Asymptotic Notation. Part II: Examples andproblems 1

Transcription

1 FORMALIZED MATHEMATICS Volume 9, Number 1, 2001 University of Białystok Asymptotic Notation. Part II: Examples andproblems 1 Richard Krueger University of Alberta Edmonton Piotr Rudnicki University of Alberta Edmonton Paul Shelley University of Alberta Edmonton Summary. The widely used textbook by Brassard and Bratley[2] includes a chapter devoted to asymptotic notation(chapter 3, pp ). We have attempted to test how suitable the current version of Mizar is for recording this typeofmaterialinitsentirety.thisarticleisafollow-upto[11]inwhichwe introduced the basic notions and general theory. This article presents a Mizar formalization of examples and solutions to problems from Chapter 3 of[2](some oftheexamplesandsolvedproblemsarealsoin[11]).notallproblemshavebeen solved as some required solutions not amenable for formalization. MML Identifier: ASYMPT 1. Thearticles[11],[10],[14],[15],[3],[4],[17],[1],[12],[13],[6],[19],[8],[9],[7], [16],[18], and[5] provide the terminology and notation for this paper. 1. Examples from the Text Weadoptthefollowingrules: c, edenoterealnumbers, k, n, m, N, n 1, M denote natural numbers, and x denotes a set. One can prove the following two propositions: 1 ThisworkhasbeensupportedbyNSERCGrantOGP c 2001 University of Białystok ISSN

2 144 richard krueger et al. (1) Let t, t 1 besequencesofrealnumbers.supposethat (i) t(0) = 0, (ii) forevery nsuchthat n > 0holds t(n) = (12 n 3 log 2 n 5 n 2 ) + (log 2 n) , (iii) t 1 (0) = 0,and (iv) forevery nsuchthat n > 0holds t 1 (n) = n 3 log 2 n. Thenthereexisteventually-positivesequences s, s 1 ofrealnumberssuch that s = tand s 1 = t 1 and s O(s 1 ). (2) Let a, bbelogbaserealnumbersand f, gbesequencesofrealnumbers. Suppose a > 1and b > 1and f(0) = 0andforevery nsuchthat n > 0 holds f(n) =log a nand g(0) = 0andforevery nsuchthat n > 0holds g(n) =log b n.thenthereexisteventually-positivesequences s, s 1 ofreal numberssuchthat s = fand s 1 = gand O(s) = O(s 1 ). Let a, b, cberealnumbers.thefunctor{a b n+c) } n N yieldsasequenceof realnumbersandisdefinedby: (Def.1) ({a b n+c) } n N )(n) = a b n+c. Let abeapositiverealnumberandlet b, cberealnumbers.onecanverify that{a b n+c) } n N iseventually-positive. The following proposition is true (3) Forallpositiverealnumbers a, bsuchthat a < bholds{b 1 n+0) } n N / O({a 1 n+0) } n N ). Thesequence{log 2 n} n N ofrealnumbersisdefinedasfollows: (Def.2) {log 2 n} n N (0) = 0 and for every n such that n > 0 holds {log 2 n} n N (n) =log 2 n. Let abearealnumber.thefunctor{n a } n N yieldingasequenceofreal numbers is defined as follows: (Def.3) {n a } n N (0) = 0andforevery nsuchthat n > 0holds{n a } n N (n) = n a. Letusmentionthat{log 2 n} n N iseventually-positive. Let abearealnumber.observethat{n a } n N iseventually-positive. We now state several propositions: (4) Let f, g be eventually-nonnegative sequences of real numbers. Then O(f) O(g)and O(f) O(g)ifandonlyif f O(g)and f / Ω(g). (5) O({log 2 n} n N ) O({n (1 2 ) } n N )and O({log 2 n} n N ) O({n (1 2 ) } n N ). (6) {n (1 2 ) } n N Ω({log 2 n} n N )and{log 2 n} n N / Ω({n (1 2 ) } n N ). (7) For every sequence f of real numbers and for every natural number k such that for every n holds f(n) = n κ=0 ({nk } n N )(κ) holds f Θ({n (k+1) } n N ). (8) Let fbeasequenceofrealnumbers.suppose f(0) = 0andforevery

