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The conventional algorithm for evaluating a polynomial anx" + an-1x"-1 + .…. + a¡x+ ao at x = c can be expressed in pseudocode by procedure polynomial(c, ao, a1, .., an : real numbers) power := 1 y := ao for i := 1 to n power := power * c y :=y+ a¡ * power return y(y 3Dа,с" + а,-1c"- +... + ajc + do } where the final value of y is the value of the polynomial at x = c. a) Evaluate 3x² +x+1 at x = 2 by working through each step of the algorithm showing the values assigned at each assignment step. b) Exactly how many multiplications and additions are used to evaluate a polynomial of degree n at x = c? (Do not count additions used to increment the loop variable.)

The conventional algorithm for evaluating a polynomial anx" + an-1x"-1 + .…. + a¡x+ ao at x = c can be expressed in pseudocode by procedure polynomial(c, ao, a1, .., an : real numbers) power := 1 y := ao for i := 1 to n power := power * c y :=y+ a¡ * power return y(y 3Dа,с" + а,-1c"- +... + ajc + do } where the final value of y is the value of the polynomial at x = c. a) Evaluate 3x² +x+1 at x = 2 by working through each step of the algorithm showing the values assigned at each assignment step. b) Exactly how many multiplications and additions are used to evaluate a polynomial of degree n at x = c? (Do not count additions used to increment the loop variable.)

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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The excerpt presents a pseudocode algorithm for evaluating a polynomial of the form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \) at \( x = c \). **Pseudocode:** ``` procedure polynomial(c, a_0, a_1, ..., a_n: real numbers) power := 1 y := a_0 for i := 1 to n power := power * c y := y + a_i * power return y {y = a_n c^n + a_{n-1} c^{n-1} + ... + a_1 c + a_0} ``` **Explanation:** The algorithm initializes `power` as 1 and `y` as \( a_0 \). It iterates from 1 to \( n \), updating `power` by multiplying it with \( c \), and `y` by adding \( a_i \times \text{power} \). Finally, it returns `y`, which represents the polynomial evaluated at \( x = c \). **Tasks:** a) Evaluate \( 3x^2 + x + 1 \) at \( x = 2 \) by stepping through the algorithm and showing the values at each assignment. b) Determine the exact number of multiplications and additions used to evaluate a polynomial of degree \( n \) at \( x = c \) (excluding additions to increment the loop variable).
Transcribed Image Text:The excerpt presents a pseudocode algorithm for evaluating a polynomial of the form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \) at \( x = c \). **Pseudocode:** ``` procedure polynomial(c, a_0, a_1, ..., a_n: real numbers) power := 1 y := a_0 for i := 1 to n power := power * c y := y + a_i * power return y {y = a_n c^n + a_{n-1} c^{n-1} + ... + a_1 c + a_0} ``` **Explanation:** The algorithm initializes `power` as 1 and `y` as \( a_0 \). It iterates from 1 to \( n \), updating `power` by multiplying it with \( c \), and `y` by adding \( a_i \times \text{power} \). Finally, it returns `y`, which represents the polynomial evaluated at \( x = c \). **Tasks:** a) Evaluate \( 3x^2 + x + 1 \) at \( x = 2 \) by stepping through the algorithm and showing the values at each assignment. b) Determine the exact number of multiplications and additions used to evaluate a polynomial of degree \( n \) at \( x = c \) (excluding additions to increment the loop variable).
### Algorithm Complexity Analysis #### 16. Maximum Solvable Problem Size in One Day Determine the largest \( n \) that can be solved within a day using an algorithm that requires \( f(n) \) bit operations, with each bit operation executed in \( 10^{-11} \) seconds. Consider the following functions \( f(n) \): - a) \( \log n \) - b) \( 1000n \) - c) \( n^2 \) - d) \( 1000n^2 \) - e) \( n^3 \) - f) \( 2^n \) - g) \( 2^{2n} \) - h) \( 2^{n^2} \) #### 17. Maximum Solvable Problem Size in One Minute Determine the largest \( n \) that can be solved within a minute using an algorithm that requires \( f(n) \) bit operations, with each bit operation executed in \( 10^{-12} \) seconds. Consider the following functions \( f(n) \): - a) \( \log \log n \) - b) \( \log n \) - c) ( \( \log n \) )\(^2\) These exercises explore the relationship between problem size and computational time, given different orders of growth of operations. Understanding these can help in selecting efficient algorithms appropriate for different computational constraints.
Transcribed Image Text:### Algorithm Complexity Analysis #### 16. Maximum Solvable Problem Size in One Day Determine the largest \( n \) that can be solved within a day using an algorithm that requires \( f(n) \) bit operations, with each bit operation executed in \( 10^{-11} \) seconds. Consider the following functions \( f(n) \): - a) \( \log n \) - b) \( 1000n \) - c) \( n^2 \) - d) \( 1000n^2 \) - e) \( n^3 \) - f) \( 2^n \) - g) \( 2^{2n} \) - h) \( 2^{n^2} \) #### 17. Maximum Solvable Problem Size in One Minute Determine the largest \( n \) that can be solved within a minute using an algorithm that requires \( f(n) \) bit operations, with each bit operation executed in \( 10^{-12} \) seconds. Consider the following functions \( f(n) \): - a) \( \log \log n \) - b) \( \log n \) - c) ( \( \log n \) )\(^2\) These exercises explore the relationship between problem size and computational time, given different orders of growth of operations. Understanding these can help in selecting efficient algorithms appropriate for different computational constraints.
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