CSE 20 Discrete Math. Perturbation method. Winter, February 23, Day 14

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1 CSE 0 Discrete Math Witer, 00 February 3, Day 14 Summatios (portios from Cocrete Mathematics by Graham, Kuth, Patashik) Iductio Give: S = 0 k a k, d last term: Perturbatio method Rewrite S+1 by splittig off first ad last term: S + a +1 = a k +1 a k = a k+1 +1 a k+1 = a k a k+1 The, work o last sum ad express i terms of S. Fially, solve for S. Istructor: Neil Rhodes S = x k 0 k Perturbatio method example Stadard closed forms Stadard Closed Forms Arithmetic series Arithmetic series Stadard Closed Forms k = 1 ( + 1) Arithmetic series Sum of squares ad cubes k = 1 ( + 1) Sums of squares ad cubes k ( + 1)( + 1) Sum of squares ad = cubes 3 ( k ( + 1)( + 1) = 4 k 3 = ( + 1) 4 Summatios (portios from Cocrete Mathematics by Graham, Kuth, Patashik) p.18 Summatios (portios from Cocrete Mathematics by Graham, Kuth, Patashik) p

2 Stadard closed forms Stadard closed form Geometric series (x 1) Harmoic series x k = x+1 1 x 1 Ifiite Geometric series ( x < 1) H = 1 k l H " x k = 1 1 x 5 Applicatio of harmoic series Give a stack of 5 playig cards (of legth uits). Ca you stack them overlappig so the top card completely overhags the table? Iductio Method of provig a statemet P() is true for a, a+1, " Base case (show P(a)) Prove true for = a (ofte, a = 1 or 0) Iductive step Show (>a) P(-1)P() - Assume true for -1 - Show true for Or, show (>a) P(0) P(1) P(-1) P() - Assume true for Show true for How do we kow that P(b) (b# a) is true? P(a) P(a) P(a+1) P(a+1) P(a+1)P(a+) P(a+) P(b-1)P(b) P(b) 7 8

3 Prove that = (+1)/ (Strog Iductio) Ay iteger > 1 is divisible by some prime Base case Base case: = 0 0 = 0(0+1)/ = 0 Iductive step: Suppose that > 0 ad = (-1)()/ (iductive hypothesis) Show that = (+1)/ Iductive step Iductive hypothesis: To show: = ( ) + ( 1) = + ( 1) + = ( 1 + ) = ( + 1) = 9 10 Sometimes, you eed to prove more tha oe base case Defie sequece a1, a, a3, : a1 = a = for k # 3:, ak = ak-1 + ak- + 8 Prove that a is eve for # 1 Base case: Sometimes Easier to Prove Stroger Result I tryig to devise a proof by mathematical iductio, you may fail for two opposite reasos. You may fail because you try to prove too much: Your P() is too heavy a burde. Yet, you may also fail because you try to prove too little: Your P() is too weak a support. I geeral, you have to balace the statemet of your theorem so that the support is just eough for the burde. Polya Iductive step Iductive hypothesis To show: 11 1

4 of Stregtheig the Statemet Give: K0 = 1 K +1 = 1 + mi(k,3k 3 ) Show K # Base case: Prove that (-1) = Iductive step: Iductive hypothesis To show: As each of a group of busiess people arrives at a meetig, each shakes hads with all the other people preset. Use iductio to show that if people come to the meetig, the (-1)/ hadshakes occur. I a roud-robi touramet, show that the teams ca be labeled T1, T,, T such that Ti beats Ti+1 for all 1 $ i $

5 Format: //Loop precoditio: Loop // Loop ivariat: exit whe ExitCoditio Loop Cotets Ed // Loop postcoditio: To Prove: loop precoditio Loop ivariat Loop Ivariat Loop Ivariat (at oe iteratio) Loop cotets ExitCoditio Loop Ivariat (at ext iteratio) ExitCoditio is evetually true Loop ivariat ExitCoditio Loop postcoditio Loop Ivariat double total = 0; it i = 0; // loop precoditio: i = 0, m: o-egative iteger: r is real, total=0 loop // loop ivariat: total = i * r exit whe i = m total = total + r; i = i + 1; ed // loop postcoditio: total = m * r Prove that (-1) = Algorithm correctess Iductio ad Algorithms Precoditio: predicates describig the iitial state (ivolvig the iput variables) Postcoditio: predicates describig the fial state (ivolvig the iput ad output variables) : Product(it m, double r) // precoditio: m a o-egative iteger, r: a real // postcoditio: returs m * r Sort(it[] y) // precoditio: y is a array of itegers // postcoditio: y is a permutatio of the origial array such that y[i] $y[i+1] for 0 $ i < y.legth 19 0

6 Format: //Loop precoditio: Loop // Loop ivariat: exit whe ExitCoditio Loop Cotets Ed // Loop postcoditio: To Prove: loop precoditio Loop ivariat Loop Ivariat Loop Ivariat (at oe iteratio) Loop cotets ExitCoditio Loop Ivariat (at ext iteratio) ExitCoditio is evetually true Loop ivariat ExitCoditio Loop postcoditio Loop Ivariat double total = 0; it i = 0; // loop precoditio: i = 0, m: o-egative iteger: r is real, total=0 loop // loop ivariat: total = i * r exit whe i = m total = total + r; i = i + 1; ed // loop postcoditio: total = m * r 1