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6) There are other recursive functions such as the Lucas Sequence, how can you apply what you have learned to it? a) Iteratively computation b) Recursive computation

6) There are other recursive functions such as the Lucas Sequence, how can you apply what you have learned to it? a) Iteratively computation b) Recursive computation

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Question #6 parts A and B. Please help me. 

Study the Fibonacci number sequence in the following forms: (1) iterative (sequential), (2) recursive, (3) and closed form solution. 1) Examine the theoretical measure of time complexity of each. a) Using theory compare the number of operations and time taken to compute Fibonacci numbers recursively versus that needed to compute them iteratively. b) How many prime Fibonacci numbers are there, and how many can you find? c) Find the smallest Fibonacci number > 1,000,000 d) Find the smallest Fibonacci number > 1,000,000,000 2) Implement each Fibonacci form (1, 2, and 3 see above) in the C++ language (C++ 17). Write four functions: a) An iterative function that accepts a non-negative integer n and returns the nth Fibonacci number. b) A recursive function that inputs a non-negative integer n and returns the nth Fibonacci number. c) An improved recursive function that employs dynamic programming to improve the efficiency. d) The closed form of the Fibonacci sequence (be sure to reference the golden ratio) 3) Using your software, compute a set of Fibonacci numbers employing the four techniques (c.f. 2, above); for example, use n: {10, 15, 40, 41, 42, 50, 60, 100, 500, 600}. a) Measure the time for each computation-send the results to a file. * Recommend using the Chrono library from C++ b) Make certain you have "sufficient" data for convincing results. 4) Import your data (from 3) to excel for easy graphing 5) Compare the theoretical curves of Big-O with the empirical results. a) Try running on machines with [significantly] different speeds. b) Compare methods to answer the find an answer to question 1.c, above. 6) There are other recursive functions such as the Lucas Sequence, how can you apply what you have learned to it? a) Iteratively computation b) Recursive computation c) Recursive computation with dynamic programming.
Transcribed Image Text:Study the Fibonacci number sequence in the following forms: (1) iterative (sequential), (2) recursive, (3) and closed form solution. 1) Examine the theoretical measure of time complexity of each. a) Using theory compare the number of operations and time taken to compute Fibonacci numbers recursively versus that needed to compute them iteratively. b) How many prime Fibonacci numbers are there, and how many can you find? c) Find the smallest Fibonacci number > 1,000,000 d) Find the smallest Fibonacci number > 1,000,000,000 2) Implement each Fibonacci form (1, 2, and 3 see above) in the C++ language (C++ 17). Write four functions: a) An iterative function that accepts a non-negative integer n and returns the nth Fibonacci number. b) A recursive function that inputs a non-negative integer n and returns the nth Fibonacci number. c) An improved recursive function that employs dynamic programming to improve the efficiency. d) The closed form of the Fibonacci sequence (be sure to reference the golden ratio) 3) Using your software, compute a set of Fibonacci numbers employing the four techniques (c.f. 2, above); for example, use n: {10, 15, 40, 41, 42, 50, 60, 100, 500, 600}. a) Measure the time for each computation-send the results to a file. * Recommend using the Chrono library from C++ b) Make certain you have "sufficient" data for convincing results. 4) Import your data (from 3) to excel for easy graphing 5) Compare the theoretical curves of Big-O with the empirical results. a) Try running on machines with [significantly] different speeds. b) Compare methods to answer the find an answer to question 1.c, above. 6) There are other recursive functions such as the Lucas Sequence, how can you apply what you have learned to it? a) Iteratively computation b) Recursive computation c) Recursive computation with dynamic programming.
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