f2006lecNotesWeek05

f2006lecNotesWeek05 - 14 14.1 Recurrence Relations...

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14 Recurrence Relations 14.1 Recursion general idea: something is defned in terms oF a simpler version oF itselF the simplest case(s) are defned explicitly they have already seen recursively defned sequences e.g. t 1 =3 ,t n =2 t n - 1 iF n> 1 ±ibonacci sequence: t 1 =1 2 n = t n - 1 + t n - 2 iF 2 recursion is an extremely important idea in computer science they will be seeing lots oF it in APS105 many diﬀerent applications an example: a recursive algorithm For counting the number oF digits in a positive integer 14.2 Recurrence relations in this course, we will Focus on one Form oF recursion — the recurrence relation defnition oF a recurrence relation an equation that expresses the n th term oF a sequence, t n , in terms oF one or more oF the preceding terms the initial conditions oF a recurrence relation give specifc values oF one or more oF t 1 2 ,... relate defnition to general Form oF recursion to solve a recurrence relation is to fnd an explicit (non-recursive) expression For t n why might we want to have an explicit expression? explain generally, problem oF fnding a solution to a recurrence relation is not easy sometimes not possible 14.3 Solving recurrence relations using patterns fnd frst Few terms look For a pattern try to prove correctness using mathematical induction an example u n = u n - 1 +3 ,n> 0 u 0 14.4 Classifying recurrence relations it turns out that classiFying recurrence relations is useFul because solution methods are known For some categories ( cf. integration) frst order f n f n - 1 second order g n = g n - 1 + g n - 2 third order h n = h n - 3 +5 31

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linear c n = c n - 1 +2 non-linear d n = d 2 n - 1 homogeneous s n = s 2 n - 1 + s n - 2 non-homogeneous t n = t n - 1 +3
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f2006lecNotesWeek05 - 14 14.1 Recurrence Relations...

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