609 4545 y Fm(5.4)112 b(Isospatial)37 b(scalabilit)m(y)609 4698 y Fr(When)23 b(w)n(e)f(use)h(the)g(isospatial)e(scalabilit)n(y)g(w)n(e)i(m)n(ust)f (main)n(tain)g(the)h(size)g(of)f(the)609 4798 y(problem)e(in)g(eac)n(h) f(pro)r(cessor.)32 b(In)20 b(the)g(case)f(of)h(the)g(ScaLAP)-7 b(A)n(CK)19 b(library)-7 b(,)20 b(if)h(w)n(e)609 4897 y(hold)h(constan)n(t)e(the)i(ratio)f Fn(n)1496 4867 y Fg(2)1533 4897 y Fn(=p)g Fr(while)g(increasing)f(the)i(n)n(um)n(b)r(er) f(of)h(pro)r(cessors,)609 4997 y(the)32 b(e\016ciency)f(of)h(the)f (driv)n(er)f(routines)h(is)h(almost)e(main)n(tained.)48 b(When)32 b(this)609 5097 y(condition)c(holds,)f(it)h(is)f(said)g(that) h(these)g(routines)f(scale)g(iso)r(e\016cien)n(tly)-7 b(.)734 5196 y(In)41 b(\014gure)f(8)h(w)n(e)g(sho)n(w)f(the)h(p)r (erformance)f(obtained)h(when)g(w)n(e)g(use)g(an)609 5296 y(isospatial)32 b(scaling)g(in)h(our)f(parallel)g(algorithm.)52 b(First,)34 b(w)n(e)f(can)f(see)h(a)g(clear)609 5396 y(decrease)d(of)h(the)h(MFlops/s.)47 b(p)r(er)32 b(no)r(de)f(when)h(w)n (e)f(increase)f(the)h(n)n(um)n(b)r(er)h(of)1831 5652 y Ft(14)p eop %%Page: 15 15 15 14 bop 609 548 a Fr(pro)r(cessors.)42 b(This)30 b(b)r(eha)n(viour)e (is)i(due)g(to)g(the)h(fact)f(that)g(w)n(e)f(are)g(not)h(using)g(a)609 648 y(ScaLAP)-7 b(A)n(CK)29 b(routine,)i(but)g(a)f(com)n(bination)f(of) i(some)e(of)i(them,)g(with)g(some)609 747 y(routines)g(implemen)n(ted)i (sp)r(eci\014cally)e(for)g(this)h(algorithm.)48 b(Besides)32 b(w)n(e)f(p)r(er-)609 847 y(form)38 b(some)g(redistributions)g(of)h (the)g(data)f(in)h(eac)n(h)f(iteration)g(that)h(clearly)609 946 y(a\013ect)28 b(the)g(global)e(scalabilit)n(y)h(of)g(the)h (algorithm.)734 1046 y(On)j(the)h(other)e(hand,)j(w)n(e)e(m)n(ust)g(p)r (oin)n(t)h(out)f(that)h(the)f(biggest)g(reduction)609 1146 y(o)r(ccurs)c(when)h(w)n(e)f(go)g(from)g(the)i(sequen)n(tial)e(v)n (ersion)f(to)h(the)h(parallel)f(v)n(ersion.)609 1245 y(If)d(w)n(e)f(increase)f(the)i(n)n(um)n(b)r(er)f(of)g(pro)r(cessors)e (in)j(the)f(parallel)f(v)n(ersion,)h(the)h(p)r(er-)609 1345 y(formance)j(is)g(almost)g(main)n(tained.)37 b(This)27 b(b)r(eha)n(viour)g(is)g(due)h(to)f(the)h(o)n(v)n(erload)609 1445 y(of)23 b(the)h(parallelization,)e(and)h(mainly)-7 b(,)24 b(to)f(the)h(comm)n(unication)e(cost.)35 b(Besides,)609 1544 y(when)26 b(w)n(e)g(b)r(egin)g(with)h(large)d(matrices,)i(the)h (decrease)d(of)i(the)h(p)r(erformance)e(is)609 1644 y(not)32 b(so)f(large)f(and)h(w)n(e)g(go)g(from)g(64)g(MFlop/s.)48 b(p)r(er)31 b(no)r(de)h(in)f(the)h(sequen)n(tial)609 1743 y(case)27 b(to)g(33)g(MFlop/s.)36 b(p)r(er)28 b(no)r(de)f(using)h (25)e(pro)r(cessors.)459 3281 y @beginspecial 81 @llx 94 @lly 763 @urx 492 @ury 3401 @rwi 1700 @rhi @setspecial %%BeginDocument: fig8.eps %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 