The polynomial is of the generic form: p(x) = anx"+an-1 Xn-¹+an-2 X-2+...+a2 x²+a₁ x+ao We will assume that the coefficient values a0 through an are all known and constant and stored in an array. This means that the evaluation of the polynomial has only the value x as its input and will return the resulting polynomial value as it output. The standard evaluation algorithm is straightforward: Evaluate (x) { //x the value to use for evaluation of the polynomial result=a[0]+ a[1]"x xPower=x for i=2 to n do xPower=xPower"x| result=result +a[i]"xPower end for return result. 1- Find the number of additions and multiplications outside the for loop 2- Find the number of additions inside the for loop 3- Find the number of multiplications inside the for loop 4- Find the number of times the for loop executes 5- Find the number of operations of the function Evaluate(x)
The polynomial is of the generic form: p(x) = anx"+an-1 Xn-¹+an-2 X-2+...+a2 x²+a₁ x+ao We will assume that the coefficient values a0 through an are all known and constant and stored in an array. This means that the evaluation of the polynomial has only the value x as its input and will return the resulting polynomial value as it output. The standard evaluation algorithm is straightforward: Evaluate (x) { //x the value to use for evaluation of the polynomial result=a[0]+ a[1]"x xPower=x for i=2 to n do xPower=xPower"x| result=result +a[i]"xPower end for return result. 1- Find the number of additions and multiplications outside the for loop 2- Find the number of additions inside the for loop 3- Find the number of multiplications inside the for loop 4- Find the number of times the for loop executes 5- Find the number of operations of the function Evaluate(x)
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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Hello can you please help me do this exercise. Can you do part 1, 2, 3, 4 and 5 (marked in red please)
The polynomial is of the generic form:
\[ p(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \ldots + a_2 x^2 + a_1 x + a_0 \]
We will assume that the coefficient values \( a_0 \) through \( a_n \) are all known and constant and stored in an array. This means that the evaluation of the polynomial has only the value \( x \) as its input and will return the resulting polynomial value as its output.
The standard evaluation algorithm is straightforward:
```
Evaluate (x) {
// x the value to use for evaluation of the polynomial
result = a[0] + a[1] * x
xPower = x
for i = 2 to n do
xPower = xPower * x
result = result + a[i] * xPower
end for
return result.
}
```
1. **Find the number of additions and multiplications outside the for loop.**
2. **Find the number of additions inside the for loop.**
3. **Find the number of multiplications inside the for loop.**
4. **Find the number of times the for loop executes.**
5. **Find the number of operations of the function Evaluate(x).**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F549651ba-b33d-474d-abf4-48863f7a8bc4%2F81238564-75bb-4ca2-9eef-89758af7bc01%2F1t7rqgi_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 4:**
The following algorithm computes the sum of a polynomial of power n.
For an explanation of what a polynomial is, consult:
[Polynomial Explanation](https://www.mathcentre.ac.uk/resources/uploaded/mc-ty-polynomial-2009-1.pdf)
The polynomial is of the generic form:
\[ p(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \ldots + a_2 x^2 + a_1 x + a_0 \]
We will assume that the coefficient values \( a_0 \) through \( a_n \) are all known and constant and stored in an array. This means that the evaluation of the polynomial has only the value \( x \) as its input and will return the resulting polynomial value as its output.
The standard evaluation algorithm is straightforward:
```
Evaluate (x) {
// x the value to use for evaluation of the polynomial
result = a[0] + a[1] * x
xPower = x
for i = 2 to n do
xPower = xPower * x
result = result + a[i] * xPower
end for
return result.
}
```
1. **Find the number of additions and multiplications outside the for loop.**
2. **Find the number of additions inside the for loop.**
3. **Find the number of multiplications inside the for loop.**
4. **Find the number of times the for loop executes.**
5. **Find the number of operations of the function Evaluate(x).**
![**Polynomial Evaluation Using Factorization**
To efficiently evaluate a polynomial, it can be factored into the following form:
\[ p(x) = ((\ldots((a_nx + a_{n-1})x + a_{n-2})x + \ldots + a_2)x + a_1)x + a_0 \]
This can be expressed algorithmically as:
**BetterEvaluate(x)**
- **// x**: the value to use for evaluation of the polynomial
- **result = a[n]**
For **i = n-1** down to **0** do
- **result = result * x**
- **result = result + a[i]**
End For
Return **result**.
**Exercises**
1. Answer questions 1-5 regarding the algorithm BetterEvaluate(x).
2. How many multiplications were saved by applying BetterEvaluate(x) rather than Evaluate(x)?
**Extra Credit (20 pts)**
- Implement each of the algorithms for **x = 2** and \(\text{a[10]} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F549651ba-b33d-474d-abf4-48863f7a8bc4%2F81238564-75bb-4ca2-9eef-89758af7bc01%2F8mfci6s_processed.png&w=3840&q=75)
Transcribed Image Text:**Polynomial Evaluation Using Factorization**
To efficiently evaluate a polynomial, it can be factored into the following form:
\[ p(x) = ((\ldots((a_nx + a_{n-1})x + a_{n-2})x + \ldots + a_2)x + a_1)x + a_0 \]
This can be expressed algorithmically as:
**BetterEvaluate(x)**
- **// x**: the value to use for evaluation of the polynomial
- **result = a[n]**
For **i = n-1** down to **0** do
- **result = result * x**
- **result = result + a[i]**
End For
Return **result**.
**Exercises**
1. Answer questions 1-5 regarding the algorithm BetterEvaluate(x).
2. How many multiplications were saved by applying BetterEvaluate(x) rather than Evaluate(x)?
**Extra Credit (20 pts)**
- Implement each of the algorithms for **x = 2** and \(\text{a[10]} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\).
Expert Solution
Step 1: Introduce polynomial evoulution:
Polynomial evaluation is a fundamental mathematical operation where a polynomial expression is computed for a given value of its variable. The generic form of a polynomial consists of coefficients stored in an array, and a standard evaluation algorithm is used to calculate the polynomial's value efficiently. Understanding the number of additions, multiplications, and loop iterations in the evaluation process is essential for optimizing computational efficiency.
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