Estimating Powers with Base Close to Unity and Large Exponents

Transcription

1 Divulgacions Mamáicas Vol. 3 No. 2005), pp Esimaing Powrs wih Bas Clos o Uniy and Larg Exponns Esimacón d Poncias con Bas Crcana a la Unidad y Grands Exponns Vio Lampr Vio.Lampr@fgg.uni-lj.si) FGG, Jamova 2 Univrsiy of Ljubljana, Slovnia 386. Absrac In his papr w driv h rlaion xp h h2 + h) 2 ε) < < xp h h2 2 + ε), valid for ε 0,), > 0 and 0 < h ε. Ths inqualiis sima h ras of convrgnc of lim + x = x and lim x + x = x x 2 2 and nabl numrical compuaion of a powr wih a bas clos o uniy and larg xponn. Ky words and phrass: approximaion of powrs, asympoic inqualiis, compuaion of powrs wih larg xponn and bas clos o uniy, xponnial funcion, simaion of powrs. Rsumn En s arículo s obin la rlación xp h h2 + h) 2 ε) < < xp h h2 2 + ε), válida para ε 0, ), > 0 y 0 < h ε. Esas dsigualdads siman las asas d convrgncia d lím + x = x y x x 2 2 Rcibido 2003//05. Rvisado 2005/03/2. Acpado 2005/03/28. MSC 2000): 26D07, 33B0, 33F05, 4A25, 4A60, 65D20. lím x + x =

2 22 Vio Lampr y prmin l cálculo numérico d una poncia d bas crcana a la unidad y gran xponn. Palabras y frass clav: aproximación d poncias, dsigualdads asinóicas, cálculo d una poncia d bas crcana a la unidad y gran xponn, función xponncial, simación d poncias. Inroducion How o compu a for a clos o and bing vry larg? Such qusion occurs whn w ) wan o obain numrical valu of h soluion x n = x 0 +h) n = x 0 + nh n n of diffrnc quaion xn x n = hx n, h = cons. 0), which is frqunly rplacd by is coninuous vrsion, namly h diffrnial quaion, dx d = hx, having soluion x = x0)h. Alhough h compuaion of a is usually an asy ask, spcially in h ag of compurs, h qusion is no as simpl as i sms. For xampl, using calculaors o compu singular powrs such as α = ) and β = ) 059. w do no obain corrc rsul du o ovrflow problms. For h sam rason, h compuaion of numbrs α and β abov is no qui an asy ask, vn for powrful mah sofwar, lik Mahmaica [5], for xampl. Morovr, if powrs α and β ar subsiud by mor singular powrs, vn Mahmaica dos no giv a usful rsul. Howvr, i is wll known ha ) + x x for larg, according o ) convrgnc lim + x = x. Bu, o us his approximaion for numrical compuaion of powrs lik α and β, w nd simpl bounds for h rrors, i.. w nd simpl funcions A, x) and B, x), clos o for larg, such ha A, x) x ) + x B, x) x for larg. To his ffc l us go back o h dfiniion of a powr o find such funcions A, x) and B, x). Svral auhors inroduc powr wih posiiv bas and ral xponn by allowing for xponn firs posiiv ingr valus and hn gnraliz h noion of h powr from h cas whn h xponn is ngaiv ingr o h cass whn h xponn is raional and ral. This rquirs a lo of im and a fair of ffor o prov h addiiviy and diffrniabiliy propris of ral xponnial funcion. Hnc, w no rcommnd his approach o powrs, no vn from only h horical poin of viw. In addiion, his way is no produciv for our purpos as wll. Forunaly, hr xiss an asir mhod basd on h dfini ingral. Choosing his mhod, h logarihmic and xponnial funcions can b inroducd asily, and hir fundamnal propris can b drivd in a simpl mannr s for xampl [, p. 409] or Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34