81 94 763 492 %%Title: ([invtoep.wbk]Escalado isoespacial MPI GM Gr\207fico 2) %%Creator: (Microsoft Excel: LaserWriter 8 E1-8.6) %%CreationDate: (09:51 mi\216rcoles, 15 septiembre 1999) %%For: (badia) %%Routing: (mailto:\000email@domain.com) %%Pages: 1 %%DocumentFonts: Times-Bold Times-Roman %%DocumentNeededFonts: Times-Bold Times-Roman %%DocumentSuppliedFonts: %%DocumentData: Clean7Bit %%PageOrder: Ascend %%Orientation: Landscape %%DocumentMedia: Default 841.92 594.96 0 () () %RBINumCopies: 1 %RBINupNess: 1 1 %ADO_ImageableArea: 12.24 13.44 829.92 581.76 %%EndComments %%BeginDefaults %%ViewingOrientation: 0 -1 1 0 %%EndDefaults userdict/dscInfo 5 dict dup begin /Title([invtoep.wbk]Escalado isoespacial MPI GM Gr\207fico 2)def /Creator(Microsoft Excel: LaserWriter 8 E1-8.6)def /CreationDate(09:51 mi\216rcoles, 15 septiembre 1999)def /For(badia)def /Pages 1 def end put %%BeginProlog /md 201 dict def md begin/currentpacking where {pop /sc_oldpacking currentpacking def true setpacking}if %%BeginFile: lw8_feature-1.01 %%Copyright: Copyright 1990-1998 Adobe Systems Incorporated and Apple Computer Incorporated. All Rights Reserved. /bd{bind def}bind def /ld{load def}bd /xs{exch store}bd /Z{0 def}bd /T true def /F false def /level2 /languagelevel where { pop languagelevel 2 ge }{ F }ifelse def /odictstk Z /oopstk Z /fcl { count oopstk sub dup 0 gt { {pop}repeat }{ pop }ifelse countdictstack odictstk sub dup 0 gt { {end}repeat }{ pop }ifelse }bd /sfcl2 { /odictstk countdictstack store count/oopstk xs }bd /efcl2 { stopped{$error/newerror F put}if fcl }bd /noload Z /startnoload { {/noload save store}if }bd /endnoload { {noload restore}if }bd /setcopies{ level2 { 1 dict begin/NumCopies exch def currentdict end setpagedevice }{ userdict/#copies 3 -1 roll put }ifelse }def level2 startnoload /ststpgdev{}def /dopgdev{}def /stpgdev{}def /buf Z /didstop T def /sfcl { /didstop T store /odictstk countdictstack store count/oopstk xs currentfile cvx stopped { $error/newerror F put didstop { save/didstop xs /buf vmstatus exch sub exch pop dup 0 lt{pop 0}if dup 64000 gt{pop 64000}if string store { currentfile buf readline { (}efcl)eq{exit}if }{ /UnexpectedEOF errordict/rangecheck get exec }ifelse }loop didstop restore }if }if fcl }bd /efcl { /didstop F store exec stop }bd level2 endnoload level2 not startnoload /setpagedevice where{pop/realstpgdev/setpagedevice ld}if /SC_topddict Z /SC_spdict Z /$spusrdict F def /dopgdev { userdict/setpagedevice undef $spusrdict { userdict/setpagedevice/realstpgdev load put /$spusrdict F store }if SC_topddict realstpgdev }bd /stpgdev { SC_topddict dup 3 -1 roll { SC_spdict 2 index known { SC_spdict 2 index get dup 3 -1 roll { put dup }forall pop put dup }{ put