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Exercises generate, create qr none on .net projects Visual Studio .NET Introduction 1. Using dynamic programming is one way to speed up the computation of the Fibonacci numbers, but another is to use a different algorithm. A much more efficient algorithm than F can be designed, based on the following identities. F2 n F2 n.

2 Fn 2 Fn 1 2 Fn ,. 2 Fn , for n for n Program F using these identities. 2. You can still speed up the code for generating Fibonacci numbers by using dynamic programming. Do so, and construct tables, like those in this section, giving the 7 Recursion number of additions performed for various n by the two programs you have just written. 3. Calculation of the Collatz numbers, as described in Exercise 5 from Section 5.3, can be implemented usi ng recursion and sped up by using dynamic programming. Using recursion and dynamic programming, create the function collatz[n,i], which computes the ith iterate of the Collatz sequence starting with integer n. Compare its speed with that of your original solution.

. 7.7 Higher-order functions and recursion As a final wrap-up on recursion, we note that many of the built-in functions discussed in 4 could be written as user-defined functions using recursion. Although they may not be as efficient as the built-in functions, creating them will give you good practice with recursion and should also give you some insight into how these functions operate. Our first example of programming some built-in functions in a recursive style is Map.

We will call our version map. map[f ,lis] applies f to each element of the list lis. This is a simple recursion on the tail of lis: if we assume that map[f, Rest[lis]] works, map[f ,lis] is easily obtained from it by joining f[First[lis]] to the beginning.

. In[1]:=. map f_, : map f_, x_, y___. : Join , map f, y We can quickly check that our map does what it was intended to. In[3]:= Out[3]=. map f, 1, 2, 3 f 1 ,f 2 ,f 3 Like many of the f unctions in 4, this function has a function as an argument. This is the first time we have seen user-defined higher-order functions. We will give one more example of a built-in function that can be defined using recursion, and leave the rest as exercises.

Nest[f ,x,n] applies f to x, n times. The recursion is, obviously, on n..

In[4]:=. nest f_, x_, 0 : x nest f_, x_, n_ : f nest f, x, n Here is an example of the use of this function. In[6]:= Out[6]=. nest Sin, , 4 Sin Sin Sin Sin An Introduction to Programming with Mathematica Before leaving thi .NET qrcode s topic, we note that, beyond a basic exercise in recursion, it is sometimes quite useful to write your own higher-order functions. Given a function f whose argument must be an integer in the range 1, , 1000, and whose result is also in that range, answer the following question: on average, for a number n1 , how many times can f be applied before it repeats itself That is, on average, if we form the sequence n1 , n2 f n1 , n3 f n2 , , what is the smallest i such that ni nj for some j i Assume f is so expensive to compute that we prefer to approximate this average by just checking ten randomly chosen numbers.

This technique, known as random sampling, is used in many areas where statistical analysis of data is required. If we had a function repeatCount[n] to answer this question for a particular n, then we might answer the question in this way:. Sum repeatCount Ra ndom Integer, 1,1000 10 , 10. So how do we write repeatCount We will define our own higher-order function. In[7]:= In[8]:=. repeat f_, lis_, p Denso QR Bar Code for .NET red_ : lis ; pred Drop lis, 1 , Last lis repeat f_, lis_, pred_ : repeat f, Append lis, f Last lis. , pred repeat takes an ar .net vs 2010 qr bidimensional barcode gument list lis, and repeatedly applies f to its last element, and adds that new value to the end, until the predicate pred returns True. repeatCount becomes:.

In[9]:= In[10]:= I n[11]:= Out[11]=. repeatCount f_, n_ .net vs 2010 QR Code ISO/IEC18004 : repeat f, n , MemberQ plus4mod20 x_ : Mod x 4, 20. repeatCount plus4m od20, 0 0, 4, 8, 12, 16, 0. Exercises 1. Write recursive definitions for Fold, FoldList, and NestList. 2.

Recall the notion of a random walk on a two-dimensional lattice from 3.. Use repeat to defi QR for .NET ne a special kind of random walk, one which continues until it steps on to a location it had previously visited. That is, define landMineWalk as a function of no arguments which produces the list of the locations visited in such a random walk, starting from location (0, 0).

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