(2) (a) Refer to the course notes on Algorithm Stability and note the definition for En· Using an integration by parts approach we were able to show that, En = 1 nEn-1. Once we were able to figure out a value for E₁, the above recurrence formula suggested a straightforward way to obtain whatever En values we wanted, for n > 1. However, the numerical results showed that something went terribly wrong with the computation. Explain in your own words why the computation goes terribly wrong. (b) In the course notes, we saw that a simple rewriting of the recurrence formula, and a fairly rough approximation for En, for a larger value of n, leads to an algorithm which provides very accurate results for smaller values of n. Explain in your own words why this approach work so well. (c) Implement the two algorithms in Fortran to reproduce the numerical results shown in the notes. Verify that you get the same results. (Start by editing the Stirling Approx- imation program for n! from Assignment #1.)

(2) (a) Refer to the course notes on Algorithm Stability and note the definition for En· Using an integration by parts approach we were able to show that, En = 1 nEn-1. Once we were able to figure out a value for E₁, the above recurrence formula suggested a straightforward way to obtain whatever En values we wanted, for n > 1. However, the numerical results showed that something went terribly wrong with the computation. Explain in your own words why the computation goes terribly wrong. (b) In the course notes, we saw that a simple rewriting of the recurrence formula, and a fairly rough approximation for En, for a larger value of n, leads to an algorithm which provides very accurate results for smaller values of n. Explain in your own words why this approach work so well. (c) Implement the two algorithms in Fortran to reproduce the numerical results shown in the notes. Verify that you get the same results. (Start by editing the Stirling Approx- imation program for n! from Assignment #1.)

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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Solve the following recurrence equations by expanding the formulas (also called the 'iteration method' on slides). Specifically, you should get T(n) = O(f(n)) for a function f(n). You may assume that T(n) = O(1) for n = O(1). You should not use the Master Theorem. (a) T(n) = 2T (n/3) + 1. (b) T(n) = 7T(n/7) + n. (c) T(n) = T(n − 1) + 2.
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4. Practice with the iteration method. We have already had a recurrence relation ofan algorithm, which is T(n) = 4T(n/2) + n log n. We know T(1) ≤ c.(a) Solve this recurrence relation, i.e., express it as T(n) = O(f(n)), by using the iteration method.Answer:(b) Prove, by using mathematical induction, that the iteration rule you have observed in 4(a) is correct and you have solved the recurrence relation correctly. [Hint: You can write out the general form of T(n) at the iteration step t, and prove that this form is correct for any iteration step t by using mathematical induction.Then by finding out the eventual number of t and substituting it into your generalform of T(n), you get the O(·) notation of T(n).]
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Solve the following Problem in Octave or Matlab Take the function fox) = x² = log (x² + 27 of which we want to compute its beras. O By plotting the graph of f, individuates the two real roots of f(x) 20 and the corresponding separation Intervals Inseln₂ 1 Find two Convergent iterative methods, with iteration functions 9₁ (x)/x i = 1.2. For each one determine The number of necessary iterations. Consider Kmax = 50, iterations and for the test on the as max number 2 = 1.0 e-b U of relative error tol Compare the results with the ones obtained with fzero. 0 +