**Algorithm Analysis: Counting Multiplications****Problem Statement:**Determine the exact number of multiplications performed in the following segment of an algorithm. Assume \( a_1, a_2, \ldots, a_n \) are positive real numbers, with \( n = 11 \).**Algorithm Segment:**```for i = 1 to n m_i = a_i; for j = i + 1 to n m_i = m_i a_j;```**Explanation:**1. **Outer Loop:** The variable \( i \) iterates from 1 to \( n \).2. **Initialization:** For each \( i \), the variable \( m_i \) is initialized as \( a_i \).3. **Inner Loop:** The variable \( j \) iterates from \( i + 1 \) to \( n \). - In each iteration of the inner loop, \( m_i \) is updated as \( m_i \times a_j \).**Total Multiplications:**To find the total number of multiplications:- For each iteration of \( i \), the inner loop runs \( (n - i) \) times (since \( j \) starts at \( i + 1 \)).- Therefore, the number of multiplications for each \( i \) is \( n - i \).Summing the multiplications over all iterations of the outer loop:\[\sum_{i=1}^{n} (n - i) = (n - 1) + (n - 2) + \ldots + 1 + 0\]This is the sum of the first \((n-1)\) natural numbers, which can be calculated using the formula:\[\frac{(n-1) \times n}{2}\]Substitute \( n = 11 \):\[\frac{10 \times 11}{2} = 55\]Thus, the exact number of multiplications performed is 55.

Give the exact number of multiplications performed in the following segment of an algorithm assuming a₁, ª2, ..., ª are positive real numbers and n = 11. for i=1 to n m₂ = a;; for j=i+1 to n mi = m;aj;

Give the exact number of multiplications performed in the following segment of an algorithm assuming a₁, ª2, ..., ª are positive real numbers and n = 11. for i=1 to n m₂ = a;; for j=i+1 to n mi = m;aj;

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1. For each pair of integers m and n, use the division algorithm to divide m by n. Write the result in the form m = q⋅n + r where 0 < r < n. a. m = 32 and n = 9 b. m 62 and n = 7 = c. m -45 and n = 11 d. m 3 and n = 8 2. For each pair of integers a and b, use the Euclidean Algorithm to find the GCD(a, b). Then use Theorem 92 to find the LCM(a, b) a. a = 48 and b = 54 b. a = 330 and b = 156 c. a=1188 and b = 385 3. For each pair of integers m and d, calculate m mod d. Show your work! a. m 43 and d = 9 b. m 50 and d = 7 c. m 28 and d = 5 d. m = 30 and d = 2
3. If a = 99 and b = 374. Use the Euclidean algorithm to find the gcd(a, b) and compute integers x and y for which ах + by 3 gcd(а, b).