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- a. Correctness of dynamic programming algorithm: Usually, a dynamic programming algorithm can be seen as a recursion and proof by induction is one of the easiest way to show its correctness. The structure of a proof by strong induction for one variable, say n, contains three parts. First, we define the Proposition P(n) that we want to prove for the variable n. Next, we show that the proposition holds for Base case(s), such as n = 0, 1, . . . etc. Finally, in the Inductive step, we assume that P(n) holds for any value of n strictly smaller than n' , then we prove that P(n') also holds. Use the proof by strong induction properly to show that the algorithm of the Knapsack problem above is correct. b. Bounded Knapsack Problem: Let us consider a similar problem, in which each item i has ci > 0 copies (ci is an integer). Thus, xi is no longer a binary value, but a non-negative integer at most equal to ci , 0 ≤ xi ≤ ci . Modify the dynamic programming algorithm seen at class for this…Problem 1. Prove that the following functions are Primitive Recursive. I – 1 if x > 0, (1) mPred(x) = for x € N. if x = 0. if x > 0, (2) sgn(x) = for x E N. 0 if x = 0. if x > 0, (3) sgn(x) = for x E N. 1 if x = 0.a)Write a computer program for the Gauss elimination method using the C programming language. Decide the number of significant figures yourselves. While writing your program, consider the effects of the number of significant figures, pivoting, scaling, and do not forget to check if the system is ill-conditioned.b)Repeat the same procedures for the Gauss-Jordan method.c)Solve an example system using your Gauss elimination and Gauss-Jordan method. Measure the time your computer solves the system for both programs.d)Write a report in which you discuss and compare your Gauss elimination and Gauss-Jordan programs. NOTE: BY USING C PROGRAMMING PLEASE SOLVE THIS QUESTION STEP BY STEP TO UNDERSTAND WELL.THE SUBJECT (Numerical Methods for Engineers) THANK YOU FOR HELPING AND HAVE A NICE DAY. I WILL RATE THE COMPLETE ANSWER
- 8- Determine if each of the following recursive definition is a valid recursive definition of a function f from a set of non-negative integers. If f is well defined, find a formula for f(n) where n is non- negative and prove that your formula is valid. a. f(0) = 2,f(1) = 3, f(n) = f(n-1)-1 for n ≥ 2 b. f(0) = 1,f(1) = 2, f(n) = 2f (n-2) for n = 2In programming, a recursive function calls itself. The classical example is factorial(n), which can be defined recursively as n*factorial(n-1). Nonethessless, it is important to take note that a recursive function should have a terminating condition (or base case), in the case of factorial, factorial(0)=1. Hence, the full definition is: factorial(n) = 1, for n = 0 factorial(n) = n * factorial(n-1), for all n > 1 For example, suppose n = 5: // Recursive call factorial(5) = 5 * factorial(4) factorial(4) = 4 * factorial(3) factorial(3) = 3 * factorial(2) factorial(2) = 2 * factorial(1) factorial(1) = 1 * factorial(0) factorial(0) = 1 // Base case // Unwinding factorial(1) = 1 * 1 = 1 factorial(2) = 2 * 1 = 2 factorial(3) = 3 * 2 = 6 factorial(4) = 4 * 6 = 24 factorial(5) = 5 * 24 = 120 (DONE) Exercise (Factorial) (Recursive): Write a recursive method called factorial() to compute the factorial of the given integer. public static int factorial(int n) The recursive algorithm is:…Design a recursive version of dynamic programming algorithm (Top-down) to construct the actual solution of the matrix chain multiplication problem (i.e., the parentheses order). For this problem, please write down the recursive function in pseudocode and Write down the dynamic table and matrix output on the following examples: (a) Three matrices (A, B, and C) with dimensions 10 × 50 × 5 × 100, respectively. (b) Four matrices (A, B, C, and D) with dimensions 20 × 5 × 10 × 30 × 10, respectively.
- Consider the following problems for recursive definition/solution. Answer the following questions. [Remember that a recursive definition/solution requires base case and recursive case] Consider a set of odd natural numbers. Make a recursive definition for the set.Given three sequences of length m, n, and p each, you are to design and analyze an algorithm to find the longest common subsequence (LCSS) for the three sequence. Is it possible to use dynamic programming to find the LCSS between the three sequences? If yes, provide recursive solution to the above problem.1. Compute the recursive function f(n) = 3f(n–3)– l; ƒ(0) = 1; for n=12;. Show %3D %3D %3D clearly how the function is evaluated and the return of the function for every recursive step as seen in class. Show all your work to get credit
- es remaining 8. Consider the function f:NxN-N defined recursively by: 1) Base case: Let meN and define (0,m) = 0 2) Recursive case: For any x,meN, x>0, define f(x,m) = (x-1,m) + (m+m) Prove the following theorem holds using proof by induction: Thereom: For any n,meN, m>0 we have (n.m) I m = n+n Fill in your answer here 9 Help BIU X, x L - ɔE =N E X FormatDesign , implement and test a recursive function that derives: How many sequences of 5 separate symbols (0,1,2,3,4,5) have an even number of 0's. (note:In Python)4. Write a recursive function code that finds the elements of the given sequence: a, 3a-1 + n², and a 1. What is the time complexity of your code? Explain it.