3 asymptotic notation. part ii: examples nsuchthat n > 0holds f(n) = n log 2 n.thenthereexistsaneventuallypositivesequence sofrealnumberssuchthat s = fand sisnotsmooth. Let bbearealnumber.thefunctor{b} n N yieldsasequenceofrealnumbers and is defined as follows: (Def.4) {b} n N = N b. Letusnotethat{1} n N iseventually-nonnegative. (9) Let f be an eventually-nonnegative sequence of real numbers. Then there existsanonemptyset Foffunctionsfrom Nto Rsuchthat F ={{n 1 } n N } and f F O({1} n N) iffthereexist N, c, ksuchthat c > 0andforevery n suchthat n Nholds 1 f(n)and f(n) c {n k } n N (n). 2. Problem 3.1 (10) Foreverysequence fofrealnumberssuchthatforevery nholds f(n) = ( n) + 27 n 2 holds f O({n 2 } n N ). 3. Problem 3.5 We now state three propositions: (11) {n 2 } n N O({n 3 } n N ). (12) {n 2 } n N / Ω({n 3 } n N ). (13) There exists an eventually-positive sequence s of real numbers such that s ={2 1 n+1) } n N and{2 1 n+0) } n N Θ(s). Let abeanaturalnumber.thefunctor{(n + a)!} n N yieldingasequence ofrealnumbersisdefinedby: (Def.5) {(n + a)!} n N (n) = (n + a)!. Let abeanaturalnumber.observethat{(n+a)!} n N iseventually-positive. We now state the proposition (14) {(n + 0)!} n N / Θ({(n + 1)!} n N ).

4 146 richard krueger et al. 4. Problem 3.6 The following proposition is true (15) Foreverysequence fofrealnumberssuchthat f O({n 1 } n N )holds f f O({n 2 } n N ). 5. Problem 3.7 We now state the proposition (16) There exists an eventually-positive sequence s of real numbers such that s ={2 1 n+0) } n N and 2{n 1 } n N O({n 1 } n N )and{2 2 n+0) } n N / O(s). 6. Problem 3.8 (17) Iflog 2 3 < ,then{n(log 2 3) } n N O({n ( ) } n N )and{n (log 2 3) } n N / Ω({n ( ) } n N )and{n (log 2 3) } n N / Θ({n ( ) } n N ). 7. Problem 3.11 We now state the proposition (18) Let f, gbesequencesofrealnumbers.supposeforevery nholds f(n) = nmod 2andforevery nholds g(n) = (n + 1)mod 2.Thenthereexist eventually-nonnegativesequences s, s 1 ofrealnumberssuchthat s = f and s 1 = gand s / O(s 1 )and s 1 / O(s). 8. Problem 3.19 We now state two propositions: (19) For all eventually-nonnegative sequences f, g of real numbers holds O(f) = O(g)iff f Θ(g). (20) Foralleventually-nonnegativesequences f, gofrealnumbersholds f Θ(g)iff Θ(f) = Θ(g).

5 asymptotic notation. part ii: examples Problem 3.21 The following propositions are true: (21) Let ebearealnumberand fbeasequenceofrealnumbers.suppose 0 < eand f(0) = 0andforevery nsuchthat n > 0holds f(n) = n log 2 n. Then there exists an eventually-positive sequence s of real numbers such that s = fand O(s) O({n (1+e) } n N )and O(s) O({n (1+e) } n N ). (22) Let ebearealnumberand gbeasequenceofrealnumbers.suppose 0 < eand e < 1and g(0) = 0and g(1) = 0andforevery nsuch that n > 1holds g(n) = n2 log 2 n.thenthereexistsaneventually-positive sequence sofrealnumberssuchthat s = gand O({n (1+e) } n N ) O(s) and O({n (1+e) } n N ) O(s). (23) Let fbeasequenceofrealnumbers.suppose f(0) = 0and f(1) = 0 andforevery nsuchthat n > 1holds f(n) = n2 log 2 n.thenthereexists aneventually-positivesequence sofrealnumberssuchthat s = fand O(s) O({n 8 } n N )and O(s) O({n 8 } n N ). (24) Let gbeasequenceofrealnumbers.supposethatforevery nholds g(n) = ((n 2 n) + 1) 4.Thenthereexistsaneventually-positivesequence sofrealnumberssuchthat s = gand O({n 8 } n N ) = O(s). (25) Let ebearealnumber.suppose 0 < eand e < 1.Thenthereexists an eventually-positive sequence s of real numbers such that s = {1 + e 1 n+0) } n N and O({n 8 } n N ) O(s)and O({n 8 } n N ) O(s). 10. Problem 3.22 s: (26) Let f, gbesequencesofrealnumbers.suppose f(0) = 0andforevery n suchthat n > 0holds f(n) = n log 2 n and g(0) = 0andforevery nsuchthat n > 0holds g(n) = n n.thenthereexisteventually-positivesequences s, s 1 ofrealnumberssuchthat s = fand s 1 = gand O(s) O(s 1 )and O(s) O(s 1 ). (27) Let fbeasequenceofrealnumbers.suppose f(0) = 0andforevery n suchthat n > 0holds f(n) = n n.thenthereexisteventually-positive sequences s, s 1 ofrealnumberssuchthat s = fand s 1 ={2 1 n+0) } n N and O(s) O(s 1 )and O(s) O(s 1 ). (28) Thereexisteventually-positivesequences s, s 1 ofrealnumberssuchthat s ={2 1 n+0) } n N and s 1 ={2 1 n+1) } n N and O(s) = O(s 1 ).