dup }ifelse }forall pop pop }bd /ststpgdev { /setpagedevice where { userdict eq { /$spusrdict T store }if }if userdict/setpagedevice/stpgdev load put /SC_topddict 0 dict store /SC_spdict 3 dict begin /InputAttributes 0 dict def /Policies 0 dict def /OutputAttributes 0 dict def currentdict end store }def /sfcl/sfcl2 ld /efcl/efcl2 ld level2 not endnoload %%EndFile %%BeginFile: lw8_basic-3.0 /xdf{exch def}bd /:L/lineto /lw/setlinewidth /:M/moveto /rl/rlineto /rm/rmoveto /:C/curveto /:T/translate /:K/closepath /:mf/makefont /gS/gsave /gR/grestore /np/newpath 12{ld}repeat /framewidth -1 def /QDframwid -1 def /numframes Z /mTS matrix def /$m matrix def /av 86 def /por T def /normland F def /psb-nosave{}def /pse-nosave{}def /us Z /psb{/us save store}bd /pse{us restore}bd /level3 /languagelevel where { pop languagelevel 3 ge }{ F }ifelse def level2 startnoload /setjob { statusdict/jobname 3 -1 roll put }bd /devg/DeviceGray def /devr/DeviceRGB def /devc/DeviceCMYK def level2 endnoload level2 not startnoload /setjob { 1 dict begin/JobName xdf currentdict end setuserparams }bd /devg[/DeviceGray]def /devr[/DeviceRGB]def /devc[/DeviceCMYK]def level2 not endnoload /pm Z /mT Z /sD Z /initializepage { mT $m currentmatrix mTS concatmatrix pop /pm save store mT concat }bd /endp { pm restore }bd /adjRect { dup 2 mul 6 2 roll 4 index sub exch 5 -1 roll sub exch 4 2 roll 4 index add exch 5 -1 roll add exch 4 2 roll }bd /frame1up { gS mTS setmatrix QDframwid lw /setstrokeadjust where{pop T setstrokeadjust}if clippath pathbbox 2 index sub exch 3 index sub exch currentlinewidth framewidth mul adjRect numframes dup 0 lt{pop 0}if { 4 copy rS currentlinewidth framewidth mul 4 mul adjRect }repeat pop pop pop pop gR }bd /$c devr def /rectclip where { pop/rC/rectclip ld }{ /rC { np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K clip np }bd }ifelse /rectfill where { pop/rF/rectfill ld }{ /rF { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl fill gR }bd }ifelse /rectstroke where { pop/rS/rectstroke ld }{ /rS { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K stroke gR }bd }ifelse %%EndFile level3 startnoload %%BeginFile: lw8_safeclipL12-1.0 /rectclip where { pop/rCa/rectclip ld }{ /rCa { np 0 1 index length 4 idiv { 2 copy 4 getinterval aload pop 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K 4 add }repeat clip np pop pop }bd }ifelse /savedstack Z /subsavedstack Z /execstring Z /saferCa { /execstring xs /odictstk countdictstack store /oopstk 0 store count 0 ne { savedstack 0 eq{ count 100 gt{count}{100}ifelse array/savedstack xs }{ count savedstack length gt { count array/savedstack xs }if }ifelse count savedstack 0 3 -1 roll getinterval astore/subsavedstack xs }{ /subsavedstack 0 store }ifelse