V. Adamchik 1. Recursions. Victor Adamchik Fall of x n1. x n 2. Here are a few first values of the above sequence (coded in Mathematica)

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1 V. Adamchik Recursios Victor Adamchik Fall of 2005 Pla. Covergece of sequeces 2. Fractals 3. Coutig biary trees Covergece of Sequeces I the previous lecture we cosidered a cotiued fractio for 2 : 2 This fractio ca be writte i a recursive form 2 x x 2 x 0 0 Here are a few first values of the above sequece (coded i Mathematica) x0 0; x_ : x2 ; Tablex,, 0, 0 N 0.,., , , , , 0.444, , 0.442, , This umeric experimet suggests that x x 0 Therefore, we say that a give sequece coverges if the limit exists: 2...

2 V. Adamchik 2-27: Cocepts of Mathematics lim x a The value to which a sequece coverges is called a fixed poit. For the sequece x a fixed poit is 2 lim x 2 We could prove this by the followig argumet. Let the sequece coverges to umber z, amely x z. Clearly, that x z. We fid the value of z from the sequece defiitio z z2 Solvig the equatio, we get two roots z 2, z 2 2. The positive root is the limit. Towers of Haoi Cosider the Towers of Haoi recursio Here are the first few values of the sequece x 2x x Clearx; x ; x_ : 2x Tablex,,, 7 N., 3., 7., 5., 3., 63., 27. We say that Therefore, the sequece diverges. More o the Golde Ratio lim x Cosider the followig recursio x x x 0 0

3 V. Adamchik 3 Does it coverge? What does it coverge to? Let us assume that x z, the x z ad Solvig this, yeilds z z lim x z 2 z0 where is the golde ratio. This immediately leads to a cotiued fractio for Recall the Euclidea algorithm... a bq r, 0 r b b r q 2 r 2, 0 r 2 r r r 2 q 3 r 3, 0 r 3 r r k2 r k q k r k, 0 r k r k r k r k q k 0 The cotiued fractio above implies that all quotiets i the Euclidea algorithm applied to GCD, are oes. At the same time, we kow that such property holds for GCDF, F. Therefore, we ca coject a relatio betwee F ad lim F F Exercise. Give a sequece x x x 0 0 that represets a ested radical... What does it coverge to? Exercise. Fid the fixed poit of the folllowig sequece x 2 x 4 x

4 V. Adamchik 2-27: Cocepts of Mathematics Cosider a geeral form recursio x 0 x fx If x coverges to a umber z, tha z is a fixed poit z fz Solvig this equatio, we fid z. The fuctioal aalysis questio: for what classes of fuctio f the sequece x coverges? Fractals Fractals are geometric objects that are self-similar, i.e. composed of ifiitely may pieces, all of which look the same. Some fractals are mudae

5 V. Adamchik 5 But some fractals are extremely complicated Sice producig fractals requires repeatig a certai step over a over agai o smaller ad smaller scales, it ca be easily draw by a computer. Madelbrot Set The Madelbrot set is a set of complex umbers z for which the followig recurrece coverges Let z, we get a a 2 z a 0 z

6 V. Adamchik 2-27: Cocepts of Mathematics Here are the first few value of the sequece a a 2 a 0 Clearx; x0 ; x_ : x 2 Tablex,,, 5 N 2., 5., 26., 677., The sequece does ot coverge. However, if z 5 the a 5 a a a the sequece does coverge to Here is a picture of the umber of iteratios that takes util a fixed poit has bee reached. Differet shadows of gray correspodet to a differet umber of iteratios.

7 V. Adamchik 7 Julia Set Julia sets are defied by iteratig a fuctio startig at the arbitarary poit i the plae. If after some umber of iteratios the result does ot drift to ifiity, but istead teds to a fixed poit, the that startig poit belogs to the Julia set. Here is a picture of the Julis set for x Cojugatex i Coutig Biary Trees A biary tree is made of odes, where each ode cotais a "left" referece, a "right" referece, ad a data elemet.the left ad right refereces recursively poit to smaller "subtrees" o either side. The topmost ode i the tree is called the "root". A recursive defiitio: a biary tree is either empty or cosists of a root, a left subtree ad a right subtree. I this sectio we will cout the umber of biary trees with odes. Coisder a few particluar cases. If, there is oly oe biary tree. If 2, there two trees

8 V. Adamchik 2-27: Cocepts of Mathematics If 3, there five trees I geeral case we will derive a recursive formula for the umber of trees based o the recursive defiitio of a biary tree. Let T deote the umber of biary trees with odes. Suppose the left subtree (LT) has k odes, the the right oe (RT) has k odes. Thus altogether we ca create TkTk biary trees with k odes i the left subtree. Sice the left subtree ca have ay umber of odes i the iterval 0k, we have to sum up over all such k

9 V. Adamchik 9 T TkTk k0 The solutio to this recurrece is kow as the Catala umbers after the Belgia Eugee Charles Catala: T 2 where 2 stads for biomial coefficiets. Exercise. Give a recurrece relatio for the umber of ways to climb stairs if the climber ca take either oe or two stairs at a time.