6 148 richard krueger et al. (29) Thereexisteventually-positivesequences s, s 1 ofrealnumberssuchthat s ={2 1 n+0) } n N and s 1 ={2 2 n+0) } n N and O(s) O(s 1 )and O(s) O(s 1 ). (30) There exists an eventually-positive sequence s of real numbers such that s ={2 2 n+0) } n N and O(s) O({(n + 0)!} n N )and O(s) O({(n + 0)!} n N ). (31) O({(n + 0)!} n N ) O({(n + 1)!} n N )and O({(n + 0)!} n N ) O({(n + 1)!} n N ). (32) Let gbeasequenceofrealnumbers.suppose g(0) = 0andforevery n suchthat n > 0holds g(n) = n n.thenthereexistsaneventually-positive sequence sofrealnumberssuchthat s = gand O({(n + 1)!} n N ) O(s) and O({(n + 1)!} n N ) O(s). 11. Problem 3.23 (33) Letgiven n.suppose n 1.Let fbeasequenceofrealnumbersand k beanaturalnumber.ifforevery nholds f(n) = n κ=0 ({nk } n N )(κ),then f(n) nk+1 k Problem 3.24 (34) Let f, gbesequencesofrealnumbers.suppose g(0) = 0andforevery nsuchthat n > 0holds g(n) = n log 2 nandforevery nholds f(n) = log 2 (n!).thenthereexistsaneventually-nonnegativesequence sofreal numberssuchthat s = gand f Θ(s). 13. Problem 3.26 The following proposition is true (35) Let f be an eventually-nondecreasing eventually-nonnegative sequence ofrealnumbersand tbeasequenceofrealnumbers.supposethatfor every nholdsif nmod 2 = 0,then t(n) = 1andif nmod 2 = 1,then t(n) = n.then t / Θ(f).

7 asymptotic notation. part ii: examples Problem 3.28 Let fbeafunctionfrom Ninto R andlet nbeanaturalnumber.then f(n)isafinitesequenceofelementsof R. Let nbeanaturalnumberandlet a, bbepositiverealnumbers.thefunctor Prob28(n, a,b)yieldsarealnumberandisdefinedby: (Def.6)(i) Prob28(n,a, b) = 0if n = 0, (ii) thereexistsanaturalnumber landthereexistsafunction p 28 from N into R suchthat l+1 = nandprob28(n,a, b) = π n p 28 (l)and p 28 (0) = a andforeverynaturalnumber nthereexistsanaturalnumber n 1 suchthat n 1 = n and p 28 (n + 1) = p 28 (n) 4 π n1 p 28 (n) + b (n ), otherwise. Let a, bbepositiverealnumbers.thefunctor{prob28(n,a, b)} n N yieldsa sequenceofrealnumbersandisdefinedby: (Def.7) ({Prob28(n,a, b)} n N )(n) =Prob28(n,a, b). The following proposition is true (36) For all positive real numbers a, b holds {Prob28(n,a, b)} n N is eventually-nondecreasing. 15. Problem 3.30 Thenonemptysubset{2 n : n N}of Nisdefinedby: (Def.8) {2 n : n N} ={2 n : nrangesovernaturalnumbers}. Next we state three propositions: (37) Let fbeasequenceofrealnumbers.supposethatforevery nholdsif n {2 n : n N},then f(n) = nandif n / {2 n : n N},then f(n) = 2 n. Then f Θ({n 1 } n N {2 n : n N})and f / Θ({n 1 } n N )and{n 1 } n N is smooth and f is not eventually-nondecreasing. (38) Let f, gbesequencesofrealnumbers.suppose f(0) = 0andforevery nsuchthat n > 0holds f(n) = n 2 log 2 n and g(0) = 0andforevery n suchthat n > 0holds g(n) = n n.thenthereexistsaneventually-positive sequence s of real numbers such that (i) s = g, (ii) f Θ(s {2 n : n N}), (iii) f / Θ(s), (iv) f is eventually-nondecreasing, (v) s is eventually-nondecreasing, and (vi) sisnotsmoothw.r.t.2.