execstring cvx stopped { $error/newerror F put newpath }if fcl subsavedstack 0 ne{ subsavedstack aload pop }if }bd %%EndFile level3 endnoload level3 not startnoload %%BeginFile: lw8_safeclipL3-1.0 /saferCa/pop ld %%EndFile level3 not endnoload %%BeginFile: lw8_level1_colorspace-2.0 /G/setgray ld /:F1/setgray ld /:F/setrgbcolor ld /:F4/setcmykcolor where { pop /setcmykcolor ld }{ { 3 { dup 3 -1 roll add dup 1 gt{pop 1}if 1 exch sub 4 1 roll }repeat pop setrgbcolor }bd }ifelse /:Fx { counttomark {0{G}0{:F}{:F4}} exch get exec pop }bd /$cs Z /:rg{devr :ss}bd /:sc{$cs :ss}bd /:dc { dup type/arraytype eq{0 get}if dup/DeviceCMYK eq { pop devc }{ /DeviceGray eq { devg }{ devr }ifelse }ifelse /$cs xdf }bd /:sgl{}def /:dr{}bd /:fCRD{pop}bd /:ckcs{}bd /:ss{/$c xdf}bd %%EndFile %%BeginFile: lw8_uniform_graphics-2.0 /@a { np :M 0 rl :L 0 exch rl 0 rl :L fill }bd /@b { np :M 0 rl 0 exch rl :L 0 rl 0 exch rl fill }bd /@c { moveto 0 rlineto stroke }bd /@w { moveto 0 exch rlineto stroke }bd /arct where { pop }{ /arct { arcto pop pop pop pop }bd }ifelse /x1 Z /x2 Z /y1 Z /y2 Z /rad Z /@q { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct fill }bd /@s { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct :K stroke }bd /@i { np 0 360 arc fill }bd /@j { gS np :T scale 0 0 .5 0 360 arc fill gR }bd /@e { np 0 360 arc :K stroke }bd /@f { np $m currentmatrix pop :T scale 0 0 .5 0 360 arc :K $m setmatrix stroke }bd /@k { gS np :T 0 0 :M 0 0 5 2 roll arc fill gR }bd /@l { gS np :T 0 0 :M scale 0 0 .5 5 -2 roll arc fill gR }bd /@m { np arc stroke }bd /@n { np $m currentmatrix pop :T scale 0 0 .5 5 -2 roll arc $m setmatrix stroke }bd %%EndFile %%BeginFile: lw8_basic_text-2.1 /S/show ld /A{ 0.0 exch ashow }bd /R{ 0.0 exch 32 exch widthshow }bd /W{ 0.0 3 1 roll widthshow }bd /J{ 0.0 32 4 2 roll 0.0 exch awidthshow }bd /V{ 0.0 4 1 roll 0.0 exch awidthshow }bd /fcflg T def /fc{ fcflg{ vmstatus exch sub 50000 lt{ (%%[ Warning: Running out of memory ]%%\r)print flush/fcflg F store }if pop }if }bd /$f[1 0 0 -1 0 0]def /:ff{$f :mf}bd /MacEncoding StandardEncoding 256 array copy def MacEncoding dup 9/space put dup 39/quotesingle put 96/grave put /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis 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}forall }bd /$ckeyd md def /:skey { 1 index maxlength 2 index length sub ge { begin /$mkeys 20 dict def /$mkeys load end dup/$ckeyd xs }if 3 1 roll put }bd /fD1pass { 40 $ckeyd //md ne { pop 1 }if $ckeyd exch :skey }bd /:searchdict Z /:searchdict { exch 2 copy known { get }{ exch/$mkeys get :searchdict }ifelse }bd /lU { //md exch 2 copy known { get }{ exch /$mkeys get :searchdict }ifelse }bd /:xs{ exch 0 1 2 index length 1 sub{ 3 copy gsave 1 getinterval show grestore exch