3 Esimaing Powrs wih Bas Clos o Uniy and a Larg Exponn 23 [4, p. 7]). To sum up, h logarihmic funcion ln : R + R, lnx) := x d, is diffrniabl, sricly monoonically incrasing bijcion wih drivaiv ln x) = x and wih h addiiv propry lnx x 2 ) = lnx ) + lnx 2 ), obaind by subsiuing h ingraion variabl wih a nw on. Is invrs funcion, xp := ln : R R +, calld h xponnial funcion, is consqunly also diffrniabl, sricly monoonically incrasing bijcion wih drivaiv xp x) = xpx) and wih h addiiv propry xpx + x 2 ) = xpx )xpx 2 ). Th powr of a posiiv ral numbr a is dfind by a x := xpxlna)), x R. Powrs, dfind his way, hav all h usual propris, which ar asily vrifid. Wih h Eulr numbr := xp), h idniy xpx) = x holds for vry ral x. By showing ha h drivaiv of quoin qx) := x fx) is idnically qual o 0, providd ha funcion f : R R + is diffrniabl, is drivaiv coincids wih islf and f0) =, i bcoms clar ha h xponn funcion is h uniqu diffrniabl funcion, whos drivaiv coincids wih islf and aks h valu a h poin 0. Taylor s formula and h qualiy xp x) = xpx) nabl us o mak som iniial numrically usful approximaions for h xponnial funcion. Using Taylor s formula, w can also approxima logarihms. Thrfor, compuaion of rgular powrs is no a hard work. On h ohr hand, compuaion of singular powrs, as has bn mniond abov, could b rahr problmaic. W would lik o find a way how o compu such singular powrs, as wll as w would lik o sima h xprssion + x ) for larg. 2 Monoonous convrgnc Our main concrn is h funcion + x ) =: Ex, ) ) Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34

4 24 Vio Lampr dfind on h inrvals I x :=, max{0, x}) and I + x := min{0, x}, ) for any ral x. For nonzro x w hav [ lim lnex, ) = lim x x ln + x ) ] ln + τ) ln) = x lim τ 0 τ = x ln ) = x. This mans, du o coninuiy diffrniabiliy) of logarihmic funcion, ha lim Ex, ) xiss and is qual o x for vry ral x: lim + x ) = x. 2) For vry I x I+ x w hav d Ex, ) = Lx, ) Ex, ), 3) d whr Lx, ) := ln + x ) x + x. Th funcion Lx, ) has drivaiv d Lx, ) = d + x x ) x x) 2 = x2 + x) 2. 3a) Consqunly, i is sricly monoonically incrasing on h inrval Ix and dcrasing on h inrval I x + for any x 0. Thrfor, Lx, ) > 0 for x 0 and Ix I x +, bcaus lim Lx, ) = 0 for any x. Furhrmor, sinc Ex, ) > 0, w conclud from 3) ha h funcion Ex, ) incrass sricly monoonously on boh inrvals Ix and I+ x for vry x 0. Hnc, for any x 0, h convrgnc lim + x ) = x = lim x ) 4) is sricly monoonically incrasing or dcrasing, rspcivly. Figur illusras his dynamics. Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34

5 Esimaing Powrs wih Bas Clos o Uniy and a Larg Exponn 25 ) 3 Th ra of convrgnc lim + x For any ε [0, ) h funcion f ε :, ) R, dfind by f ε τ) := ln + τ) τ + τ 2 2 ε), x x ) Figur : lim + x ) = x = lim x. has drivaiv Thrfor f ε τ) = + τ + τ ε = τ + ε)τ ε) + τ). min f ετ) = f ε 0) = 0 τ ε and consqunly f ε τ) > 0 for τ [ ε, )\{0}. Tha is τ τ2 2 ε) < + τ 5) for any ε [0, ) and for vry nonzro τ ε. Similarly, for any ε [0, ), w ra h funcion g ε :, ) R dfind by τ 2 g ε τ) := τ ln + τ). 2 + ε) Is drivaiv, g ετ) = τ + ε + τ = τ ε) τ + ε) + τ), Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34

6 26 Vio Lampr shows ha min g ετ) = g ε 0) = 0. <τ ε Hnc, g ε τ) > 0 for τ, ε]\{0}; hus + τ < τ τ2 2+ε) 5a) for vry ε [0, ) and for vry nonzro τ such ha < τ ε. L us xploi h abov rlaions 5) and 5a). Indd, for any ral x 0, ε 0, ), and x ε h numbr τ := x 0 lis on h inrval [ ε, ε]. Thrfor, according o 5) and 5a), h following rlaion holds x x 2 2 ε) 2 < + x < x x2 2+ε) 2. Taking h powrs w obain h main sima x x2 2 ε) < + x ) < x x2 2+ε), 6) valid for vry ral x 0, ε 0, ) and x /ε. From hs inqualiis w obain, aking h = x, h asympoic sima ) ) xp h h2 < + h) < xp h h2, 6a) 2 ε) 2 + ε) ru for ε 0, ), > 0 and 0 < h ε. Figur 2 illusras h sima 6) for x = and x =, and for ε = 0, whr dashd curvs rprsn lowr and uppr bounds. Considring h x x Figur 2: Lowr and uppr bounds 6) for h funcion + x ). approximaion + x ) x, 7) Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34