8 150 richard krueger et al. (39) Let gbeasequenceofrealnumbers.supposethatforevery nholdsif n {2 n : n N},then g(n) = nandif n / {2 n : n N},then g(n) = n 2. Then there exists an eventually-positive sequence s of real numbers such that s = gand{n 1 } n N Θ(s {2 n : n N})and{n 1 } n N / Θ(s)and s 2 O(s)and{n 1 } n N iseventually-nondecreasingand sisnoteventuallynondecreasing. 16. Problem 3.31 Let xbeanaturalnumber.thefunctor x yieldinganaturalnumberis defined as follows: (Def.9)(i) Thereexists nsuchthat n! xand x < (n+1)!and x = n!if x 0, (ii) x = 0, otherwise. Next we state the proposition (40) Let fbeasequenceofrealnumbers.supposethatforevery nholds f(n) = n.thenthereexistsaneventually-positivesequence sofreal numberssuchthat s = fand fiseventually-nondecreasingandforevery nholds f(n) {n 1 } n N (n)and sisnotsmooth. 17. Problem 3.34 Letusmentionthat{n 1 } n N {1} n N iseventually-positive. (41) Θ({n 1 } n N {1} n N ) + Θ({n 1 } n N ) = Θ({n 1 } n N ). 18. Problem 3.35 (42) Thereexistsanonemptyset F offunctionsfrom Nto Rsuchthat F ={{n 1 } n N }andforevery nholds{n ( 1) } n N (n) {n 1 } n N (n)and {n ( 1) } n N / F O({1} n N).

9 asymptotic notation. part ii: examples Addition The following proposition is true (43) Let cbeanonnegativerealnumberand x, fbeeventually-nonnegative sequencesofrealnumbers.given e, Nsuchthat e > 0andforevery n suchthat n Nholds f(n) e.if x O(c + f),then x O(f). 20. Potentatially Useful The following propositions are true: (44) 2 2 = 4. (45) 2 3 = 8. (46) 2 4 = 16. (47) 2 5 = 32. (48) 2 6 = 64. (49) 2 12 = (50) Forevery nsuchthat n 3holds n 2 > 2 n+1. (51) Forevery nsuchthat n 10holds 2 n 1 > (2 n) 2. (52) Forevery nsuchthat n 9holds (n + 1) 6 < 2 n 6. (53) Forevery nsuchthat n 30holds 2 n > n 6. (54) Foreveryrealnumber xsuchthat x > 9holds 2 x > (2 x) 2. (55) Thereexists Nsuchthatforevery nsuchthat n Nholds n log 2 n > 1. (56) Forallrealnumbers a, b, csuchthat a > 0and c > 0and c 1holds a b = c b log c a. (57) (4 + 1)! = 120. (58) 5 5 = (59) 4 4 = 256. (60) Forevery nholds (n 2 n) + 1 > 0. (61) Forevery nsuchthat n 2holds n! > 1. (62) Forall n 1, nsuchthat n n 1 holds n! n 1!. (63) Forevery ksuchthat k 1thereexists nsuchthat n! kand k < (n + 1)!andforevery msuchthat m! kand k < (m + 1)!holds m = n. (64) Forevery nsuchthat n 2holds n 2 < n. (65) Forevery nsuchthat n 3holds n! > n.

