Consider the Lucas numbers:

$L_n=L_{n-1}+L_{n-2},n>1$;$L_1=1$;$L_0=2$

a$)$ Write a recursive function to compute $L_n$ using the above definition directly. Implement your solution and print $L_{i**5},$ where $0i10,$ as output.

b$)$ Write a recursive function$/$procedure to compute $L_n$ with time complexity $O(n)($more precisely, the time complexity should be $O(nA(n))$ when $n$ is large, where $A(n)$ is the complexity of adding $L_{n-1}$ and $L_{n-2}).$ Implement your solution and print $L_{i**20},$ where $0i25,$ as output. This program must be able to compute $L_n$ precisely for $n500.$ Hint $1$:

Let $K_n=([L_n],[L_{n-1}])$:$K_n=([1,1],[1,0])K_{n-1},n1$;$K_0=([2],[-1])$

With this formulation, design a recursive algorithm for $K_n$ such that the algorithm will return both $L_n$ and $L_{n-1}$ with input parameter $n.$

Hint $2$: Can you use a primitive type to store $L_{500}?$

For programs in $7a)$ and $7b)$ of this question, you are NOT allowed to use Python. For C$++$ and Java, you can only use primitive types such as char, int, long and long long. You are not allowed to use large integer, such as BigInteger in Java, from the language library. You have to write your own large integer class or data type, if needed.

c$)$ Use the Unix time facility $($bash time command$)$ to track the time needed for each algorithm. Compare the results and state your conclusion in two or three sentences.

d$)$ Can you use your program in $7$ a$)$ to compute $L_{50}$ if int type of $4$ bytes is used? Briefly explain your answer. Explain why your program in $7$ b$)$ can computes $L_{500}$ precisely?

Algorithms and answers for $7$ b$)$ to $7$ d$)$ should be in asn$1$ solutions.pdf

For this question, for $C++,$ when compiling option $O2$ could be considered and a makefile should be written such that the command "make clean" will remove all the $"*.$o$"$ files, the command "make asn$1\_$a$"$ will generate an executable file "asn$1\_$a$"$ for $7$ a$)$ that can be run by typing "asn$1\_$a$"$; and the command "make asn$1.$b$"$ will generate an executable file "asn$1\_$b$"$ for $7$ b$)$ that can be run by typing "asn$1\_$b$".$ If you are using Java, you may not need the makefile. In that case, you should have shell script files, "asn$1\_$a$"$ and "asn$1\_$b$"$ such that by typing, "asn$1\_$a$"$ and "asn$1\_$b$",$ your java programs will run.