7 Esimaing Powrs wih Bas Clos o Uniy and a Larg Exponn 27 w obain from 6) h asympoic sima for h rlaiv rror x2 2+ε) < rx, ) < x2 2 ε) 8) rx, ) := x + x x, 8a) which holds wih h sam condiions as wr quod for 6). Figur 3 illusras h sima 8) for x = ± and x = ±2 a ε = 0. ) x 2, ε x 2 4, ε Figur 3: Esima 8) of h convrgnc rx, ) 0 as. From 8) w can g an addiional, lss accura, bu simplr sima for rx). To his ffc w obsrv ha h funcion ϕ : R R, dfind by ϕτ) := τ + τ, has a posiiv drivaiv for τ > 0. This mans ϕτ) > 0, i.. hr holds h sima for τ > 0; consqunly h funcion ψ:r R, whr has a drivaiv τ < τ 9) ψτ) := τ τ + τ 2 /2, 0) ψ τ) = τ + τ > 0 for τ > 0. Hnc, w hav found ha ψτ) is posiiv for τ posiiv. Thrfor, by dfiniion 0), w hav x ε) τ > τ τ2 2 = τ τ 2 ) ) for τ > 0. Combining 8) wih 9) and ), w obain h sima x ε) ) < rx, ) < x 2 2 ε), 2) Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34

8 28 Vio Lampr which holds for vry ral x 0, ε 0, ) and x /ε. Rfrring o 8a), h lf par of his rlaion is obviously inrsing only in cas > x2 4+ε), sinc rx, ) > 0 for x 0 and > 0, du o h fac ha ) + x convrgs monoonously from blow owards x as, s 2. Bounds prsnd in 2) ar clos o h bounds in 8). In Figur 4 w illusra his fac for [0, 30] by ploing graphs of diffrncs bwn lowr lf) and uppr righ) bounds d l ε, x, ) and d u ε, x, ) [ )] [ d l ε, x, ) := xp x2 x 2 x 2 )] 2 + ε) 2 + ε) 4 + ε) and d u ε, x, ) := x 2 2 ε) [ )] xp x2. 2 ε) 2x x 2, ε 0 2x x 2, ε Figur 4: Graphs of diffrncs d l lf) and d u righ). According o dfiniion 8a) w find, from rlaion 2) abov, h sima x x 2 x 2 ) < x + x ) x x 2 < 2 + ε) 4 + ε) 2 ε), 2a) valid undr h sam condiions as wr sad for 2). W also no an obvious and usful fac ha h funcion x ε) incrass monoonously on h inrval 0, ), whil h funcions ε + ε and ε ε rspcivly, dcras and incras monoonously on h inrval 0, ). Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34

9 Esimaing Powrs wih Bas Clos o Uniy and a Larg Exponn 29 Sing ε = 2 in 2) w dduc h sima x 2 ) x2 3 6 < rx, ) < x2, 2b) which is valid for x 0 and 2 x. For h sam rasons as wr sad in commn o 2), h lf par of his sima is inrsing only if x 2 /6. From 2b) w can xrac h rlaion x2 ) x < + x ) < [ x2 3 )] x2 x, 3) 6 which holds for x 0 and 2 x. Considring h rmark abov, h lf par of his rlaion is obviously inrsing only for > max{2 x, x 2 } and h righ par for > max{2 x, x 2 /6}. Puing h = x ino 3), w obain h sima h 2 ) [ )] h < + h) < h2 h2 h, 4) 3 6 valid for > 0 and 0 < h /2. Having posiiv, h lf sid of 4) is obviously inrsing only for 0 < h < min{/2, / } and h righ sid for 0 < h < min{/2, 6/}. Taking x 0 and > x, and puing ε := x in 2a), w obain h sima x x x x 2 ) [ < x + x ) ] < x x x 2 x, 5) valid for x 0 and > x. Ling o approach infiniy in 5), w obain h nx rsul [ lim x + x ) )] = x x ) Figur 5 illusras sima 5) for [0, 00]. 4 Exampls 4. L us ak x =. Choosing ε = 2, h rlaion 2a) can b applid o hos which fulfil h condiion x /ε = 2. Hnc, for all 2, h following sima holds ) ) < + < 2 2 Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34