10 152 richard krueger et al. (66) Forallnaturalnumbers m, nsuchthat m > 0holds m n isanatural number. (67) Forevery nsuchthat n 2holds 2 n > n + 1. (68) Let abealogbaserealnumberand fbeasequenceofrealnumbers. Suppose a > 1and f(0) = 0andforevery nsuchthat n > 0holds f(n) =log a n.then fiseventually-positive. (69) Foralleventually-nonnegativesequences f, gofrealnumbersholds f O(g)and g O(f)iff O(f) = O(g). (70) Forallrealnumbers a, b, csuchthat 0 < aand a band c 0holds a c b c. (71) Forevery nsuchthat n 4holds 2 n+3 < 2 n. (72) Forevery nsuchthat n 6holds (n + 1) 2 < 2 n. (73) Foreveryrealnumber csuchthat c > 6holds c 2 < 2 c. (74) Let ebeapositiverealnumberand fbeasequenceofrealnumbers. Suppose f(0) = 0andforevery nsuchthat n > 0holds f(n) =log 2 (n e ). Then f/{n e } n N isconvergentandlim(f/{n e } n N ) = 0. (75) Foreveryrealnumber esuchthat e > 0holds{log 2 n} n N /{n e } n N is convergentandlim({log 2 n} n N /{n e } n N ) = 0. (76) Foreverysequence fofrealnumbersandforevery Nsuchthatforevery nsuchthat n Nholds f(n) 0holds N κ=0 f(κ) 0. (77) Forallsequences f, gofrealnumbersandforevery Nsuchthatforevery nsuchthat n Nholds f(n) g(n)holds N κ=0 f(κ) N κ=0 g(κ). (78) Let fbeasequenceofrealnumbersand bbearealnumber.suppose f(0) = 0andforevery nsuchthat n > 0holds f(n) = b.let Nbea naturalnumber.then N κ=0 f(κ) = b N. (79) Forallsequences f, gofrealnumbersandforallnaturalnumbers N, Mholds M κ=n+1 f(κ) + f(n + 1) = M κ=n+1+1 f(κ). (80) Let f, gbesequencesofrealnumbers, Mbeanaturalnumber,and given N.Suppose N M + 1.Ifforevery nsuchthat M + 1 nand n Nholds f(n) g(n),then M κ=n+1 f(κ) M κ=n+1 g(κ). (81) Forevery nholds n 2 n. (82) Let fbeasequenceofrealnumbers, bbearealnumber,and Nbea naturalnumber.suppose f(0) = 0andforevery nsuchthat n > 0holds f(n) = b.let Mbeanaturalnumber.Then M κ=n+1 f(κ) = b (N M). (83) Let f, gbesequencesofrealnumbers, Nbeanaturalnumber,and c bearealnumber.suppose fisconvergentandlimf = candforevery n suchthat n Nholds f(n) = g(n).then gisconvergentandlimg = c. (84) Forevery nsuchthat n 1holds (n 2 n) + 1 n 2. (85) Forevery nsuchthat n 1holds n 2 2 ((n 2 n) + 1).

11 asymptotic notation. part ii: examples (86) Foreveryrealnumber esuchthat 0 < eand e < 1thereexists Nsuch thatforevery nsuchthat n Nholds n log 2 (1 + e) 8 log 2 n > 8 log 2 n. (87) Forevery nsuchthat n 10holds 22 n n! < 1. 2 n 9 (88) Forevery nsuchthat n 3holds 2 (n 2) n 1. (89) Foreveryrealnumber csuchthat c 0holds c 1 2 = c. (90) Thereexists Nsuchthatforevery nsuchthat n Nholds n n log 2 n > n 2. (91) Foreverysequence sofrealnumberssuchthatforevery nholds s(n) = (1 + 1 n+1 )n+1 holds sisnon-decreasing. (92) Forevery nsuchthat n 1holds ( n+1 n )n ( n+2 n+1 )n+1. ) ( n+1 (93) Forall k, nsuchthat k nholds ( n k n+1. (94) Foreverysequence fofrealnumberssuchthatforevery nholds f(n) = log 2 (n!)andforevery nholds f(n) = n κ=0 ({log 2 n} n N )(κ). (95) Forevery nsuchthat n 4holds n log 2 n 2 n. (96) Let a, b be positive real numbers. Then Prob28(0,a, b) = 0 and Prob28(1,a, b) = aandforevery nsuchthat n 2thereexists n 1 such that n 1 = n 2 andprob28(n,a, b) = 4 Prob28(n 1,a, b) + b n. k ) (97) Forevery nsuchthat n 2holds n 2 > n + 1. (98) Forevery nsuchthat n 1holds 2 n+1 2 n > 1. (99) Forevery nsuchthat n 2holds 2 n 1 / {2 n : n N}. (100) Forall n, ksuchthat k 1and n! kand k < (n + 1)!holds k = n!. (101) Forallrealnumbers a, b, csuchthat a > 1and b aand c 1holds log a c log b c. References [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41 46, [2] Gilles Brassard and Paul Bratley. Fundamentals of Algorithmics. Prentice Hall, [3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55 65, [4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1): , [5] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47 53, [6] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35 40, [7] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2): , [8] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3): , [9] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2): , [10] Elżbieta Kraszewska and Jan Popiołek. Series in Banach and Hilbert Spaces. Formalized Mathematics, 2(5): , 1991.

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