get 0.0 rmoveto }for pop pop }bd /:ys{ exch 0 1 2 index length 1 sub{ 3 copy gsave 1 getinterval show grestore exch get 0.0 exch rmoveto }for pop pop }bd /:xys{ exch 0 1 2 index length 1 sub{ 3 copy gsave 1 getinterval show grestore exch 2 mul 2 copy 1 add get 3 -2 roll get exch rmoveto }for pop pop }bd level2 endnoload level2 not startnoload /copyfontdict { findfont dup length dict copy begin }bd /fD1pass/def ld /lU/load ld level2{ /:xs/xshow ld /:ys/yshow ld /:xys/xyshow ld }if level2 not endnoload /fD Z /sf Z /scf Z /sf1pass { lU setfont }bd /scf1pass { exch lU exch scalefont fD }bd /scf2pass { scalefont fD }bd md/fontname known not{ /fontname/customfont def }if /Encoding Z /:mre { copyfontdict /Encoding MacEncoding def fontname currentdict end definefont :ff fD }bd /:bsr { copyfontdict /Encoding Encoding 256 array copy def Encoding dup }bd /pd{put dup}bd /:esr { pop pop fontname currentdict end definefont :ff fD }bd /ps Z /fz{/ps xs}bd /cF/currentfont ld /mbf { /makeblendedfont where { pop makeblendedfont /ABlend exch definefont }{ pop }ifelse fD }def %%EndFile /currentpacking where {pop sc_oldpacking setpacking}if end %%EndProlog %%BeginSetup md begin /fD/def ld/sf/setfont ld /scf/scf2pass ld /mT[.24 0 0 -.24 12.24 581.773]def /sD 16 dict def %%IncludeFont: Times-Bold /f133/Times-Bold :mre /f147 f133 75 scf %%IncludeFont: Times-Roman /f160/Times-Roman :mre /f175 f160 58 scf /Courier findfont[10 0 0 -10 0 0]:mf setfont %%EndSetup %%Page: 1 1 %%BeginPageSetup initializepage %%EndPageSetup gS 0 0 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%%Trailer end %%EOF %%EndDocument @endspecial 656 3477 a Ft(Figure)k(8:)41 b(Mega\015ops)31 b(p)s(er)f(pro)s(cessor)g(using)f(an)h(isospatial)f(scaling.)609 3923 y Fq(6)135 b(Conclusions)609 4105 y Fr(In)24 b(this)g(pap)r(er)f (w)n(e)h(presen)n(t)f(a)g(parallel)g(algorithm)g(that)h(solv)n(es)e (e\016cien)n(tly)i(and)609 4204 y(in)37 b(a)f(quite)h(scalable)e(w)n(a) n(y)h(the)h(in)n(v)n(erse)e(eigenproblem)g(for)h(real)g(symmetric)609 4304 y(T)-7 b(o)r(eplitz)21 b(matrices.)33 b(W)-7 b(e)21 b(sho)n(w)e(the)h(p)r(ossibilit)n(y)g(of)g(implemen)n(ting)h(this)f(t)n (yp)r(e)h(of)609 4404 y(algorithm)26 b(on)g(an)h(arc)n(hitecture)e (with)j(an)e(excellen)n(t)g(rate)h(cost/p)r(erformance.)609 4503 y(Sp)r(eci\014cally)-7 b(,)36 b(w)n(e)d(ha)n(v)n(e)g(used)g(a)h (cluster)f(of)h(p)r(ersonal)f(computers)g(connected)609 4603 y(with)28 b(a)f(high)h(p)r(erformance)e(net)n(w)n(ork.)734 4703 y(W)-7 b(e)26 b(m)n(ust)g(alw)n(a)n(ys)e(tak)n(e)h(in)n(to)h 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(ols)g(to)g(arc)n(hitectures)609 2441 y(based)i(on)g(clusters)g(of)g(p) r(ersonal)g(pro)r(cessors.)63 b(It)38 b(is)f(not)h(necessary)d(to)j (use)609 2540 y(big)24 