Give the exact number of multiplications performed in the following segment of an algorithm assuming a₁, a2,..., an are positive real numbers and n = 13. for i=1 to n for j=i+1 to n tij =iaj;
Give the exact number of multiplications performed in the following segment of an algorithm assuming a₁, a2,..., an are positive real numbers and n = 13. for i=1 to n for j=i+1 to n tij =iaj;
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:**Title: Calculation of Multiplications in Algorithm Segment**
**Description:**
This section aims to determine the exact number of multiplications performed in a specific segment of an algorithm. The algorithm deals with a series of computations involving positive real numbers \(a_1, a_2, \ldots, a_n\) where \(n\) is predefined as 13.
**Algorithm Segment:**
1. **Outer Loop:** Iterate over variable \(i\) from 1 to \(n\).
2. **Inner Loop:** For each \(i\), iterate over variable \(j\) from \(i + 1\) to \(n\).
3. **Computation:** For each combination of \(i\) and \(j\), compute \(t_{ij} = i \cdot a_j\).
**Explanation:**
- **Outer Loop (for \(i = 1\) to \(n\)):** Begins the iteration with \(i = 1\) and progresses to the final value of \(n\).
- **Inner Loop (for \(j = i + 1\) to \(n\)):** For each value of \(i\), \(j\) starts at \(i + 1\), ensuring that no multiplication occurs when \(i = j\).
- **Multiplication Count:** The computation \(t_{ij} = i \cdot a_j\) involves one multiplication for each pair of \((i, j)\) within the defined loop bounds.
To calculate the total number of multiplications:
- For \(i = 1\), \(j\) ranges from 2 to 13, resulting in 12 multiplications.
- For \(i = 2\), \(j\) ranges from 3 to 13, resulting in 11 multiplications.
- Continue this pattern until
- For \(i = 12\), \(j\) ranges from 13 to 13, resulting in 1 multiplication.
- No multiplication occurs for \(i = 13\), as \(j\) would start at 14, which is outside the loop range.
The total number of multiplications is the sum of an arithmetic series: \(12 + 11 + \ldots + 1\), which equals 78.
This simple segment provides insight into nested loops and efficient calculation within algorithmic contexts, crucial for understanding algorithm efficiency.
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