10 30 Vio Lampr x x Figur 5: Bounds 5) for h convrgnc 6). i.. or < + ) < 2.8 < + ) < 2.8. If w ak ε = 0. in 2a), hn, for 0, w obain h rlaion ) < + ) < Thus, w hav mor accura sima.20 < + ) <.5, ru for 0. Taking ε = 0.0 and 00 in 2a), w obain h sima ) < + ) < , which amouns o an vn mor accura rlaion.34 < + ) <.38, 7) valid for 00. For ε sill closr o 0, w would obain from 2a) furhr mor accura simas, which ar crainly ru for largr valus of. 4.2 Sing x =, ε = 0.0 and 00 in 2a) w obain ) < ) < , Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34

11 Esimaing Powrs wih Bas Clos o Uniy and a Larg Exponn 3 i.. for < ) < ) 4.3 To drmin powr α := 0 59.) w us 4), sing h = and = Sinc h = and h 2 = w sima and α > ) > ) 0 59 > > α < )] [ < [ )] = ) = ) 0 60 Hnc, < < ) ) xp 0 59 < α < xp or numrically α = , whr all 2 dcimal placs ar corrc. 4.4 To compu powr β := ) w pu h = and = Sinc h = and h 2 = w ar simaing, according o 4), as follows: and β < β > 0 58.) > )0 58 > > )] [ < [ )] = ) = < < , ) 0 59 Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34

12 32 Vio Lampr hus ) ) xp < β < xp or numrically β = , whr all 8 dcimal placs ar corrc. 5 Rmarks 5. Using a slighly diffrn chniqus as hos ha hav bn applid driving 7), w can obain an sima, similar o 7). Namly, puing ε = 0 in 5) w g, for τ > 0, h sima ) xp τ τ2 < + τ 2 or i.. τ 2 < ln + τ)/τ + τ) /τ < τ 2. 9) To g an opposi inqualiy, w considr h funcion having h drivaiv Fτ) := τ τ2 2 + τ3 ln + τ), 3 F τ) = τ3 + τ > 0 for τ > 0. Hnc, Fτ) > F0) = 0 for τ > 0, ha is or i.. τ τ2 2 + τ3 3 > ln + τ) τ 2 + τ2 3 > ln + τ)/τ, + τ) /τ > τ 2 + τ2 3 a any τ > 0. Combining his rlaion wih 9), w find ha h sima τ 2 3 ) τ2 < + τ) /τ < ) τ 2 20) Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34

13 Esimaing Powrs wih Bas Clos o Uniy and a Larg Exponn 33 holds for vry τ > 0. For τ 0, 3 2), h numbr dτ) := τ 2 τ2 3 = τ 2 τ ) 3 is lying on h inrval 0, 3 6) 0, ). Bu, h funcion G:d d d sricly incrass on inrval 0, ), du o is posiiv drivaiv. Thrfor, for d 0, ), w hav Gd) > G0) = 0, i.. d > d/. Hnc, according o 9), w g d < d < d for vry d 0, ). Wih his in mind, according o 20), w conclud wih rlaion ) τ 2 τ2 < + τ) /τ < τ 3 2, valid for vry τ 0, 2) 3. Consqunly, sing τ =, w obain h sima < + ) < 2, 2) ru for > 2/ Inqualiis 6) hav bn obaind alrady in [2], bu using an ingral. 6 Qusions 6. Prov or disprov h qualiy { [ x x 2 lim x + x 2 ) )]} = x x 3 3x + 8) 24 and find furhr nsd limis, oghr wih suiabl simas. 6.2 How o sima h norm + x n) n x from blow and from abov for n N and x A, A bing ral or complx unial Banach algbra, possibly B algbra or only h fild C or marix algbra C n n? Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34

14 34 Vio Lampr Rfrncs [] Egn, G.J., Hill, E., Salas and Hill s Calculus On and Svral Variabls, John Wily & Sons, Inc. 995). [2] Lampr, V., How Clos is + x ) o x?, Mahmaics and Informaics Quarrly, 6996), [3] Mirinović, D.S., Vasić, P.M.,Analyic Inqualiis, Springr 970). [4] Pror, M.H., Morry, C.B., A Firs Cours in Ral Analysis, Springr UTM 99). [5] Wolfram, S., Mahmaica, vrsion 5.0., Wolfram Rsarch, Inc., Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34