b(sup)r(ercomputers,)g(with)h(v)n(ery)e(exp)r (ensiv)n(e)h(hardw)n(are)e(and)i(sp)r(eci\014cally)g(de-)609 2640 y(signed)i(soft)n(w)n(are)e(to)i(obtain)g(go)r(o)r(d)f(result)h (in)g(the)h(solution)f(of)g(complex)f(linear)609 2740 y(algebra)h(problems.)734 2839 y(Besides,)g(w)n(e)h(ha)n(v)n(e)f (obtained)g(a)h(considerable)e(degree)h(of)h(scalabilit)n(y)f(on)h(a) 609 2939 y(cluster)22 b(of)g(p)r(ersonal)e(computers)i(connected)f (with)i(an)e(external)h(net)n(w)n(ork)e(with)609 3039 y(excellen)n(t)i(p)r(erformance)g(as)g(the)h(Myrinet.)35 b(This)22 b(result)h(is)f(v)n(ery)g(promising,)g(as)609 3138 y(it)j(pro)n(v)n(es)d(the)j(p)r(ossibilit)n(y)f(of)h(increasing)e (the)h(area)f(of)i(application)e(of)i(parallel)609 3238 y(algorithms)35 b(to)h(arc)n(hitectures)e(based)i(on)g(standard)f(comp) r(onen)n(ts)h(with)g(lo)n(w)609 3337 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b(9\(2\):401{405,)d(Marc)n(h)j(1988.)609 4053 y([17])45 b(Myricom.)66 b(The)38 b(GM)h(message-passing)c(system.) 67 b(T)-7 b(ec)n(hnical)37 b(rep)r(ort,)784 4153 y(Myricom)27 b(Inc.,)g(1998.)609 4274 y([18])45 b(VIT)-7 b(A)40 b(Standards)e(Org.) 71 b(Myrinet-on-VME)39 b(proto)r(col)f(sp)r(eci\014cation.)784 4374 y(Draft)20 b(Standard.)25 b(T)-7 b(ec)n(hnical)19 b(Rep)r(ort)h(26-199x)e(Draft)i(1.1.,)h(VIT)-7 b(A,)21 b(1998.)609 4496 y([19])45 b(Marc)d(Snir,)47 b(Stev)n(e)c(W.)g(Otto,)k (Stev)n(en)c(Huss-Lederman,)i(Da)n(vid)e(W.)784 4595 y(W)-7 b(alk)n(er,)33 b(and)f(Jac)n(k)f(Dongarra.)49 b Fb(MPI:)36 b(the)e(c)l(omplete)h(r)l(efer)l(enc)l(e)p Fr(.)52 b(MIT)784 4695 y(Press,)26 b(Cam)n(bridge,)g(MA,)i(USA,)h (1996.)609 4816 y([20])45 b(William)31 b(F.)h(T)-7 b(renc)n(h.)46 b(Numerical)31 b(solution)f(of)h(the)h(in)n(v)n(erse)d(eigen)n(v)-5 b(alue)784 4916 y(problem)29 b(for)f(real)h(symmetric)g(T)-7 b(o)r(eplitz)29 b(matrices.)41 b Fb(SIAM)31 b(Journal)g(on)784 5015 y(Scienti\014c)f(Computing)p Fr(,)e(18\(6\):1722{1736,)22 b(No)n(v)n(em)n(b)r(er)27 b(1997.)609 5137 y([21])45 b(Rob)r(ert)34 b(A.)g(v)-5 b(an)34 b(de)g(Geijn.)56 b Fb(Using)35 b(PLAP)-6 b(A)n(CK:)36 b(Par)l(al)t(lel)i(Line)l(ar)e(A)n (l-)784 5236 y(gebr)l(a)g(Package)p Fr(.)60 b(MIT)34 b(Press,)h(Cam)n(bridge,)f(MA,)h(USA,)g(1997.)56 b(With)784 5336 y(con)n(tributions)27 b(b)n(y)g(Philip)g(Alpato)n(v)g(and)h (others.)1831 5652 y Ft(17)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF --------------------------------------------------------------------- José M. Badía PhD Universitat Jaume I Informatica José M. Badía PhD Universitat Jaume I Informatica Campus Riu Sec s/n Fax: +34 964 72 84 35 Castellon Work: +34 964 72 82 95 12071 Conference Software Address España Specific Directory Server Additional Information: Last Name Badía First Name José M. Version 2.1